Network data analysis has wide applications in computational social science, computational biology, online social media, and data visualization. For many of these network inference problems, the brute-force (yet statistically optimal) methods involve combinatorial optimization, which is computationally prohibitive when we are faced with large scale networks. Therefore, it is important to understand the effect of computational constraints on statistical inference.

In this talk, we will discuss three closely related statistical models for different network inference problems. These models answer inference questions on cliques, communities, and ties, respectively. For each particular model, we will describe the statistical model, propose new computationally efficient algorithms, and study the theoretical properties and numerical performance of the algorithms. Further, we will quantify the computational optimality through describing the intrinsic barrier for certain efficient algorithm classes, and investigate the computational-to-statistical gap theoretically. A key feature shared by our studies is that, as the parameters of the model changes, the problems exhibit different phases of computational difficulty.

The {\itshape Zimmer Program} is a collection of conjectures and questions regarding actions of lattices in higher-rank simple Lie groups on compact manifolds. For instance, it is conjectured that all non-trivial volume-preserving actions are built from algebraic examples using standard constructions. In particular, on manifolds whose dimension is below the dimension of all algebraic examples, {\itshape Zimmer's conjecture} asserts that every action is finite.

I will present some background, motivation, and selected previous results in the Zimmer program. I will then explain two of my own results within the Zimmer program:
(1) a solution to Zimmer's conjecture for actions of cocompact lattices in $SL(n,R), n>=3$ (joint with D. Fisher and S. Hurtado);
(2) a classification (up to topological semiconjugacy) of lattice actions on tori whose induced action on homology satisfies certain criteria (joint with F. Rodriguez Hertz and Z. Wang).

Cohomology jump loci are generalizations of usual cohomology groups
of a topological space. In the first part of the talk, I will give a
survey on the recent development of the theory of cohomology jump loci
of complex algebraic varieties. In the second part of the talk, I will
use some concrete examples of (real) 6-dimensional symplectic-complex
Calabi-Yau manifolds to illustrate how cohomology jump loci can give
new constraints on the topology of compact Kahler manifolds.

One of the main goals of any resource theory such as entanglement, quantum thermodynamics, quantum coherence, and asymmetry, is to find necessary and sufficient conditions (NSC) that determine whether one resource can be converted to another by the set of free operations. In this talk I will present such NSC for a large class of quantum resource theories which we call affine resource theories (ARTs). ARTs include the resource theories of athermality, asymmetry, and coherence, but not entanglement. Remarkably, the NSC can be expressed as a family of inequalities between resource monotones (quantifiers) that are given in terms of the conditional min entropy. The set of free operations is taken to be (1) the maximal set (i.e. consists of all resource non-generating (RNG) quantum channels) or (2) the self-dual set of free operations (i.e. consists of all RNG maps for which the du al map is also RNG). As an example, I will discuss the applications of the results to quantum thermodynamics with Gibbs preserving operations, and several other ARTs. Finally, I will discuss the applications of these results to resource theories that are not affine.

The Birch and Swinnerton-Dyer conjecture is a central open problem in Mathematics, and Euler Systems have been at the source of much of the progress to date in this direction. In my talk, I will give a motivated introduction to the BSD conjecture, survey what we know about it, and highlight some recent advances arising from the construction of new Euler Systems with a bearing on the arithmetic of elliptic curves.

The Truncated Moment Problem seeks conditions on an n-dimensional multisequence of degree $m$,
$y \equiv (y_i)_{|i| ≤ m}$, such that there exists a positive Borel measure $\mu$ on $\mathbb{R}^n$ satisfying $y_i = \int \xi d \mu \, (|i| ≤ m)$ (where $x = (x_1, \ldots, x_n)$, $i = (i_1, \ldots, i_n)$). In previous work we associated to $y$ an algebraic variety in $\mathbb{R}^n$ , the core variety $V = V(y)$, and showed that if $V$ is nonempty, then the Riesz functional $L$ corresponding to $y$ is strictly V-positive, i.e., if $p(x) := \Sigma a_i x_i \, (|i| ≤ m)$ is nonnegative on $V$, and $p|_V$ is not identically $0$, then $L(p) := \Sigma a_i y_i > 0$. In current work with G. Blekherman, we prove that if $L$ is strictly $K$-positive for any closed subset $K$ of $\mathbb{R}^n$, then $y$ has a representing measure $\mu$ (as above) whose support is contained in $K$. As a consequence, we prove that $y$ has a representing measure if and only if $V(y)$ is nonempty, in which case $V(y)$ coincides with the union of the supports of all representing measures. As a corollary, we obtain a new proof of the Bayer-Teichmann Theorem on multivariable cubature.

It is a theorem of de Bruijn and Erdos that $n$ points in the plane
determine at least $n$ lines, unless all the points lie on a line. This
is one of the earliest results in enumerative combinatorial geometry.
We will present a higher dimensional generalization of this theorem,
which confirms a “top-heavy” conjecture of Dowling and Wilson in 1975.
I will give a sketch of the key idea of the proof, which uses the hard
Lefschetz theorem and the decomposition theorem in algebraic geometry.
I will also talk about a log-concave conjecture on the number of
independent sets. This is joint work with June Huh.

Let $G$ be uniformly distributed on the set of all simple $d$-regular graphs on $n$ vertices, and assume $d$ is bigger than some (small) power of $n$. We show that the second largest eigenvalue of $G$ is of order $\sqrt{d}$ with probability close to one. Combined with earlier results covering the case of sparse random graphs, this settles the problem of estimating the magnitude of the second eigenvalue, up to a multiplicative constant, for all values of $n$ and $d$, confirming a conjecture of Van Vu. Joint work with Pierre Youssef.

Let $E$ be an elliptic curve defined over ${Q}$. The torsion subgroup
of $E$ over the compositum of all quadratic extensions of ${Q}$ was
studied by Michael Laska, Martin Lorenz, and Yasutsugu Fujita. Laska
and Lorenz described a list of $31$ possible groups and Fujita proved
that the list of $20$ different groups is complete.

In this talk, we will generalize the results of Laska, Lorenz and
Fujita to the elliptic curves defined over a quadratic cyclotomic
field i.e. $Q(i)$ and $Q(\sqrt{-3})$.

While the topic of geometric incidences has existed for
several decades, in recent years it has been experiencing a
renaissance due to the introduction of new polynomial methods. This
progress involves a variety of new results and techniques, and also
interactions with fields such as algebraic geometry and harmonic
analysis.

A simple example of an incidences problem: Given a set of $n$ points and
set of n lines, both in $R^2$, what is the maximum number of point-line
pairs such that the point is on the line. While this may seem as a
simple problem, incidence problems often have a deep underlying
theory, which may involve the uncovering of hidden structure and
symmetries.

In this talk we introduce and survey the topic of geometric
incidences, focusing on the recent polynomial techniques and results
(some by the speaker). We will see how various algebraic and analytic
tools can be used to solve such combinatorial problems.

In joint work with G. Blekherman we proved that a truncated multisequence $y$ of degree $m$ has a representing measure in the Truncated Moment Problem if and only if its core variety $V(y)$ is nonempty, in which case $V(y)$ coincides with the union of the supports of all representing measures. In general, for a given numerical sequence $y$, it may be quite difficult to compute $V(y)$ or even to determine if it is nonempty. We illustrate some cases where we can compute $V(y)$ or can otherwise describe it concretely.

Hyperkahler (HK) manifolds appear in many fields of mathematics, such
as differential geometry, mathematical physics, representation theory,
and algebraic geometry. Compact HK manifolds are one of the building
blocks for algebraic varieties with trivial first Chern class and
their role in algebraic geometry has grown immensely over the last 20
year. In this talk I will give an overview of the theory of compact HK
manifolds and then focus on some of my work, including a recent joint
work with R. Laza and C. Voisin.

We consider the problem of classifying completely unlabelled data using convex inequalities that contain absolute values of the data. This allows each data point to belong to either one of two classes by entering the inequality with a plus or minus value. Using such absolute value inequalities in support vector machine classifiers, unlabelled data can be successfully partitioned into two classes that capture most of the correct labels dropped from the data. Inclusion of partially labelled data leads to a semisupervised classifier. Computational results include unsupervised and semisupervised classification of the Wisconsin Breast Cancer Wisconsin (Diagnostic) Data Set.

This talk is about representation learning with a nontrivial geometry of
variables. A convolutional neural network can be viewed as a statistical
machine to detect and count features in an image progressively through a
multi-scale system. The constructed features are insensitive to nuance
variations in the input, while sufficiently discriminative to predict
labels. We introduce the Haar scattering transform as a model of such
a system for unsupervised learning. Employing Haar wavelets makes it
applicable to data lying on graphs that are not necessarily pixel grids.
When the underlying graph is unknown, an adaptive version of the
algorithm infers the geometry of variables by optimizing the
construction of the Haar basis so as to minimize data variation. Given
time, I will also mention an undergoing project of flow cytometry data
analysis, where histogram-like features are used for comparing empirical
distributions. After ``binning'' samples on a mesh in space, the problem
can be closely related to feature learning when a variable geometry is
present.

Following Colding and Minicozzi, we consider the entropy of (hyper)-surfaces in Euclidean space. This is a numerical measure of the geometric complexity of the surface. In addition, this quantity is intimately tied to to the singularity formation of the mean curvature flow which is a natural geometric heat flow of submanifolds. In the talk, I will discuss several results that show that closed surfaces for which the entropy is small are simple in various senses. This is all joint work with L. Wang.

High-dimensional temporal dependent data arise in a wide range of disciplines.
The fact that the classical CLT for i.i.d. random vectors may fail in
high dimensions makes high-dimensional inference notoriously difficult.
More challenges are imposed by temporal and cross-sectional dependence.
In this talk, I will introduce the high-dimensional CLT for temporal dependent
data. Its validity depends on the sample size $n$, the dimension $p$, the
moment condition and the dependence of the underlying processes. An example
is taken to appreciate the optimality of the allowed dimension $p$. Equipped
with the high-dimensional CLT result, we have a new sight on many problems
such as inference for covariances of high-dimensional time series which can
be applied in the analysis of network connectivity, inference for multiple
posterior means in MCMC experiments as well as Kolmogorov-Smirnov test for
high-dimensional dependent data. I will also introduce an estimator for
long-run covariance matrices and two resampling methods, i.e., Gaussian
multiplier resampling and subsampling, to make the high-dimensional CLT more
applicable. Our work is then corroborated by a simulation study with a
hierarchical model.

A basic problem in Kahler geometry, going back to Calabi in the 50's, is to find Kahler metrics with the best curvature properties, e.g., Einstein metrics. Such special metrics are minimizers of well known functionals on the space of all Kahler metrics H. However these functionals become convex only if an adequate geometry is chosen on H. One such choice of Riemannian geometry was proposed by Mabuchi in the 80's, and was used to address a number of uniqueness questions in the theory. In this talk I will present more general Finsler geometries on H, that still enjoy many of the properties that Mabuchi's geometry has, and I will give applications related to existence of special Kahler metrics, including the recent resolution of Tian's related properness conjectures.

A number of questions in Galois theory can be phrased in the following
way: how large (in various senses) can the Galois group G of an
extension of the rational numbers be, if the extension is only allowed
to ramify at a small set of primes? If we assume that G is abelian,
class field theory provides a complete answer, but the question is
open is almost every nonabelian case, since there is no known way to
systematically and explicitly construct such extensions in full
generality.

However, there have been some programs that are gaining ground on this
front. While the problem above is natural and the objects are
classical, we will see that to answer certain questions about the
“largeness” of this Galois group, it seems necessary to use techniques
involving automorphic forms and their representation-theoretic
avatars. In particular, it will turn out that some recent results on
“harmonic” families of automorphic forms translate to the fact that
such number fields, despite not being explicitly constructible by
known methods, turn out to “exist in abundance” and allow us to find
bounds on the sizes of such Galois groups.

Massive data are often contaminated by \underline{outliers} and heavy-tailed errors. To address this challenge, we propose the adaptive Huber regression for robust estimation and inference. The key observation is that the robustification parameter should adapt to sample size, dimension and moments for optimal \underline{tradeoff} between bias and robustness. Our framework is able to handle heavy-tailed data with bounded $(1+\delta)$-th moment for any $\delta>0$. We establish a sharp phase transition for robust estimation of regression parameters in both finite dimensional and high dimensional settings: when $\delta \geq 1$, the estimator achieves sub-Gaussian rate of convergence without sub-Gaussian assumptions, while only a slower rate is available in the regime $0<\delta <1$ and the transition is smooth and optimal. As a consequence, the \underline{nonasymptotic Bahadur} representation for finite-sample inference can only be derived when the second moment exists. Numerical experiments lend further support to our obtained theories.

Tautological relations are certain equations in the Chow ring of
the moduli space of curves. I will discuss a family of such relations,
first conjectured by A. Pixton, that arises by studying moduli spaces of
ramified covers of the projective line. These relations can be used to
recover a number of well-known facts about the moduli space of curves, as
well as to generate very special equations known as topological recursion
relations. This is joint work with various subsets of S. Grushevsky, F.
Janda, X. Wang, and D. Zakharov.

This is part of a series lectures studying stability of symplectic forms
on noncompact manifolds. The case of compact manifolds is well understood
thanks to seminal work of Jurgen Moser in the 1960s.

First suggested by Witten in the early 1990's, the
Landau-Ginzburg/Calabi-Yau correspondence studies a relationship between
spaces of maps from curves to the quintic 3-fold (the Calabi-Yau side) and
spaces of curves with 5th roots of their canonical bundle (the
Landau-Ginzburg side). The correspondence was put on a firm mathematical
footing in 2008 when Chiodo and Ruan proved a precise statement for the
case of genus-zero curves, along with an explicit conjecture for the
higher-genus correspondence. In this talk, I will begin by describing the
motivation and the mathematical formulation of the LG/CY correspondence,
and I will report on recent work with Shuai Guo that verifies the
higher-genus correspondence in the case of genus-one curves.

Some of the outstanding and still classical problems in Algebraic Combinatorics concern understanding the Kronecker coefficients of the symmetric group, the multiplicities describing the decomposition of tensor products of representations into irreducibles, which are nonnegative integers lacking a positive combinatorial formula for over 75 years. Recently they appeared in Geometric Complexity Theory (GCT), a program aimed to distinguish computational complexity classes (like the P vs NP problem) and prove complexity theoretic bounds using Algebraic Geometry and Representation Theory.

On the combinatorial side, we \lbrack{Pak-P}\rbrack will show various bounds on Kronecker coefficients via character evaluations and partition enumeration, and use them to extend Sylvester and Stanley's theorem on the unimodality of partitions inside a rectangle and find asymptotic bounds. On the GCT side, using algebraic and combinatorial methods, we \lbrack{Burgisser-Ikenmeyer-P, Ikenmeyer-P}\rbrack show that the relevant Kronecker and plethysm coefficients of the general linear group are positive, thereby disproving a Mulmuley and Sohoni conjecture on the existence of ``occurrence obstructions" and practically showing that the 'P vs NP' problem is even more difficult. In the reverse direction, GCT arguments show that rectangular Kronecker coefficients are larger than plethysm coefficient in a stable range \lbrack{Ikenmyer-P}\rbrack, establishing a connection between apriori unrelated and greatly mysterious multiplicities.

Initial data in general relativity must satisfy certain underdetermined differential equations called the constraint equations. A natural problem is to find a parameterization of all possible initial data. A standard method for this is called the conformal method. In this talk, we'll discuss the successes and failures of this method, and future directions for research.

I will present recent joint work with Arnaud Brothier on actions of locally compact groups on von Neumann algebras. We study algebraic properties of the associated crossed-product algebras, and prove among other things a correspondance result between certain subalgebras of this crossed-product and closed subgroups of the acting group. This generalizes results of Izumi-Longo-Popa. I will explain our (very different) approach, and give related questions and conjectures.

One of the outstanding open problems in the study of fluids is the global regularity of smooth solutions to the three-dimensional incompressible Euler equation. I will begin by introducing the incompressible Euler equation as well as some classical well-posedness results. Then I will discuss various attempts to understand the global regularity problem and related problems moving into recent results. One popular attempt to understand the global regularity problem is to study simplified lower dimensional models that can be satisfactorily solved. A major issue with studying simplified models is that they may have no bearing on the dynamics of the actual 3d Euler equation--especially since the closer a model gets to modeling the dynamics of 3d Euler the more challenging understanding the dynamics of the model is. In recent works with I. Jeong, we derived a “good” model through the use of symmetry properties of the equation. In particular, we proved that if singularity formation can be established for a particular two-dimensional equation, then there is singularity formation for the full 3d Euler equation for finite-energy solutions lying in a critical space where there is local well-posedness. Similar results can be proven for the surface quasi-geostrophic (SQG) equation arising in atmospheric dynamics and there a one-dimensional model is derived. I will then discuss recent results on some of these models and their implications.

At the beginning of this talk, we will introduce the background of tensor network states (TNS) in various areas such as quantum physics, quantum chemistry and numerical partial differential equations. Famous TNS includes tensor trains (TT), matrix product states (MPS), projected entangled pair states (PEPS) and multi-scale entanglement renormalization ansatz (MERA). Then we will explain how to define TNS by graphs and we will define tensor network ranks which can be used to measure the complexity of TNS. We will see that the notion of tensor network ranks is an analogue of tensor rank and multilinear rank. We will discuss basic properties of tensor network ranks and the comparison among tensor network ranks, tensors rank and multilinear rank. If time permits, we will also discuss the dimension of tensor networks and the geometry of TNS. This talk is based on papers joined with Lek-Heng Lim.

Artin's conjecture on primitive roots, which was originally formulated for multiplicative groups, has a natural analogue for elliptic curves. In this survey talk, I will discuss the analogy and focus on``new'' phenomena such as the existence of ``never-primitive'' points.

Schubert curves are the spaces of solutions to certain one-dimensional Schubert problems involving flags osculating the rational normal curve. The real locus of a Schubert curve is known to be a natural covering space of $RP^1$, so its real geometry is fully characterized by the monodromy of the cover. It is also possible, using K-theoretic Schubert calculus, to relate the real locus to the overall (complex) Riemann surface.

We present a local algorithm for computing the monodromy operator in terms of Jeu de Taquin-like operations on certain skew Young tableaux, and use it to provide purely combinatorial proofs of some of the connections to K-theory. We will also explore partial progress in this direction in the Type C setting of the orthogonal grassmannian. This is joint work with Jake Levinson.

This is part of a series lectures studying stability of symplectic forms
on noncompact manifolds. The case of compact manifolds is well understood
thanks to seminal work of Jurgen Moser in the 1960s.

Artists, Coders, and other creative professionals are never on the cutting edge of research. However, their use cases are often the most memorable and effective tools for communicating what mathematics, and applied mathematics especially, is capable of. When Google shows off its neural networks, it makes your selfies look like Van Gogh drew them. When IBM wants to prove its advancements in NLP, it plays Jeopardy. This talk aims to present a handful of examples of computational art - and to explore both its technical background and it's impact. We will discuss the ways in which mathematics impacts these projects, and the ways in which the art communicates mathematics.

A ${\rm II}_1$ factor $M$ is called prime if it cannot be decomposed as a tensor product of ${\rm II}_1$ subfactors. Naturally, if $M$ is not prime, one asks if $M$ can be uniquely factored as a tensor product of prime subfactors. The first result in this direction is due to Ozawa and Popa in 2003, who gave a large class of groups $\mathcal{C}$ such that for any $\Gamma_1, \dots, \Gamma_n \in \mathcal{C}$, the associated von Neumann algebra $L(\Gamma_1) \,\overline{\otimes}\, \cdots \,\overline{\otimes}\, L(\Gamma_n)$ is uniquely factored in a strong sense. This talk will consider the case where $\Gamma$ is icc group that is measure equivalent to a product of non-elementary hyperbolic groups. In joint work with Daniel Drimbe and Adrian Ioana, we show that any such $\Gamma$ admits a unique decomposition $\Gamma = \Gamma_1 \times \Gamma_2 \times \cdots \times \Gamma_n$ such that $L(\Gamma) = L(\Gamma_1) \,\overline{\otimes}\, \cdots \,\overline{\otimes}\, L(\Gamma_n)$ is uniquely factored in sense of Ozawa and Popa. Using this, we provide the first examples of prime ${\rm II}_1$ factors arising from lattices in higher rank Lie groups.

Heavy-tailed distributions are ubiquitous in modern statistical analysis and machine learning problems. This talk gives a simple principle for robust high-dimensional statistical inference via an appropriate shrinkage on the data. This widens the scope of high-dimensional techniques, reducing the moment conditions from sub-exponential or sub-Gaussian distributions to merely bounded second moment. As an illustration of this principle, we focus on robust estimation of the low-rank matrix from the trace regression model. It encompasses four popular problems: sparse linear models, compressed sensing, matrix completion, and multi-task regression. Under only bounded $2+\delta$ moment condition, the proposed robust methodology yields an estimator that possesses the same statistical error rates as previous literature with sub-Gaussian errors. We also illustrate the idea for estimation of large covariance matrix. The benefits of shrinkage are also demonstrated by financial, economic, and simulated data.

The analog of the Riemann curvature tensor for noncommutative tori manifests itself in the term $a_4$ appearing in the heat kernel expansion of the Laplacian of curved metrics. This talk presents a joint work with Alain Connes, in which we obtain an explicit formula for the $a_4$ associated with a general metric in the canonical conformal structure on noncommutative two-tori. Our final formula has a complicated dependence on the modular automorphism of the state or volume form of the metric, namely in terms of several variable functions with lengthy expressions. We verify the accuracy of the functions by checking that they satisfy a family of conceptually predicted functional relations. By studying the latter abstractly we find a partial differential system which involves a natural flow and action of cyclic groups of order two, three and four, and we discover symmetries of the calculated expressions with respect to the action of these groups. At the end, I will illustrate the application of our results to certain noncommutative four-tori equipped with non-conformally flat metrics and higher dimensional modular structures.

Let $P: ... \to C_2 \to C_1 \to P^1$ be a $\mathbb{Z}_p$-cover of the
projective line over a finite field of characteristic $p$ which ramifies
at exactly one rational point. In this talk, we study the $p$-adic
Newton slopes of L-functions associated to characters of the Galois
group of $P$. It turns out that for covers $P$ such that the genus of $C_n$ is a quadratic polynomial in $p^n$ for $n$ large, the Newton slopes are uniformly distributed in the interval $[0,1]$. Furthermore, for a large class of such covers $P$, these slopes behave in an even more regular way. This is joint work with Hui June Zhu.

Recently, Venkatesh introduced the derived Hecke algebra to explain extra endomorphisms on the cohomology of arithmetic manifolds: the crucial local construction is a derived version of the spherical Hecke algebra of a reductive p-adic group. Working with p-torsion coefficients, we will describe a Satake homomorphism for the derived spherical Hecke algebra of a p-adic group. This will allow us to understand its structure well enough to attack some global questions, which are work in progress.

This is part of a series lectures studying stability of symplectic forms
on noncompact manifolds. The case of compact manifolds is well understood
thanks to seminal work of Jurgen Moser in the 1960s.

The Hodge theory of surfaces provides a link between enumerative geometry and modular forms, via the cohomological theta correspondence. I will present an approach to studying the Gromov-Witten invariants of Weierstrass fibrations over $P^2$, proving part of a conjectural formula coming from topological string theory.

Yes, the title makes sense: there are (a lot of) connections between logic and operator algebras! And in this talk we are going to cover some. The first goal will be to introduce a generalization of classical first order logic, a ``continuous logic'' which can be used to describe continuous structures like metric spaces, operator algebras, etc. We will see how this can help us to study some questions concerning such structures. In particular, we will cover two natural questions that arise in the context of operator algebras which turn out to be independent of ZFC - our everyday axiom system all of set theory is based on!

As shown by N. Brown in 2011, for a separable $II_1-factor N$, the invariant $Hom(N,R^U)$ given by unitary equivalence classes of embeddings of $N$ in to $R^U$--an ultrapower of the separable hyperfinite $II_1-factor$--takes on a convex structure. This provides a link between convex geometric notions and operator algebraic concepts; e.g. extreme points are precisely the embeddings with factorial relative commutant. The geometric nature of this invariant provides a familiar context in which natural curiosities become interesting new questions about the underlying operator algebras. For example, such a question is the following. ``Can four extreme points have a planar convex hull?''

The goal of this talk is to present a recent result generalizing the characterization of extreme points in this convex structure. After introducing and discussing this convex structure, we will see that the dimension of the minimal face containing an equivalence class $[\pi]$ is one less than the dimension of the center of the relative commutant of $\pi$. This result also establishes the ``convex independence'' of extreme points, providing a negative answer to the above question. Along the way we make use of a version of Schur's Lemma for this context. No prior knowledge of this convex structure will be assumed.

Coupling is a way of constructing Markov processes with prescribed laws on the same probability space. It is known that the rate of coupling (how fast you can make two processes meet) of elliptic/Riemannian diffusions is connected to the geometry of the underlying space. In this talk we consider coupling of hypoelliptic diffusions (diffusions driven by vector fields satisfying Hoermander's condition). S. Banerjee and W. Kendall constructed successful Markovian couplings for a large class of hypoelliptic diffusions. We use a non-Markovian coupling of Brownian motions on the Heisenberg group, and then use this coupling to prove analytic gradient estimates for harmonic functions for the sub-Laplacian.

This talk is based on the joint work with Sayan Banerjee and Phanuel Mariano.

We will discuss various ways to produce statistics for families of curves over finite fields. This gives us a window into the greater world of arithmetic statistics and some of the tools used in the field, from geometric invariant theory to automorphic forms.

Elliptic curves defined over the rational numbers are of great
interest in modern number theory. The rank of an elliptic curve is a
crucial invariant, with many open questions about its behavior.

In particular, there is great interest in the ``average'' rank of an
elliptic curve. The minimalist conjecture is that the average rank
should be 1/2. In 2007, Bektemirov, Mazur, Stein, and Watkins [BMSW],
using well-known databases of elliptic curves, set out to numerically
compute the average rank of elliptic curves, ordered by conductor.
They found that ``there is a somewhat more surprising interrelation
between data and conjecture: they are not exactly in open conflict one
with the other, but they are no great comfort to each other either.''

In joint work with Ho, Kaplan, Spicer, Stein, and Weigandt, we have
assembled a new database of elliptic curves ordered by height. I will
describe the database and examine some of the questions posed by
[BMSW]. I will also discuss ongoing work by a team of undergraduates
at Oxford on similar questions about families of elliptic curves.

This is part of a series lectures studying stability of symplectic forms
on noncompact manifolds. The case of compact manifolds is well understood
thanks to seminal work of Jurgen Moser in the 1960s.

About 50 billion cells in your body go through mitosis each day, a process that requires a mother cell replicating and splitting its DNA among two daughter cells. We know that DNA is super-coiled and is very knotted in the nucleus for space purposes. Yet, with overwhelming probability, DNA manages to split evenly even in this highly tangled state. As it turns out, there is an enzyme called Topoisomerase II which cuts and glues DNA to let other strands of DNA pass through. No one is quite sure how this enzyme works or makes decisions on when to cut. In this talk, we'll explore a model that assumes Topo II makes strand cuts based off local topological properties. We'll also look at some results of numerical simulations to see how well this model mimics the true behavior of Topo II.

Initial data in general relativity must satisfy certain underdetermined differential equations called the constraint equations. A natural problem is to find a parameterization of all possible initial data. A standard method for this is called the conformal method. In this talk, we'll discuss the successes and failures of this method, and future directions for research.

Compressive sensing in its most practical form aims to recover a function that exhibits sparsity in a given basis from as few function samples as possible. One of the fundamental results of compressive sensing tells us that $O(s \log^4 N)$ samples suffice in order to robustly and efficiently recover any function that is a linear combination of $s$ arbitrary elements from a given bounded orthonormal set of size $N > s$. Furthermore, the associated recovery algorithms (e.g., Basis Pursuit via convex optimization methods) are efficient in practice, running in just polynomial-in-$N$ time. However, when $N$ is very large (e.g., if the domain of the given function is high-dimensional), even these runtimes may become infeasible.

If the orthonormal basis above is Fourier, then the sparse recovery problem above can also be solved using Sparse Fourier Transform (SFT) techniques. Though these methods aim to solve the same problem, they have a different focus. Principally, they aim to reduce the runtime of the recovery algorithm as much as absolutely possible, and are willing to sample the function a bit more often than a compressive sensing method might in order to achieve that objective. By doing so, one can indeed achieve similar recovery guarantees to Basis Pursuit, but with radically reduced runtimes that depend only logarithmically on $N$. However, SFTs are highly adapted to the special properties of the Fourier basis, making their extension to other orthonormal bases difficult.

In this talk we will present a general framework that can be used in order to construct a highly efficient SFT algorithm. The framework abstracts many of the components required for SFT design in an attempt to simplify the application of SFT ideas to other basis choices. Extension of arbitrary SFTs to the Chebyshev and Legendre polynomial bases will also be discussed.

We will discuss, using explicit examples, how dynamical systems can be used to study certain problems in number theory and geometry.

It is a famous conjecture that any one variable polynomial satisfying some simple conditions should take infinitely many prime values. Unfortunately, this isn't known in any case except for linear polynomials - the sparsity of values of higher degree polynomials causes substantial difficulties. If we look at polynomials in multiple variables, then there are a few polynomials known to represent infinitely many primes whilst still taking on `few' values; Friedlander-Iwaniec showed $X^2+Y^4$ is prime infinitely often, and Heath-Brown showed the same for $X^3+2Y^3$. We will demonstrate a family of multivariate sparse polynomials all of which take infinitely many prime values.

A planar set that contains a unit segment in every direction is called a Kakeya set. These sets have been studied intensively in geometric measure theory and harmonic analysis since the work of Besicovich (1919); we find a new connection to game theory and probability. A hunter and a rabbit move on an n-vertex cycle without seeing each other until they meet. At each step, the hunter moves to a neighboring vertex or stays in place, while the rabbit is free to jump to any node. Thus they are engaged in a zero sum game, where the payoff is the capture time. We show that every rabbit strategy yields a Kakeya set; the optimal rabbit strategy is based on a discretized Cauchy random walk, and it yields a Kakeya set K consisting of 4n triangles, that has minimal area among such Kakeya sets. (Talk based on joint work with Y. Babichenko, R. Peretz, P. Sousi and P. Winkler).

Holomorphic symplectic manifolds are the higher-dimensional
analogs of K3 surfaces and their local and global deformation theories
enjoy many of the same nice properties. By work of Namikawa, some aspects
of the story generalize to singular symplectic varieties, but the lack of
a well-defined period map means the moduli theory is badly behaved. In
joint work with C. Lehn, we consider locally trivial
deformations---deformations along which the singularities don't
change---and show that in this context most of the results from the smooth
case extend. In particular, we prove a version of the global Torelli
theorem and derive some applications to the geometry of birational
contractions of moduli spaces of vector bundles on K3 surfaces.

Fusion categories are generalizations of the representation categories of finite groups. One source of new fusion categories are subfactors, inlusions of von Neumann algebras with trivial center. The search for exotic subfactors led to new interesting fusion categories. One can study chiral conformal field theory via so-called conformal nets. I will explain how conformal nets give rise to fusion categories via its (higher) representation theory. It is an open question if all unitary fusion categories come from conformal nets. I will give examples of families of fusion categories for which one can reconstruct a conformal net.

In this talk, we consider a class of orthogonal constrained optimization problems, the feasible region of which is called the Stiefel manifold. Our new proposed framework combines a function value reduction step with a multiplier correction step. Different with the existing approaches, the function value reduction is conducted in the Euclidean space instead of the Stiefel manifold or its tangent space. We construct two types of algorithms based on! this new framework. The first type is gradient reduction based algorithms which consists of gradient reflection (GR) and gradient projection (GP) two implementations. The other one adopts a column-wise block coordinate descent (CBCD) scheme with a novel idea for solving the corresponding CBCD subproblem inexactly. Theoretically, we can prove that both GR/GP with a fixed stepsize and CBCD belong to our framework, and any clustering point of the iterates generated by the proposed framework is a first-order stationary point. Preliminary experiments illustrate that our new framework is of great potential.

We construct a pair of related diffusions on a space of partitions of the unit interval whose stationary distributions are the complements of the zero sets of Brownian motion and Brownian bridge respectively. Our methods can be extended to construct a class of partition-valued diffusions obtained by decorating the jumps of a spectrally positive Levy process with independent squared Bessel excursions. The processes of ranked interval lengths of our partition-valued diffusions are members of a two parameter family of infinitely many neutral allele diffusion models introduced by Ethier and Kurtz (1981) and Petrov (2009). Our construction is a step towards describing a diffusion on the space of real trees, stationary with respect to the law of the Brownian CRT, whose existence has been conjectured by Aldous. Based on joint work with N. Forman, S. Pal, and M. Winkel.

The Riemann Mapping Theorem (RMT) is a staple in complex analysis in one variable: {\it Any simply connected domain in the plane (other than the plane itself) is biholomorphically equivalent to the unit disk.} A direct analog is not true in two dimensions and higher. As discovered by Poincar\'e, the unit ball in $C^2$ is not biholomorphic to the bidisk. The reason is that in higher dimensions the boundary of a domain inherits a non-trivial structure---a CR structure--- from the ambient complex structure. We will discuss how one can formulate a version of the RTM that holds in higher dimensions as well. After this introduction, we shall mention some current fundamental problem in this area.

The l-adic cohomology of Rapoport-Zink spaces is expected to realize local Langlands correspondences in many cases. Along this line is a conjecture by Harris, which roughly says that when the underlying Rapoport-Zink space is not basic, the l-adic cohomology of the space is parabolically induced. In this talk, we will discuss a result on this conjecture when the Rapoport-Zink space is of Hodge type and ``Hodge-Newton reducible''. The main strategy is to embed our Rapoport-Zink space to an appropriate space of EL type, for which the conjecture is already known to hold. If time permits, we will also discuss other applications of this strategy.

This is not a mathematics talk but it is a talk for mathematicians. Too often, we think of historical mathematicians as only names assigned to theorems. With vignettes and anecdotes, I'll convince you they were also human beings and that, as the Chinese say, ``May you live in interesting times'' really is a curse.

Let $f: C' -> C$ be a cyclic cover of smooth projective curves.
Its Prym variety is by definition the complement of the pullback of the
Jacobian of $C$ in the Jacobian of $C'$. It is an abelian variety with a
polarization depending on the genus of $C$, the degree of $f$ and the
ramification type of the covering $f$. This gives a map from the moduli
space of coverings of this type into the moduli space of abelian varieties
of the corresponding type with endomorphism structure induced by the
automorphism given by $f$, called Prym map. In many cases the Prym map is
generically injective. Particularly interesting are the cases where the
Prym map is finite and dominant. In this talk these cases will be worked
out for covers of degree a prime number and twice an odd prime. In some
cases the degree of the Prym map is determined. This is joint work with
Angela Ortega.

We'll give multiple approaches to the problem of representing an integer as the sum of two squares. With this concrete motivation, we'll see examples of some important objects and theorems from the past 200+ years of number theory.

Bi-free probability is a generalization of free probability to study pairs of left and right faces in a non-commutative probability space. In this talk, I will demonstrate a characterization of bi-free independence inspired by the ``vanishing of alternating centred moments'' condition from free probability. I will also show how these ideas can be used to introduce a bi-free unitary Brownian motion and a liberation process which asymptotically creates bi-free independence.

The number of standard tableaux of a given shape and major index $r$ mod $n$ give the irreducible multiplicities of certain induced or restricted representations. We give simple necessary and sufficient conditions classifying when this number is zero. This result generalizes the $r=1$ case due essentially to Klyachko (1974) and proves a recent conjecture due to Sundaram (2016) for the $r=0$ case. Indeed, we prove a stronger asymptotic uniform distribution result for ``almost all'' shapes.

We'll discuss aspects of the proof, including a representation-theoretic formula due to Desarmenien, normalized symmetric group character estimates due to Fomin-Lulov, and new techniques involving ``opposite hook lengths'' for classifying $\lambda \vdash n$ where $f^\lambda \leq n^d$ for fixed $d$.

The solution set of a linear matrix inequality (LMI) is known as a spectrahedron. Free spectrahedra, obtained by substituting matrix tuples instead of scalar tuples into an LMI, arise canonically in the theory of operator algebras, systems and spaces and the theory of matrix convex sets. Indeed, free spectrahedra are the prototypical examples of matrix convex sets, set with are closed with respect to taking matrix convex combinations. They also appear in systems engineering, particularly in problems governed by a signal flow diagram.

Extreme points are an important topic in convexity; they lie on the boundary of a convex set and capture
many of its properties. For matrix convex sets, it is natural to consider matrix analogs of the notion of an extreme point. These notions include, in increasing order of strength, Euclidean extreme points, matrix extreme points, and Arveson boundary points. This talk will, in the context of matrix convex sets over $\mathbb{R}^g$, provide geometric unified interpretations of Euclidean extreme points, matrix extreme points, and Arveson boundary points. Additionally, methods for computing Arveson boundary points of free spectrahedra will be discussed.

In this talk, a representation of local and regular Dirichlet forms on real line, which are associated with symmetric linear diffusions, will be given and based on this, several related problems will be discussed.

In this paper, we propose a new weighted square-root LASSO procedure to estimate the regression coefficient matrix in sparse multivariate regression model with high-dimensional responses. The key advantage of the methodology is that it does not require the knowledge of the error term and has the tuning-insensitive property. To account for the within-subject correlation between responses, we use a working precision matrix which can be easily obtained in practice. Oracle inequalities of the estimators are derived. The performance of our proposed methodology is illustrated via extensive simulation studies. An application to DNA methylation data is also provided.

I will survey recent progress aimed at understanding the tautological rings of the moduli spaces of curves and K3 surfaces.

Eikonal equations arise in the fields of computer vision, image processing, geoscience, seismic tomography, to name a few. In some applications, the equation needs to be solved on a billion-point grid, and for tens of thousand times. In this talk, I will first introduce the most popular Fast Marching Method (FMM) by Sethian in 1996 and Fast Sweeping Method (FSM) by Zhao in 2005. Then I will briefly survey some modern variants and many parallelization techniques. In the last, I will describe a significant improvement when the algorithm is applied locally.

A del Pezzo surface of degree $d$ over a finite field of size $q$ has at most $q^2+(10-d)q+1$ $\mathbb{F}_q$-rational points. A surface attaining this maximum is called ‘split’, and if all of these rational points lie on the exceptional curves of the surface, then it is called ‘full’. Can we count and classify these extremal surfaces? We focus on del Pezzo surfaces of degree 3, cubic surfaces, and of degree 2, double covers of the projective plane branched over a quartic curve. We will see connections to the geometry of bitangents of plane quartics, counting formulas for points in general position, and error-correcting codes.

Wadspurger has shown that the genuine automorphic cuspidal representations of the metaplectic cover S of $SL_2$ are divided naturally into packets, and that thse packets are indexed by the cuspidal automorphic representations of $PGL_2$. We construct packets of Lambda-adic modular forms of half integral weight, indexed by Lambda-adic forms on $PGL_2$. The elements of the Lambda-adic packets are nonzero, but they have specializations that vanish, owing to a trivial zero phenomenon and the sign of a complex root number. This is in contrast to the usual trivial zero phenomenon which arises from the vanishing of a p-adic factor.

Combinatorics is rich with objects counted by the Catalan numbers. One such set of objects is the set of noncrossing partitions of the numbers 1 through n. There is a natural generalization in which one considers the set of noncrossing partitions of kn with block sizes each divisible by k. In this talk, we'll consider a rational generalization of noncrossing partitions and discuss current research in this subject.

Solving an unconstrained quadratic program means solving a linear equation where the matrix is symmetric and positive definite. This is a fundamental subproblem in nonlinear optimization. We discuss the behavior of the method of conjugate gradients and quasi-Newton methods on a quadratic problem. We show that by interpreting the method of conjugate gradients as a particular exact line search quasi-Newton method, necessary and sufficient conditions can be given for an exact line search quasi-Newton method to generate a search direction which is parallel to that of the method of conjugate gradients. The analysis gives a condition on the quasi-Newton matrix at a particular iterate, the projection is inherited from the method of conjugate gradients. We also analyze update matrices and show that there is a family of symmetric rank-one update matrices that preserve positive definiteness of the quasi-Newton matrix. This is in contrast to the classical symmetric-rank-one update where there is no freedom in choosing the matrix, and positive definiteness cannot be preserved.

Optimization is an indispensable tool in planning of intensity-modulated radiotherapy therapy (IMRT) planning. Conventional planning objectives are designed to minimize the violation of so-called dose-volume histogram (DVH) thresholds using penalty functions. In this study, we abandon the usual penalty-function framework and propose planning objectives that more explicitly relate to DVH statistics. The proposed planning objectives are based on mean-tail-dose, resulting in optimization problems of linear programming type. We investigate the potential of the proposed planning objectives as tools for optimizing DVH statistics through juxtaposition with the conventional planning objectives on two patient cases. An obstacle when changing from quadratic penalty functions to optimization of mean-tail-dose is that the dimension of the optimization problem is increased by several orders of magnitude. We demonstrate how to adapt a higher-order interior method to make this problem size manageable.

We will present a quantitative equidistribution result for adelic homogeneous subsets whose stabilizer is maximal and semisimple. Some number theoretic applications will also be discussed. This is based on a joint work with Einsiedler, Margulis and Venkatesh.

Energy minimization is the process by which nature selects the configuration of a system and balances the relevant forces. But not all of the current standard energy functionals in electrostatics are bounded below, leading to a contradiction. By recognizing Legendre transforms at the stationary points (or maximums) of non-convex (or concave) functionals, one can rewrite the functionals in terms of the transformed variable such that the new functionals are convex, hence better suited to standard optimization techniques. In this talk I will describe how Legendre transforms can be applied to reformulate the electrostatic free energy functional and present a proof of the equivalence of the new formulation with the classical one. I will also discuss how the Legendre transformed functional can be applied to the dielectric boundary problem in molecular solvation.

After a brief introduction to general relativity, I will discuss Penrose's celebrated strong cosmic censorship conjecture. This conjecture can be viewed as a global uniqueness conjecture, which states that for generic initial data, the solution is uniquely determined. This is in contrast to the phenomenon present in some special explicit black hole solutions such that determinism breaks down. I will then discuss some recent mathematical progress regarding this conjecture.

I will describe an ongoing project to study slices to Schubert varieties in the affine Grassmannian. These are Poisson varieties, and we will be mainly interested in quantizing them. The resulting algebras, called truncated shifted Yangians, have a beautiful representation theory. We will discuss this and also mention some connections to Nakajima quiver varieties which were recently discovered by Braverman-Finkelberg-Nakajima and Webster. In the pre-talk I'll define the affine Grassmannian and discuss its role in geometric representation theory.

Over 75$\%$ of ovarian cancers are detected in late stage disease with poor prognosis while if detected in early stage prognosis is often excellent. Despite therapeutic advances the mortality rate has not changed over the past 50 years. This makes early detection an appealing approach to investigate for its potential to reduce ovarian cancer mortality.

Screening trials starting in 1985 tested serum CA125 annually, a newly discovered blood test for monitoring ovarian cancer therapy. Women with CA125 exceeding a threshold were referred to transvaginal ultrasound and additional CA125 tests. This multi-modal approach attained an acceptable positive predictive value (PPV) however greater sensitivity for early stage disease remained a significant concern. Statistical analysis of longitudinal CA125 values from these trials indicated that most cases had exponentially rising CA125 from a baseline while most non-cases had relatively flat CA125 profiles. The challenge for the statistician was to devise a screening approach that leveraged the information in longitudinal CA125 values to increase sensitivity while maintaining the same PPV. Statistical modeling of these data led to a calculation of the risk of having a change-point given age and one or more serial CA125 values, essentially using each woman as her own control. The basis for the risk calculation was a hierarchical longitudinal change-point mixture model. This risk estimate incorporating longitudinal information is a surrogate for the risk of having undetected ovarian cancer which is the optimal information on which screening decisions should be based.

In 1996, the first of five screening trials implemented this risk calculation in general population postmenopausal women and in women at increased genetic risk. Trials in the general population measured CA125 annually and the algorithm referred women with intermediate risks to an additional CA125 test in 3 months, and women with elevated risks to an immediate ultrasound. All published trials implementing the risk of ovarian cancer algorithm showed an increase in early stage detection. No other ovarian cancer screening trials in the general or high risk populations have achieved this result.

Statisticians were also crucial in the design and analyses of these screening trials. The largest trial had ovarian cancer mortality as the endpoint and showed a mortality difference (p $\leq$ 0.05) with a 28$\%$ reduction in the second half. This reduction was seen in the 80\% of cases where the first CA125 test preceded the change-point (incident cases) enabling such cases to be their own control. However, further follow-up is needed for definitive conclusions. (Jacobs Menon et. al. The Lancet 2015).

One way of modeling phenomena in typical physical settings is to study PDEs in random environments. The subject of stochastic homogenization is concerned with identifying the asymptotic behavior of solutions to PDEs with random coefficients. Specifically, we are interested in the following: if the random effects are microscopic compared to the length scale at which we observe the phenomena, can we predict the behavior which takes place on average? For certain models of PDEs and under suitable hypotheses on the environment, the answer is affirmative. In this talk, I will focus on the stochastic homogenization for reaction-diffusion equations with both KPP and ignition nonlinearities. In the large-scale-large-time limit, the behavior of typical solutions is governed by a simple deterministic Hamilton-Jacobi equation modeling front propagation. In particular, we prove the existence of deterministic asymptotic speeds of propagation for reaction-diffusion equations in random media with both compactly supported and front-like initial data. Such models are relevant for predicting the evolution of a population or the spread of a fire in a heterogeneous environment. This talk is based on joint work with Andrej Zlatos.

In this talk, I will introduce a new geometric inequality: the Sphere Covering Inequality. The inequality states that the total area of two {\it distinct} surfaces with Gaussian curvature less than 1, which are also conformal to the Euclidean unit disk with the same conformal factor on the boundary, must be at least $4 \pi$. In other words, the areas of these surfaces must cover the whole unit sphere after a proper rearrangement. We apply the Sphere Covering Inequality to show the best constant of a Moser-Trudinger type inequality conjectured by A. Chang and P. Yang. Other applications of this inequality include the classification of certain Onsager vortices on the sphere, the radially symmetry of solutions to Gaussian curvature equation on the plane, classification of solutions for mean field equations on flat tori and the standard sphere, etc. The resolution of several open problems in these areas will be presented. The talk is based on joint work with Amir Moradifam from UC Riverside.

The study of partial sums of iid sequences is tightly connected to that of iterated convolutions. In this talk, I will discuss results that resemble local limit theorems for iterated convolution of complex valued functions in the case of $\mathbb Z$ and $\mathbb Z^d$. Similarities and differences with the probability densities will be in the spotlight.

The Faltings height is a useful invariant for addressing questions in arithmetic geometry. In his celebrated proof of the Mordell and Shafarevich conjectures, Faltings shows the Faltings height satisfies a
certain Northcott property, which allows him to deduce his finiteness
statements. In this work we prove a new Northcott property for the
Faltings height. Namely we show, assuming the Colmez Conjecture and the
Artin Conjecture, that there are finitely many CM abelian varieties of a
fixed dimension which have bounded Faltings height. The technique
developed uses new tools from integral p-adic Hodge theory to study the
variation of Faltings height within an isogeny class of CM abelian
varieties. In special cases, we are able to use these techniques to
moreover develop new Colmez-type formulas for the Faltings height.

In recent years the sequence of integers Cat(n)=(n choose 2)/(n+1) called ``Catalan numbers'' has been extended to a family of integers Cat(a,b)=(a+b choose a)/(a+b) that is parametrized by rational numbers a/b. These numbers originally showed up as the number of lattice paths in an aXb rectangle that stay above the diagonal. In 2002, Jaclyn Anderson gave a bijection between these paths and so-called (a,b)-core partitions. These are integer partitions in which no cell has hook length divisible by a or b. This result unlocked many new ideas in the area between combinatorics and representation theory. On the one hand, there have been many combinatorial conjectures and slightly fewer proofs. On the other hand, it seems that the numbers Cat(a,b) ultimately come from the representation theory of rational Cherednik algebras. The existence of a symmetric (q,t)- graded version of the numbers Cat(a,b) suggests that there should be a ``rational'' generalization of Mark Haiman's results on the Hilbert scheme of points in $C^2$.

We give a survey on recent work of Bao, Chan, Gaddis, He, Kirkman, Moore, Walton, Won, and others in noncommutative invariant theory.

I will speak on a recent joint study with J. Marzuola and D. Tataru which proves low regularity local well-posedness for quasilinear Schroedinger equations. Similar results were previously proved by Kenig, Ponce, and Vega in much higher regularity spaces using an artificial viscosity method. Our techniques, and in particular the spaces in which we work, are motivated by those used by Bejenaru and Tataru for semilinear equations.

Free Araki-Woods factors (FAWF) were introduced by Shlyakhtenko in 1996. In some sense they are free probability analogs of the hyperfinite factors. Shlyakhtenko showed that they are typically von Neumann algebras of type $III_1$, and moreover he constructed a one parameter family of non isomorphic type $III_1$ FAWF. In this talk I will discuss about the complexity of the classification problem of FAWF from the descriptive set theory point of view.

I present recent work on the design of scalable optimization algorithms for aiding in the big data task of subspace clustering. In particular, I will describe three approaches that we recently developed to solve optimization problems constructed from the so-called self-expressiveness property of data that lies in the union of low-dimensional subspaces. Sources of data that lie in the union of low-dimensional subspaces include multi-class clustering and motion segmentation. Our optimization algorithms achieve scalability by leveraging three features: a rapidly adapting active-set approach, a greedy optimization method, and a divide-and-conquer technique. Numerical results demonstrating the scalability of our approaches will be presented.

I will discuss a far reaching conjectural link between special
values of global and p-adic L-functions which unifies and refines some
classical conjectures of Stark, Gross, and Rubin. I will report on
progress in establishing this link emerging from my joint work with
Greither in equivariant Iwasawa theory and relate that to recent
results in this area due to Dasgupta and his collaborators.

The (Wednesday) RTG Colloquium lecture will be introductory in nature. It
will be followed by a more technical (Thursday) Number Theory Seminar
lecture where the main techniques and results will be explained in more
detail.

Multivariate polynomial optimization where variables and data are complex numbers is a non-deterministic polynomial-time hard problem that arises in various applications such as electric power systems, signal processing, imaging science, automatic control, and quantum mechanics. Complex numbers are typically used to model oscillatory phenomena which are omnipresent in physical systems. We propose a complex moment/sum-of-squares hierarchy of semidefinite programs to find global solutions with reduced computational burden compared with the Lasserre hierarchy for real polynomial optimization. We apply the approach to large-scale sections of the European high-voltage electricity transmission grid. Thanks to an algorithm for exploiting sparsity, instances with several thousand variables and constraints can be solved to global optimality.

I will discuss a far reaching conjectural link between special values of global and p-adic L-functions which unifies and refines some classical conjectures of Stark, Gross, and Rubin. I will report on progress in establishing this link emerging from my joint work with Greither in equivariant Iwasawa theory and relate that to recent results in this area due to Dasgupta and his collaborators.

Note: This is the second (more technical) lecture of a two lecture series. The first (introductory) lecture will be given in the RTG
Colloquium on April 5.

Introduced by Hamilton about thirty-five years ago, Ricci flow has developed into an integral and vital part of the geometric analysis. Some of its spectacular successes include the resolution of the Poincare conjecture for three manifolds and the complete classification of quarter pinched Riemannian manifolds. Ricci solitons, as self-similar solutions to Ricci flow, play an important role in understanding the singularity formation and long time dynamics of the flow. The talk will focus on the so-called shrinking solitons. We will review their classification in dimension two and three case, and mention some recent progress made jointly with Ovidiu Munteanu concerning their geometry in dimension four.

In this talk, we explore the relationship between Symplecticity and the
preservation of quadratic invariants. Symplectic Runge-Kutta methods,
the bread and butter of numerical geometric integration, are exactly
the ones that preserve all quadratic first integrals of a system. But
when we expand our focus to larger classes of methods, we will find a
more nuanced connection. It is also known that for symplectic RK
methods, the action of discretization commutes with forming variational
equations, and we will discuss the expansion of this result to a larger
class of methods.

We generalize the notion of a de Bruijn sequence to a ``multi de
Bruijn sequence'': a cyclic or linear sequence that contains every
$k$-mer over an alphabet of size $q$ exactly $m$ times. For example,
over the binary alphabet $\{0,1\}$, the cyclic sequence $(00010111)$
and the linear sequence $000101110$ each contain two instances of each
$2$-mer $00,01,10,11$. We derive formulas for the number of such
sequences. The formulas and derivation generalize classical de Bruijn
sequences (the case $m=1$). We also determine the number of multisets
of aperiodic cyclic sequences containing every $k$-mer exactly $m$
times; for example, the pair of cyclic sequences $(00011)(011)$
contains two instances of each $2$-mer listed above. This uses an
extension of the Burrows-Wheeler Transform due to Mantaci et al., and
generalizes a result by Higgins for the case $m=1$.

In their celebrated work, P. Li and S.-T. Yau proved the famous Li-Yau gradient bound for positive solutions of the heat equation on manifolds with Ricci curvature bounded from below. Since then, Li-Yau type gradient bounds has been widely used in geometric analysis, and become a powerful tool in deriving geometric and topological properties of manifolds.

In this talk, we will present our recent works on Li-Yau type gradient bounds for positive solutions of the heat equation on complete manifolds with certain integral curvature bounds, namely, $\vert Ric_\vert$ in $L^p$ for some $p>n/2$ or certain Kato type of norm of $\vert Ric_\vert$ being bounded together with a Gaussian upper bound of the heat kernel. These assumptions allow the lower bound of the Ricci curvature to tend to negative infinity, which is weaker than the assumptions in the known results on Li-Yau bounds. These are joint works with Qi S. Zhang.

I present a survey on the use of polynomial optimization in game theory. In recent years, there has been increased interest in applying methods from polynomial optimization to game theory. This research often studies games where individuals have polynomial utility functions or polynomial approximations to continuous utility functions. The presentation will include a discussion of research on zero sum games, generalized Nash equilibrium, principal-agent problems, and potential games.

We investigate in this talk the average root number (i.e. sign of the functional equation) of
non-isotrivial one-parameter families of elliptic curves (i.e elliptic curves over Q(t), or elliptic
surfaces over Q). For most one-parameter families of elliptic curves, the average root number
is predicted to be 0. Helfgott showed that under Chowla's conjecture and the square-free
conjecture, the average root number is 0 unless the curve has no place of multiplicative
reduction over Q(t). We then build non-isotrivial families of elliptic curves with no place
of multiplicative reduction, and compute the average root number of the families. Some
families have periodic root number, giving a rational average, and some other families have
an average root number which is expressed as an innite Euler product.
We then prove several density results for the average root number of non-isotrivial families
of elliptic curves, over Z and over Q (the previous density results found in the literature were
for isotrivial families). We also exhibit some surprising examples, for example, non-isotrivial
families of elliptic curves with rank r over Q(t) and average root number $-(-1)^r$, which
were not found in previous literature.

Since the seminal work of Yudovich in 1963, it has been known that for a given uniformly bounded and compactly supported initial vorticity profile, there exists a unique global solution to the 2d incompressible Euler equation. A special class of Yudovich solutions are so-called vortex patch solutions where the vorticity profile is the characteristic function of an (evolving) bounded set in $\mathbb{R}^2.$ In 1993 Chemin and Bertozzi-Constantin proved that sufficiently high regularity of the boundary is propagated for all time. Since then, there have been numerous numerical and rigorous works on understanding the long-time dynamics of smooth vortex patches as well as the short time dynamics of vortex patches with corners. In this work, we consider two regimes; one where we prove well-posedness and the other where we prove ill-posedness. First, for vortex patches with corners enjoying a certain symmetry property at the corners, we prove global propagation of the corners; we also give examples where these vortex patches cusp in infinite time. Second, we prove that vortex patches with a single corner (which do not satisfy the symmetry condition) immediately cease to have a corner. This is joint work with I. Jeong.

With respect to the numerical solution of partial differential equations, we explore the simple idea of treating time
as a space variable, and not employing the usual
method of lines time stepping approach. While this increases the space
dimension of a given PDE problem by one, it introduces a
static convection term that can be treated by a variety of
techniques. This approach can be especially beneficial
in the setting of parallel adaptive finite element
computations.

In a series of recent works Blomer, Fouvry, Kowalski, Milicevic, Sawin and myself have been able to solve the vexing problem of evaluating asymptotically the second moment of the central L-values of character twists (of large prime conductor) of a fixed modular form; the solution combines the spectral theory of modular forms, bounds for bilinear sums of Kloosterman sums and advanced methods in l-adic cohomology. We will describe the proof and especially the second and third ingredient which is joint work with E. Kowalski an W. Sawin.
References: https://arxiv.org/abs/1411.4467 and https://arxiv.org/abs/1511.01636.

With the increasing interest in applying the methodology of difference-of-convex (dc) optimization to diverse problems in engineering and statistics, we show that many well-known functions arising therein can be represented as the difference of two convex functions. These include a univariate folded concave function commonly employed in statistical learning, the value function of a copositive recourse function in two-stage stochastic programming, and many composite statistical functions in risk analysis, such as the value-at-risk (VaR), conditional value-at-risk (CVaR), expectation-based, VaR-based, and CVaR-based random deviation functionals. We also discuss decomposition methods for computing directional stationary points of a class of nonsmooth, nonconvex dc programs that combined the Gauss-Seidel idea, the alternating direction method of multipliers, and a special technique to handle the negative of a pointwise max function.

I will talk about joint work with Raymond Heitmann that gives a construction of big Cohen-Macaulay algebra in mixed characteristics following the recent breakthroughs on the direct summand conjecture by Andr\'{e} and Bhatt. In fact, we prove a weakly functorial version for certain surjective ring homomorphism that leads to the solution of the vanishing conjecture for maps of Tor in mixed characteristic. Our work also gives a simplified proof of the direct summand conjecture, and that direct summand of regular rings are Cohen-Macaulay.

On a compact Riemannian manifold with boundary having positive mean
curvature, a fundamental result of Shi and Tam states that, if the
manifold has nonnegative scalar curvature and if the boundary is
isometric to a strictly convex hypersurface in the Euclidean space,
then the total mean curvature of the boundary is no greater than the
total mean curvature of the corresponding Euclidean hypersurface. In
3-dimension, Shi-Tam's result is known to be equivalent to the
Riemannian positive mass theorem.

In this talk, we will discuss a supplement to Shi-Tam's theorem
by including the effect of minimal hypersurfaces on a chosen boundary
component. More precisely, we consider a compact manifold with
nonnegative scalar curvature, whose boundary consists of two parts,
the outer boundary and the horizon boundary. Here the horizon
boundary is the union of all closed minimal hypersurfaces in the
manifold and the outer boundary is assumed to be a topological
sphere. In a relativistic context, such a manifold represents a body
surrounding apparent horizon of black holes in a time symmetric
initial data set. By assuming the outer boundary is isometric to a
suitable 2-convex hypersurface in a Schwarzschild manifold of
positive mass m, we establish an inequality relating m, the area of
the horizon boundary, and two weighted total mean curvatures of the
outer boundary and the hypersurface in the Schwarzschild manifold. In
3-dimension, our result is equivalent to the Riemannian Penrose
inequality. This is joint work with Siyuan Lu.

I will present a recent work (joint with Sung-Jin Oh) on the strong
cosmic censorship conjecture for the
Einstein-Maxwell-(real)-scalar-field system in spherical symmetry for
two-ended asymptotically flat data. For this model, it was previously
proved (by M. Dafermos and I. Rodnianski) that a certain formulation
of the strong cosmic censorship conjecture is false, namely, the
maximal globally hyperbolic development of a data set in this class
is extendible as a Lorentzian manifold with a C0 metric. Our main
result is that, nevertheless, a weaker formulation of the conjecture
is true for this model, i.e., for a generic (possibly large) data set
in this class, the maximal globally hyperbolic development is
inextendible as a Lorentzian manifold with a C2 metric.

We study reaction-diffusion equations $u_t = L u + f(u)$ with homogeneous reactions f and diffusion operators L arising from the theory of Levy processes, with emphasis on propagation phenomena. The classical diffusion case (L = Laplacian) has been well-studied, including questions about traveling fronts, wavefront propagation, existence of spreading speeds, etc. After a brief review of the one-dimensional theory, we will concentrate on the case of nonlocal diffusions in several dimensions. We will discuss questions concerning long time dynamics of solutions, including spreading vs. quenching and existence of spreading speeds.

Initial data in general relativity must satisfy certain underdetermined differential equations called the constraint equations. A natural problem is to find a parameterization of all possible initial data. A standard method for this is called the conformal method. In this talk, we'll discuss the successes and failures of this method, and future directions for research.

We consider a reduced rank vector autoregressive model where one or several of the roots of the determinant of the characteristic polynomial have modulus stricter than one. We prove a noncausal Johansen-Granger representation for such time series and discuss how the parameters can be estimated. The asymptotic distribution of the trace statistic is also considered. Some Monte Carlo simulations and a small illustration are presented.

Since the discovery of the Jones polynomial in the 1980's, it is
well-known that subfactors of von Neumann factors are intimately
related to quantum topology. A subfactor is said to have infinite
representation theory, if its standard representation generates
infinitely many non-equivalent irreducibles. Such subfactors are
quite hard to come by, and very few methods are known to produce
interesting examples. I will highlight one such procedure, due to
Vaughan Jones and myself. The construction yields new C$^*$-tensor
categories and solutions of the quantum Yang-Baxter equation. If
time allows, I will also talk about invariants for subfactors
beyond the standard invariant.

We will start with a discussion of a hierarchy for convex optimization problems based on the classes of convex cones used in representations. Then, we will move to reviewing some of the recent developments in primal-dual interior-point algorithms for convex optimization problems in the context of this hierarchy. We will make connections to various areas of mathematics and mention some open problems.

I will talk about a result that is motivated by the work of De Simoi-Kaloshin-Wei. It concerns inverse spectral problems for strictly convex domains with one reflectional symmetry. The two key ingredients are wave trace formulas of Guillemin-Melrose and asymptotic of periodic billiard orbits of rotation numbers 1/q.

In this talk we will cover the topic of Hamiltonian variational
Integrators and a framework for adaptive symplectic integrators. First, I
will introduce variational integrators from the perspective of duality in
optimization. Then a variety of results will be presented on Hamiltonian
variational integrators and the role they can play in developing adaptive
symplectic integrators.

Formal schemes are, for admissible rings, the analogues of schemes for
ordinary rings. Formal schemes were originally introduced alongside
ordinary schemes in EGA, largely as tools for understanding the latter.
While they served well in such applications, more recent developments,
particularly in rigid analytic geometry, have motivated a study of
formal schemes in their own right. In this talk I will discuss some
vanishing theorems for formal schemes as work towards establishing a
minimal model program for (some suitable class of) them.

Tensors are multiway arrays, and tensor decompositions are powerful tools for data analysis and compression. In this talk, we demonstrate the wide-ranging utility of both the canonical polyadic (CP) and Tucker tensor decompositions with examples in neuroscience, chemical detection, and combustion science. The CP model is extremely useful for interpretation, as we show with an example in neuroscience. However, it can be difficult to fit to real data for a variety of reasons. We present a novel randomized method for fitting the CP decomposition to dense data that is more scalable and robust than the standard techniques. The Tucker model is useful for compression and can guarantee the accuracy of the approximation. We show that it can be used to compress massive data sets by orders of magnitude; this is done by determining the latent low-dimensional multilinear manifolds.

This talk features joint work with Woody Austin (University of Texas), Casey Battaglino (Georgia Tech), Grey Ballard (Wake Forrest), Alicia Klinvex (Sandia), Hemanth Kolla (Sandia), and Alex Williams (Stanford University).

A geometric flow is a way of evolving a geometry on a manifold.
The hope is that as time goes on, the geometry converges to something
recognizable. I will talk about what's known, and what's not known, for two
geometric flows in three dimensions. The first flow is the Ricci flow, used by
Perelman to prove Thurston's geometrization conjecture. The second flow
is the Einstein flow, which generates solutions of the vacuum Einstein equations
on a four dimensional spacetime. No prior knowledge of Ricci flow or the
Einstein equations will be assumed.

Suppose $\Omega$ is a nonempty open set with Lipschitz continuous boundary in $\mathbb{R}^n$. There are certain exponents $e\in \mathbb{R}$ and $q\in (1, infty)$ for which $\displaystyle \frac{\partial}{\partial x^j}: W^{e,q}(\Omega) \rightarrow W^{e-1,q}(\Omega)$ is NOT a well-defined continuous operator. Now suppose $M$ is a compact smooth manifold. In this talk we will try to discuss the
following questions:
\begin{enumerate}
\item How are Sobolev spaces of sections of vector bundles on $M$ defined?
\item Is it possible to extend $d: C^\infty(M)\rightarrow C^\infty(T^{*}M)$ to a continuous linear map from $W^{e,q}(M)$ to $W^{e-1,q}(T^{*}M)$ for all $e\in \mathbb{R}$ and $q\in (1,\infty)$?
\item Why are we interested in the above question?
\end{enumerate}

In recent years there has been growing interest in foliations
in complex algebraic geometry, both as a tool to study algebraic varieties
and as objects in their own right.
I will describe some recent work on applying the ideas of Mori theory to
foliations, and in particular, work on developing a foliated minimal model
program (MMP)
and a foliated Kodaira-Enriques classification.

We investigate Riemannian (non-Kahler) Ricci flow solutions that develop finite-time singularities
with the property that parabolic rescalings at the singularities converge to singularity
models that are shrinking Kahler--Ricci solitons, specifically, the conjecturally stable
``blowdown soliton'' discovereed by Feldman, Ilmanen and Knopf. This is a joint work with Isenberg and Knopf.

I will first talk about a recent project focusing on regulation of cell aging:

Cell aging is a universal biological phenomenon, but mechanisms that regulate aging remain unclear. Using novel microfluidic technologies, we tracked the replicative aging of single yeast cells and found a dramatic loss of heterochromatin silencing leading to cell aging and death. The dynamics of silencing loss during aging can be dissected into an early phase with sporadic silencing loss, followed by sustained silencing loss preceding cell death. Although the length of the later phase is relatively constant, the length of the early phase is highly variable among cells and largely determines lifespan. The intermittent silencing dynamics during the early phase depends on a conserved histone deacetylase Sir2 and is important for longevity, whereas either sustained silencing or loss of silencing shortens lifespan. These findings reveal that the temporal dynamics of key molecular processes can directly influence cell aging.

I will then show a couple of examples of projects, in which we are facing challenges in single-cell imaging analysis.

One important technical ingredient in many arithmetic statistics papers is upper bounds for finite exponential sums which arise as Fourier transforms of characteristic functions of orbits. This is typical in results obtaining power saving error terms, treating ``local conditions'', and/or applying any sort of sieve. In my talk I will explain what these exponential sums are, how they arise, and what their relevance is.

I will outline a new method for explicitly and easily evaluating them, and describe some pleasant surprises in our end results. I will also outline a new sieve method for efficiently exploiting these results, involving Poisson summation and the Bhargava-Ekedahl geometric sieve. For example, we have proved that there are ``many'' quartic field discriminants with at most eight prime factors. This is joint work with Takashi Taniguchi.

Abstract: Turán and Szemerédi-type theorems prove the existence of certain well-behaved patterns in dense graphs and hypergraphs. Here we show much stronger/larger patterns exist for graphs and hypergraphs that arise from geometry or algebra. The talk is based on recent joint works with Jacob Fox, János Pach, Adam Sheffer, and Josh Zahl.

Guided by the framework of conformal geometry in four dimensions, we consider several global invariants for CR geometry in 3-D that has implications for the embedding problem as well as the rigidity questions for CR structures.

I will talk about Gromov-Witten theory of Calabi-Yau one-folds
and Fan-Jarvis-Ruan-Witten theory of counterpart Landau-Ginzburg models.
The GW invariants and FJRW invariants are enumerative counting of stable
maps and sections of certain orbifold line bundles. We prove these
invariants are coefficients of expansions of appropriate quasi-modular
forms at different points, thus can be related by Cayley transformations.

We study the quasilinear wave equation $\Box_g u=0$, where the metric $g$ depends on $u$ and equals the Schwarzschild metric when $u$ is identically 0. Under a couple of assumptions on the metric $g$ near the trapped set and the light cone, we prove global existence of solutions. This is joint work with Hans Lindblad.

We explore two problems with applications in detonation shock dynamics and
molecular dynamics. First, we discuss level set methods, which are a
popular approach to modeling evolving interfaces. We present a level set
advection solver in two and three dimensions using high-order finite
elements. Our approach leads to stable front propagation and convergence
on high-order, curved, unstructured meshes. The solver's ability to
implicitly track moving fronts lends itself to a number of applications;
in particular, we highlight applications to high-explosive (HE) burn and
detonation shock dynamics (DSD).

In the second half, we look at molecular dynamics (MD) simulations, which
are widely used to study the motion and thermodynamic properties of
molecules. Computational limitations and the complexity of problems,
however, result in the need for error quantification. We examine the
inherent two-scale nature of MD to construct a large-scale dynamics
approximation as a means of error estimation.

Conical singularities occur quite often and naturally. For example, according to Cheeger-Colding, under Ricci curvature lower bounds, the limit spaces will generally carry singularity of conical type. This process of a family of smooth metrics limiting to a singular metric of conical type will be called conical degeneration. Again by Cheeger-Colding, under rather general conditions, the basic analytic quantities such as the eigenvalues and eigenfunctions will converge. So will the heat kernels (Ding). It is rather different story for global geometric invariants defined in terms of the eigenvalues. We will discuss some recent work in this direction.

We compute the Picard group of the moduli stack of genus zero stable quotients to projective space, Grassmannians, and any SL flag variety in the case of more than 2 markings. Furthermore, in the case of exactly 2 markings, we calculate the Picard group of the moduli stack of genus zero stable quotients to projective space, Grassmannians, and to partial flag varieties where the ranks of the subspaces differ by more than 1. Along the way we establish projectivity of the coarse moduli space.

A number of problems in statistical signal processing model the data as a Wide-Sense
Stationary (WSS) time series, whose power spectrum (or equivalently, the covariance
matrix) acts as a sufficient statistic for inferring parameters of interest. The covariance
matrix of such data exhibit Toeplitz structure, which can be leveraged to design highly
efficient compressive samplers to reduce the dimension of the WSS data, without
requiring it to have a sparse representation. Unlike existing results in Compressed
Sensing, the goal here is to recover the high dimensional covariance matrix (or infer
parameters of interest from it), instead of reconstructing the data itself.

Inspired by our past work on nested sensor arrays, I will describe a new sampling
techniques, known as the ``Generalized Nested Sample'' (GNS), to acquire compressive
measurements in such a way that it becomes possible to perfectly reconstruct the original
high dimensional covariance matrix from these compressed sketches. I will focus on lowrank
Toeplitz covariance matrices and develop an efficient GNS-based sampler which
allows the recovery of a rank-r Toeplitz covariance matrix from a compressed sketch of
size $O(√r) x O(√r)$, where the size of the sketch has no dependence on the ambient large
dimension N. Our reconstruction technique will use a regularizer-free framework,
combined with the ability to extrapolate additional “unobserved” entries of the NxN
covariance matrix. The algorithm has significantly lower computational complexity (that
does not scale with N) compared to recent nuclear-norm based compressive covariance
estimators, and is provably robust against bounded errors. Finally, I will consider the
special case of rank-1 and sparse covariance matrices that arise in the important problem
of “phase retrieval” in optical imaging. The role of 2nd order difference sets in complex
phase retrieval will be demonstrated, inspiring the design of a new class of non-uniform
Fourier sampler, that can provably recover a complex signal (upto a global phase
ambiguity) from its amplitude measurements with near-minimal number of samples. An
interesting connection with the so-called 4N-4 conjecture will also be established, which
hypothesizes 4N-4 to be the minimum number of measurements necessary to ensure
injectivity in N dimensions for complex phase retrieval.
Joint work with Heng Qiao, graduate student, University of California, San Diego.

I will summarize the work I have done both myself and with my coauthors during my Ph.D. studies. This includes the estimates of moduli of continuity for viscosity solutions in domains in Euclidean spaces and on manifolds, asymptotic behavior of nonparametric hypersurfaces moving by powers of Gauss Curvature, and the classification of four-dimensional shrinking gradient Ricci solitons with positive isotropic curvature.

Previous works have shown that arctic circle phenomenons and limiting behaviors of some integrable discrete systems can be explained by a variational principle. In this talk we will present the first results of the same type for a non-integrable discrete system: graph homomorphisms form $Z^d$ to a regular tree. We will also explain how the technique used could be applied to other non-integrable models.

When utilizing the finite element method in two dimensions, one requires a suitable mesh of the domain they wish to solve on. In this talk we will go over the term suitable, an in depth approach to arrive at this goal, and strategies for programming implementations. This talk will additionally demonstrate the workings behind the culmination of this research: a program which allows users to create 2D triangle meshes for any domain they desire.

Recently, Mendel and Naor, Dumitriu and Radcliffe, and Radcliffe and Williamson have begun the study of what could be termed the geometric Fiedler vector (or spectral gap) for graphs. Their principle observation is that the functional form associated with the graph can be expressed in terms of the distance function on $\mathbb{R}$. We give a partial structural characterization of when the geometric Fiedler vector can be extended to a geometric spectrum. Additionally, we provide applications of the geometric spectrum to community detection in graphs. This is joint work with Tobias Hagge, Patrick Mackey, Kathleen Nowak, Carlos Ortiz Marrero, and Jenny Webster.

The growth of bacterial colony exhibits striking patterns that are determined by the interactions among individual, growing and dividing bacterial cells, and that between cells and the surrounding nutrient and waste. Understanding the principles that underlie such growth has far-reaching consequences in biological and health sciences. In this work, we construct a multi-scale model of the growth of E. coli cells on agar surface. Our model consists of detailed, microscopic descriptions of the cell growth, cell division with fluctuations, and cell movement due to the cell-cell and cell-environment mechanical interactions, and macroscopic diffusion equations for the nutrient and waste. Our large-scale simulations reproduce experimentally observed growth scaling laws, strip patterns, and many other features of an E. coli colony. This work is the first step toward detailed multi-scale computational modeling of three-dimensional bacterial growth with mechanical and chemical interactions.

I will discuss recent joint work with Pankaj Vishe, in which we are able to say something
about the naive moduli space of rational curves on arbitrary smooth hypersurfaces
of sufficiently low degree, by invoking methods from analytic number theory.

One possible path forward for practical quantum computation is
the development of a topological quantum phase. In principle, such as
system will be more resistant to decoherence because of its inherently
topological nature. We highlight progress on two combinatorial questions
which arise naturally in the study of these systems.
Joint work with Paul Bruillard and Kathleen Nowak.

Clustering objects into similar clusters is an important practical problem in a wide variety of fields, including statistics, physics, bioinformatics, articial intelligence, and data mining. My thesis focuses on feature selection and semiparametric modeling in clustering. In this talk, I will present three of the projects I have done with my advisor during my PhD studies. The first one proposes a hill-climbing approach to sparse clustering, which has been shown to be competitive with existing methods in literature on simulated and real-world datasets. The second one considers a semiparametric mixture model for clustering and we propose a semiparametric EM algorithm to fit the model. The third one discusses the difficulty we uncovered in clustering with radial distributions. Under mild conditions, we prove that the magnitudes of the norm of observations sampled from a radial distribution are highly concentrated as the dimension becomes large.

In the classification theory of higher dimensional algebraic varieties a central role is played by the canonical divisor $K_X$ and its multiples. A famous theorem of Y. T. Siu states that if ${\pi:X\longrightarrow}$ T is a smooth projective family of varieties, then the plurigenera of the fibres $\lbrace h^0(X_t, mK_{X_t})\rbrace_{m\geq 0}$ are constant in t. Despite being an algebraic problem, Siu's proof employs methods which are essentially analytic in nature.
After giving an overview of the techniques involved, we outline a path to a possible algebraic proof, based on a reduction to the general type case via the Iitaka fibration.

Compressed sensing or compressive sampling (CS) is a signal
processing technique for efficiently acquiring and reconstructing
sparse signals by solving underdetermined linear systems. In practice,
CS needs to be accompanied by a quantization process. That is, after
sampling the signals, we represent the measurements using discrete
data, e.g. 0s and 1s, and recover the signals from the quantized
measurements. In this talk, I will discuss how to extend the
noise-shaping quantization methods beyond the case of Gaussian
measurements to structured random measurements, including random
partial Fourier and random partial Circulant measurements. This is
joint work with Rayan Saab

Unit root process, as a process with stochastic trend and a generalization
from random walk, is pervasive in physics, economics, and finance. In the
hypothesis test for unit root, bootstrap methods have earned a great deal
of attention. This talk will present three applications of bootstrap
methods. While the first application focuses on classic unit root process,
which contains only stochastic trend, the second and the third tackle
seasonal unit root process, which include simultaneously stochastic trend
and stochastic seasonality.

I will address two conjectures: a generalization of
Kurihara's conjecture on higher Fitting ideals, and a special case of the
Breuil-Schneider conjecture for certain de Rham representations.

We present a proof of the fact that given a Hodge
isometry Psi between the rational second cohomology of two Kahler K3
surfaces $S_1$ and $S_2$, we can find a finite sequence of K3 surfaces and
analytic (2, 2)-classes supported on successive products, such that
the isometry Psi is the convolution of these classes. The proof of
this fact implies that for projective K3 surfaces $S_1$, $S_2$ the class
of Psi is algebraic. This proves a conjecture of I. Shafarevich.

In this talk I will introduce the notion of
multivariable $(\phi, \Gamma)$-modules and explain their relationship
to representations of direct products of Galois groups. I will then
describe my recent work with Kiran Kedlaya aimed at proving that all
such multivariable $(\phi, \Gamma)$-modules are overconvergent.

In this talk, we provide a brief description of the stock market and internet search engines. We then suggest some efficient methods to gathering historical stock price data and Google search data. Additionally, we propose using a test that we created to explore the relationship, if any, of stock prices and the popularity of Google searches. Finally, we share our results from the test and discuss the possibility of using the popularity of Google searches to predict future stock price movement.

In this talk I will discuss some of the details of the proof that all ordinary $(\phi, \Gamma)$-modules are overconvergent and examine what additional arguments are necessary to prove this statement in the multivariable setting. Along the way we will encounter a restatement of Drinfeld's Lemma in the context of diamonds. Part 1 of this talk is being given during this week's RTG colloquium.

I will discuss recent approaches to working with formal schemes, including details from both proofs in my thesis and in the work of others.

Variational integrators preserve geometric and topological
structure when applied to Hamiltonian systems. Most of the research into
variational integrators has focused upon their derivation by discretizing
Hamilton's principle as a type I generating function of the symplectic
map. In this talk we examine the derivation of variational integrators
from a type II generating function. Even when the maps resulting from
different generating functions are analytically equivalent there can be
important numerical differences.
We introduce a new class of variational integrators based on the
Taylor method and an augmented shooting method. The role of automatic
differentiation for an efficient implementation is discussed. Finally,
a new framework for adaptive variational integrators is presented,
which is dependent upon Hamiltonian variational integrators.

Tensor categories have myriad uses in mathematics and physics, for
instance they appear as algebraic data associated to topological quantum
field theories and provide the framework for topological quantum
computation. What are all the tensor categories with given fusion rules?
This question can't be answered in full generality at the moment (by me)
but in this talk we discuss the classification of braided tensor
categories whose fusion rings are those of the representation rings of
classical Lie groups.

A few years ago in a landmark paper, Guionnet and Shlyakhtenko proved the existence of free monotone transport maps from the free group factors to von Neumann algebras generated by elements which have a joint law ``close'' to that of the free semicircular law. In this talk, I will discuss how to modify their idea to obtain similar results for interpolated free group factors using an operator-valued framework. This is joint work with Brent Nelson.

For a bounded convex domain on a Riemannian manifold, the fundamental gap is the difference of the first two non-trivial Dirichlet eigenvalues. In their celebrated work, B. Andrews and J. Clutterbuck proved the fundamental gap conjecture for convex domains in the Euclidean space, showing that the gap is at least as large as the one for a one-dimensional model. They also conjectured that similar results hold for spaces with constant sectional curvature. Very recently, on the unit sphere, Seto-Wang-Wei proved that the fundamental gap is greater than the gap of the one dimensional sphere model, in particular, $\geq 3\frac{\pi^2}{D^2}$($n \geq 3$), provided the diameter of the domain $D \leq \pi/2$. In a joint work with Guofang Wei at UCSB, we extend Seto-Wang-Wei's lower bound estimate to all convex domains in the hemisphere.

Recent advances in single cell technology enable researchers to study heterogeneity of cell populations and dynamics of gene expression in individual cell level. One of the main challenges is to extract the salient features in a manner that reveals the underlying dynamics process.

An optimization method, Single-cell Low Rank Similarity-based Method (ScLRSM), is proposed for identifying cell types associated with cell differentiation and detecting cell lineage from single-cell gene expression data. ScLRSM constructs structured cell-to-cell similarity matrix based on a low rank optimization model and cell types can be derived directly through the similarity matrix using non-negative matrix factorization.
The number of cell types is determined automatically via computing the eigenvalue gaps of the constructed consensus matrix while the vast majority of algorithms require the prior knowledge of such a number. In particular, the temporal order of cells is estimated by the non-negative rank one approximation of the cell-to-cell similarity matrix, which captures the global structure of the whole data. Cell lineage is inferred by constructing the minimum spanning tree of the weighted cluster-to-cluster graph.

We applied our method to three different single cell data sets with known lineage and developmental time information from both mouse early embryo and human early embryo. ScLRSM successfully identifies the cell subpopulations within different developmental time stages and reconstructs cell differentiation trajectories which is agreed with the previously experiments. The current results demonstrate the potential and high accuracy of the proposed method in determining cellular differentiation states and reconstructing cell lineages from single cell gene expression data.

The aim of this work (joint with Amy Ward [USC, Marshall School of Business]) is to determine fluid asymptotically optimal policies for many server queues with general reneging distributions. For exponential reneging distributions, it has been shown that static priority policies are optimal in a variety of settings, that include generally distributed interarrival and service times. Moreover, in these cases, the priority ranking is determined by a simple rule known as the c-mu-theta rule. For non-exponential reneging distributions, the story is more complex. We study reneging distributions with monotone hazard rates. For reneging distributions with bounded, nonincreasing hazard rates, we prove that static priority is not necessarily asymptotically optimal. We identify a new class of policies, which we are calling Random Buffer Selection and prove that these are asymptotically optimal in the fluid limit. We further identify a fluid approximation for the limiting cost as the optimal value of a certain optimization problem. For reneging distributions with nondecreasing hazard rates, our work suggests that static priority policies are in fact optimal, but the rule for determining the priority ranking seems more complex in general. It is work in progress to prove this.

Free entropy dimension and the first $L^2$ Betti number are both numeric invariants of discrete groups; one comes from Voiculescu’s free probability theory and is defined by using finite matrices to ``approximate’’ the group, while the other comes from geometric group theory and is of cohomological nature. Somewhat surprisingly, the two numbers are related. I will describe this connection and talk about some applications to von Neumann algebras.

Let $*$ be a binary operation on a set $X$ and let $x_0,x_1,\ldots,x_n$ be $X$-valued indeterminate.
Define two parenthesizations of $x_0*x_1*\cdots*x_n$ to be equivalent if they give the same function from $X^{n+1}$ to $X$. Under this equivalence relation, we study the number $C_{*,n}$ of equivalence classes and the largest size $\widetilde C_{*,n}$ of an equivalence class. We have $1\le C_{*,n}\le C_n$ and $1\le \widetilde C_{*,n}\le C_n$, where $C_n := \frac{1}{n+1}{2n\choose n}$ is the ubiquitous Catalan number. Moreover, $C_{*,n}=1 \Leftrightarrow$ $*$ is associative $\Leftrightarrow \widetilde C_{*,n}=C_n$. Thus $C_{*,n}$ and $\widetilde C_{*,n}$ measure how far the operation $*$ is away from being associative. In this talk we will present various results on the nonassociativity measurements $C_{*,n}$ and $\widetilde C_{*,n}$, and show their connections to many known combinatorial results, assuming $*$ satisfies some multiparameter generalizations of associativity.

The Graber-Harris-Starr theorem says that any family of smooth
rationally connected varieties over a complex curve has a section. A
natural analogue of this statement in mixed characteristic would be that
every rationally connected variety over the maximal unramified extension
of a p-adic field has a rational point. I will discuss a geometric
approach to this problem, as well as a proof of this statement for del
Pezzo surfaces (for ${p>3}$). This is joint work with David Zureick-Brown.

We apply spectral theory to study random processes involving directed graphs. In the first half of this talk, we apply spectral tools to study orientations of graphs. We focus on counting orientations yielding strongly connected directed graphs, called strong orientations. Namely, we show that under a mild spectral and minimum degree condition, a possibly irregular, sparse graph has ``many'' strong orientations. Furthermore, we provide constructions that show our conditions are essentially best possible. In the second half, we examine random walks on directed graphs, which is rooted in the study of non-reversible Markov chains. We prove bounds on key spectral invariants which play a role in bounding the rate of convergence of the walk and capture isoperimetric properties of the directed graph. These invariants include the principal ratio of the stationary distribution and the first nontrivial Laplacian eigenvalue. Finally, we conclude by briefly exploring future related work.

The problem of classifying modular categories is motivated by applications to
topological quantum computation as algebraic models for topological phases of matter.
These categories have also applications in different areas of mathematics like topological
quantum field theory, von Neumann algebras, representation theory, and others.

In this talk, we will start by introducing some of the basic definitions and properties of fusion, braided, and modular categories, and we will also give some concrete examples to have a better understanding of their structures. We will empathize some of the interesting properties that modular categories carry with them. We will give a brief overview on the situation of the classification program for this kind of categories.

In our talk we will discuss the old and celebrated question regarding the pointwise behavior of Fourier Series near $L^1$. This presentation will include

\begin{itemize}

\item the resolution of Konyagin's conjecture (ICM, Madrid 2006) on the pointwise convergence of Fourier Series along lacunary subsequences;

\item the $L^1$-strong convergence of Fourier Series along lacunary subsequences.

\item recent progress on the $L^1$-strong convergence of (the full) Fourier Series.

\end{itemize}

We end with several considerations on the relevance/impact of the above items on the subject of the pointwise convergence of Fourier Series.

In this talk, a new sequential quadratically constrained quadratic programming (SQCQP) algorithm is presented for nonlinear programming. At each iteration of an SQCQP method, a quadratically constrained quadratic program (QCQP) subproblem is solved followed by a line search. If an l-infinity penalty function is used as a merit function, this method is shown to have global convergent property under the MFCQ and other mild conditions. No convexity assumptions are made concerning the objective and constraints. Finally numerical results from the CUTEst test collection will be given to justify our theoretical prediction.

We investigate the collinearity of vector time series in the frequency domain,
by examining the
rank of the spectral density matrix at a given frequency of interest. Rank reduction
corresponds to collinearity at the given frequency. When the time series data
is nonstationary and has been differenced to stationarity, collinearity corresponds to
co-integration
at a particular frequency. We pursue a full understanding of rank through the
Schur complements of
the spectral density matrix, and test for rank reduction via assessing the positivity of
these Schur complements,
which are obtained from a nonparametric estimator of the spectral density. We provide new
asymptotic results
for the Schur complements, under the fixed bandwidth ratio paradigm. The test statistics
are $O_P (1)$ under
the alternative, but under the null hypothesis of collinearity the test statistics are
$O_P (T^{-1})$, and the
limiting distribution is non-standard. Subsampling is used to obtain the limiting null
quantiles. Simulation study and an empirical illustration for six-variate time series
data are provided.

Alongside derivative-based methods, which scale better to higher-dimensional problems, derivative-free methods play an essential role in the optimization of many practical engineering systems, especially those in which function evaluations are determined by statistical averaging, and those for which the function of interest is nonconvex in the adjustable parameters. This talk focuses on the development of a new family of surrogate-based derivative-free optimization schemes, namely Delta-DOGS schemes. The idea unifying this efficient and (under the appropriate assumptions) provably-globally-convergent family of schemes is the minimization of a search function which linearly combines a computationally inexpensive ''surrogate`` (that is, an interpolation or in some cases a regression, of recent function evaluations - we generally favor some variant of polyharmonic splines for this purpose), to summarize the trends evident in the data available thus far, with a synthetic piecewise-quadratic ''uncertainty function`` (built on the framework of a Delaunay triangulation of existing datapoints), to characterize the reliability of the surrogate by quantifying the distance of any given point in parameter space to the nearest function evaluations.

This talk introduces a handful of new schemes in the Delta-DOGS family:

(a) Delta-DOGS(Omega) designs for nonconvex (even, disconnected) feasible domains defined by computable constraint functions within a bound search domain.

(b) Delta-DOGS(Lambda) accelerates the convergence of Delta-DOGS family by restricting function evaluations at each iteration to lie on a dense lattice (derived from an n-dimensional sphere packing) in a linear constraint search domain. The lattice size is successively refined as convergence is approached.

(c) gradient-based acceleration of Delta-DOGS combines derivative-free global exploration with derivative-based local refinement.

(d) alpha-DOGSX designs to simultaneously increase the sampling time, and refine the numerical approximation, as convergence is approached.

This talk also introduces a method to scale the parameter domain under consideration based on the adaptive variation of the seen data in the optimization process, thereby obtaining a significantly smoother surrogate. This method is called the Multivariate Adaptive Polyharmonic Splines (MAPS) surrogate model. The judicious use of MAPS to identify variation of the objective function over the parameter space in some of the iterations results in neglecting the less significant parameters, thereby speeding up convergence rate.

These algorithms have been compared with existing state-of-the-art algorithms, particularly the Surrogate Management Framework (SMF) using the Kriging model and Mesh Adaptive Direct Search (MADS), on both standard synthetic and computer-aided shape designs such as the design of airfoils and hydrofoils. We showed that in most cases, the new Delta-DOGS algorithms outperform the existing ones.

In the 1960s, Benjamin and Feir, and Whitham, discovered that a Stokes wave would be unstable to long wavelength perturbations, provided that (the carrier wave number) x (the undisturbed water depth) $>$ 1.363.... In the 1990s, Bridges and Mielke studied the corresponding spectral instability in a rigorous manner. But it leaves some important issues open, such as the spectrum away from the origin. The governing equations of the water wave problem are complicated. One may resort to simpler approximate models to gain insights.

I will begin by Whitham's shallow water equation and the modulational instability index for small amplitude and periodic traveling waves, the effects of surface tension and vorticity. I will then discuss higher order corrections, extension to bidirectional propagation and two-dimensional surfaces. This is partly based on joint works with Jared Bronski (Illinois), Mat Johnson (Kansas), and Ashish Pandey (Illinois).

One way to measure the entanglement of a (pure) quantum state ${\Psi}$ is the Geometric Measure of Entanglement ${E(\Psi)}$, related to the spectral norm of tensor product states. The maximal possible value of ${E(\Psi)}$ on an $m$-qubit state ${\Psi}$ is $m$; it is $0$ only for product states.

In quantum computation, it is tempting to think ``the more entanglement'', the better. In 2009, Gross, Flammia, and Eisert showed that this intuition is incorrect. They proved that if ${\Psi}$ is an $m$-qubit state with near maximal entanglement, ${E(\Psi)>m-\delta}$, and if an NP problem can be solved by a computer with the power to perform local measurements on ${\Psi}$, then there is a purely classical algorithm that can solve the same problem (with positive probability) in a time only about ${2^\delta}$ times longer.

This suggests states with low entanglement are needed to get the exponential speed-up quantum computation is supposed to offer. However, as Gross et. al. also show, the situation seems hopeless: with respect to the Haar probability measure on all $m$ qubit states, ${E(\Psi)}$ is bigger than $m$ minus log factors with very high probability as $m$ grows.

Fortunately, this analysis ignores one key fact: the real quantum states that any proposed quantum computers use are Boson (symmetric states), since they are built out of photons. Hence, the results on entanglement of generic states do not apply.

In this talk, I will discuss my recent work with Shmuel Friedland, where we prove that the maximal possible entanglement for an $m$-qubit Boson state is ${\log_2(m+1)}$. Moreover, we show the same concentration phenomenon in this sphere: up to {\em double} log factors, with very high probability Boson quantum states are maximally entangled.

Transport phenomena in fluid flows are observed ubiquitously in nature
such as smoke rings in the air, pollutants in
the aquifers, plankton blooms in the ocean,
flames in combustion engines, and stirring a few
drops of cream in a cup of coffee.
We begin with examples of two dimensional Hamiltonian systems
modeling incompressible planar flows, and illustrate the transition
from ordered to chaotic flows as the Hamiltonian becomes more time dependent.
We discuss diffusive, sub-diffusive, and residual diffusive behaviors, and their
analysis via stochastic differential equation and a so called elephant random walk model.
We then turn to level-set Hamilton-Jacobi models of the flames, and
properties of the effective flame speeds in fluid flows under smoothing (such as
regular diffusion and curvature) as well as stretching.

The talk will indicate the key features in the proof of the
minimal model theorem for foliations by curves, which despite their
possibly chaotic nature more closely parallels semi-stable reduction of
curves (in arbitrary dimension) rather than the MMP for varieties. Indeed
since vanishing theorems are false, it is ironically Mori Theory as Mori
intended since everything must be done via the study of invariant rational
curves. Highlights include simple local criteria for canonical foliation
singularities, a simple classification of (foliated) Fano objects, and an
explicit (foliated) flip theorem by way of the study of formal
neighbourhoods of extremal rays.

In this talk we will investigate some topological properties of the space Fk(M) of ordered k-tuples of distinct points in a manifold M. For a fixed manifold M, as k increases, we might expect the topology of the configuration spaces Fk(M) to become increasingly complicated. Church and others showed, however, that when M is a connected and open, there is a representation-theoretic sense in which these configuration spaces stabilize. In this talk I will explain these stability patterns, and how they generalize classical notions of homological stability proved by McDuff and Segal in the 1970s. I will describe higher-order ``secondary stability'' phenomena established in recent work joint with Jeremy Miller. The project is inspired by work of Galatius--Kupers--Randal-Williams.

The moduli space of Riemann surfaces of fixed genus is one of the hubs of modern mathematics and physics. We will tell the story of how simple sounding problems about polygons, some of which arose as toy models in physics, became intertwined with problems about the geometry of moduli space, and how the study of these problems in Teichmuller dynamics lead to connections with homogeneous spaces, algebraic geometry, dynamics, and other areas. The talk will mention joint works with Alex Eskin, Simion Filip, Curtis McMullen, Maryam Mirzakhani, and Ronen Mukamel.

In celebration of Samuel Buss's 60th birthday, we are organizing an Omni Buss celebration. As Sam's work has had major impact on many areas of mathematics and computer science, including logic, proof complexity computational complexity, algorithms and graphics, the celebration will feature an eclectic combination of speakers.

Program
\indent 10:00 AM - Welcome

\indent 10:15 AM - Toniann Pitassi (University of Toronto) - Title TBA

\indent 11:00 AM - Coffee Break

\indent 11:15 AM - Arnold Beckmann (Swansea University) - Bounded Arithmetic a la Sam

\indent 11:45 AM - Steve Rotenberg (UC San Diego) - Interactive 3D Modeling of Roads

\indent 12:30 PM - 2:00 PM - Lunch Break

\indent 2:00 PM - Russell Impagliazzo (UC San Diego) - Title TBA

\indent 2:30 PM - Maria Luisa Bonet Carbonell (Universidad Politecnica de Cataluna) - Title TBA

\indent 3:00 PM - Jonathan Buss (University of Waterloo) - Some Corners of Computational Complexity

\indent 3:30 PM - Coffee Break

\indent 3:45 PM - Ryan Williams (MIT) - Sam and I versus P versus NP

\indent 4:30 PM - Finale a sorpresa

Despite the outstanding success of deep neural networks in real-world
applications, most of the related research is
empirically driven and a mathematical foundation is almost completely
missing. One central task of a neural network
is to approximate a function, which for instance encodes a
classification task. In this talk, we will be concerned
with the question, how well a function can be approximated by a neural
network with sparse connectivity. Using methods
from approximation theory and applied harmonic analysis, we will derive
a fundamental lower bound on the sparsity of
a neural network. By explicitly constructing neural networks based on
certain representation systems, so-called
$\alpha$-shearlets, we will then demonstrate that this lower bound can
in fact be attained. Finally, we present
numerical experiments, which surprisingly show that already the standard
backpropagation algorithm generates deep
neural networks obeying those optimal approximation rates. This is joint
work with H. Bolcskei (ETH Zurich), P. Grohs
(Uni Vienna), and P. Petersen (TU Berlin).

Let $G = PSL(2,\mathbb{R})$, let $\Gamma$ be a lattice in $G$, and let $\mathcal{H}$ be an irreducible unitary representation of $G$ with square-integrable matrix coefficients. A theorem in Goodman--de la Harpe--Jones (1989) states that the von Neumann dimension of $\mathcal{H}$ as a $W^*(\Gamma)$-module is equal to the formal dimension of the discrete series representation $\mathcal{H}$ times the covolume of $\Gamma$, both calculated with respect to the same Haar measure. We will present two results which take inspiration from this theorem. In the first part of the talk, we will show that there is a representation of $W^*(\Gamma)$ on a subspace of cuspidal automorphic functions in $L^2(\Lambda \backslash G)$, where $\Lambda$ is any other lattice in $G$ (and $W^*(\Gamma)$ acts on the right); and this representation is unitarily equivalent to one of the representations in Goodman--de la Harpe--Jones. In the second part of the talk, we will explain how the proof of the theorem in Goodman--de la Harpe--Jones carries over to a wider class of groups, including the situation where $G$ is $PGL(2,F)$, for $F$ a local non-archimedean field of characteristic $0$, and $\Gamma$ is a torsion-free lattice in $PGL(2,F)$, which, by a theorem of Ihara, is a free group. To give a simple example, we will focus on the case when $\mathcal{H}$ is the Steinberg representation (as opposed to a supercuspidal representation), and we will calculate its von Neumann dimension as a $W^*(\Gamma)$-module. This yields representations of free group factors that are not unitarily equivalent to those representations obtained in the setting of $PSL(2,\mathbb{R})$.

With the development of nonconventional oil and gas exploration, microscopic analysis of mineral distributions in shale receives much more attention in recent years. Meanwhile X-ray computerized tomography (CT) based on synchrotron radiation (SR), as a non-destructive technique, become an important tool and can be applied to the study of morphology, microstructure, transport properties and fracturing of shale. Traditional methods such as optical and scanning electron microscopy (SEM) are common tools for providing valuable information of microstructures; however, those surface observations are often inadequate in obtaining detailed 3D information of the sample, such as compositional distribution inside the shale. Moreover, samples of shale are usually damaged during serial sectioning. Therefore two scientific issues rose: one is how to generate high level reconstructed image data using SR-CT, another is how to use these CT image data to analyzing compositional microstructures. Therefore, these two issues lead to two kinds of inverse problems.

For X-ray tomographic imaging, the filtered backprojection (FBP) is the conventional algorithm for image reconstruction. There are several variations of the FBP, all of them rely on the Fourier central slice theorem of CT, and all of them incorporate a (linear) filtering operation and a backprojection operation. However, the FBP algorithm is sensitive to noise and is inflexible.

We study both inverse problems in our recent research projects. For the former issue, we consider the sparse regularization technique. For the later one, we consider the CT-data constrained minimization technique. Model-based optimization methods and neural networks-based big data analysis will be addressed.

I will discuss the recent work with Gang Liu on the diameter rigidity for Kahler manifolds with positive bisectional curvature.

The Fargues-Fontaine curve is a fundamental object in p-adic Hodge theory. I will mention some of the important properties of the curve, then introduce the rings that are used to build the curve. These are the extended Robba rings; they share many properties with the one dimensional affinoid algebras of rigid analytic geometry. I will discuss these similarities, point out some key differences, and explain how to adapt some proofs from rigid geometry to bypass these differences. The pre-talk will give an overview of some of the basics of rigid analytic geometry.

Geometric mechanics involves the use of differential geometry and symmetry techniques to study mechanical systems. In particular, it deals with global invariants of the motion, and how they can be used to describe and understand the qualitative properties of complicated dynamical systems, without necessarily explicitly solving the equations of motion. This approach parallels the development of geometric numerical methods in numerical analysis, wherein numerical algorithms for the solution of differential equations are constructed so as to exactly conserve the invariants of motion of the continuous dynamical system.

This talk will provide a gentle introduction to the role of geometric methods in understanding nonlinear dynamical systems, and why it is important to develop numerical methods that have good global properties, as opposed to just good local behavior.

First we focus on the size-Ramsey number of a path $P_n$ on $n$
vertices. In particular, we show that $5n/2 - 15/2 \le \hat{r}(P_n)\le
74n$ for $n$ sufficiently large. This improves the previous lower bound
due to Bollob\'{a}s, and the upper bound due to Letzter.

\medskip

Next we study long monochromatic paths in edge-colored random graph
$G(n,p)$ with $pn\to\infty$. Recently, Letzter showed that a.a.s. any
2-edge coloring of $G(n, p)$ yields a monochromatic path of length
$(2/3-o(1))n$, which is optimal. Extending this result, we show that
a.a.s. any 3-edge coloring of $G(n, p)$ yields a monochromatic path of
length $(1/2-o(1))n$, which is also optimal. We also consider a related
problem and show that for any $r\ge 2$, a.a.s. any $r$-edge coloring of
$G(n, p)$ yields a monochromatic connected subgraph on $(1/(r-1)-o(1))n$
vertices, which is also tight.

\medskip

Finally, we discuss some extensions of the above results for random
hypergraphs. In particular, we obtain a random analog of a result of
Gy\'arf\'as, S\'ark\'ozy, and Szemer\'edi on the longest monochromatic
loose cycle in $2$-colored complete $k$-uniform hypergraphs.

\medskip

This is joint work with Pawel Pralat and also with Patrick
Bennett, Louis DeBiasio, and Sean English.

We prove a quenched invariance principle and local limit theorem
for a random walk in an ergodic balanced time dependent environment
on the lattice. Our proof relies on the parabolic Harnack inequality
for the adjoint operator. This is joint work with X. Guo.

Lots of progress have been made in the recent years on the birational geometry of surfaces and 3-folds in positive characteristic over algebraically closed field. The same can not be said about the varieties over imperfect fields. These varieties appear naturally in positive characteristic while studying fibrations (as a generic fiber). Recently the minimal model program (MMP) for surfaces over excellent base scheme was successfully carried out by Tanaka. He also showed that the abundance conjecture holds for surfaces over imperfect fields. His results have become one of main tools for studying fibrations in positive characteristic. One of the things that is not covered in Tanaka's papers is the del Pezzo surfaces (a regular surface with -$K_X$ ample) over imperfect fields. One interesting feature of del Pezzo surfaces is that over an algebraically closed field they satisfy the Kodaira vanishing theorem. This makes the theory of del Pezzo surfaces quite interesting. However, over imperfect fields it was known for a while that in char 2, Kodaira vanishing fails for del Pezzo surfaces, due to (Schroer and Maddock). It is only very recently that some positive results started to show up. In a recent paper by Patakfalvi and Waldron it was shown that the Kodaira vanishing theorem holds for del Pezzo surfaces over imperfect fields in char $p>3$. In this talk I will show that in fact the Kawamata-Viehweg vanishing theorem holds for del Pezzo surfaces over imperfect fields in char $p>3$. I will also report on a project which is a work in progress (joint with Joe Waldron) on the minimal model program for 3-folds over imperfect fields and the BAB conjecture for del Pezzo surfaces over imperfect fields.

We will discuss a historical theorem of Gelfand and Neumark
characterizing abelian C*-algebras. The main indication for the importance of
this result is the fact that it relates two areas in math, namely topology and
operator algebras. We will take the time to discuss the ideas behind the proof
of this beautiful theorem as well as some other applications and related ideas.

Interior methods provide an effective approach for the treatment of inequality constraints in nonlinearly constrained optimization. A new primal-dual interior method is proposed that has favorable global convergence properties, yet, under suitable assumptions, is equivalent to the conventional path-following interior method in the neighborhood of a solution. The method may be combined with a primal-dual shifted penalty function for the treatment of equality constraints to provide a method for general optimization problems with a mixture of equality and inequality constraints.

Error-correcting codes are often used when data is transmitted over
a channel in which noise can occur, thereby damaging some of the data. There
are several types of error-correcting codes. In this talk, we will discuss an errorcorrecting
code that is defined in terms of a particular finite geometry. This finite
geometry comes from the incidence matrix of the so-called Wenger graphs. These
graphs are well-known to those working in extremal graph theory. The talk will
begin with a brief introduction to error-correcting codes, followed by linear codes.
We will then define the finite geometry, and discuss some progress on an open
problem of Cioab\u{a}, Lazebnik, and Li.

\'{S}niady constructed a random matrix model whose empirical eigenvalue distribution converges to the $q$-Gaussian random variable. In this talk, we prove that the Segal-Bargmann transform defined on the \'{S}niady random matrix model converges to the $q$-Segal-Bargmann transform.

Ancient solution is a type of Ricci flow that plays a fundamental role in singularity analysis. We introduce some results for ancient solutions, especially the classification of three-dimensional Type I ancient solutions, and a rigidity theorem for the four-dimensional shrinking cylinder.

This talk introduces a new kernel-based Maximum Mean Discrepancy (MMD) statistic for measuring the distance between two distributions given finitely-many multivariate samples. When the distributions are locally low-dimensional, the proposed test can be made more powerful to distinguish certain alternatives by incorporating local covariance matrices and constructing an anisotropic kernel. The kernel matrix is asymmetric; it computes the affinity between n data points and a set of $n_R$ reference points, where $n_R$ can be drastically smaller than n. While the proposed statistic can be viewed as a special class of Reproducing Kernel Hilbert Space MMD, the consistency of the test is proved, under mild assumptions of the kernel, as long as $\Vert p-q \Vert \sim$ O($n^{-1/2+\delta})$ for any $\delta>$ 0 based on a result of convergence in distribution of the test statistic. Applications to flow cytometry and diffusion MRI datasets are demonstrated, which motivate the proposed approach to compare distributions.

Many problems in stochastic modeling come down to study the crossing time of a certain stochastic process through a given boundary, lower or upper. Typical fields of application are in risk theory, epidemic modeling, queueing, reliability and sequential analysis. The purpose of this talk is to present a method to determine boundary crossing probabilities linked to stochastic point processes having the order statistic property. A very well-known boundary crossing result is revisited, a detailed proof is given. the same arguments may be used to derive results in trickier situations. We further discuss the practical implications of this classical.

I will explain the arithmetic analogue of the Gan-Gross-Prasad conjecture for unitary groups. I will also explain how to use arithmetic theta lift to prove certain endoscopic cases of it.

Fogarty showed in the 1970s that the Hilbert scheme of n points on a smooth surface is itself smooth. Interest in these Hilbert schemes has grown since it has been shown they arise in hyperkahler geometry, geometric representation theory, and algebraic combinatorics. In this talk we will explore the geometry of certain tautological bundles on the Hilbert scheme of points. In particular we will show that these tautological bundles are (almost always) stable vector bundles. We will also show that each sufficiently positive vector bundle on a curve C is the pull back of a tautological bundle from an embedding of C into the Hilbert scheme of the projective plane.

Functional time series has become a recent focus of statistical research.
In this talk, we will discuss the applications of kernel methods in the analysis of nonparametric functional time series. In the first half, we propose the kernel estimates for the autoregressor in a nonparametric functional autoregression model. It consistency is proved and a valid bootstrap procedure is provided to construct the prediction regions. In the second half of the talk, we propose a class of estimators for the spectral density kernel, which is a key element encapsulates the second-order dynamics of a functional time series. The new class of estimators employs the inverse Fourier transform of a flat-top function as the weight function employed to smooth the periodogram. It is shown that using a flat-top kernel yields a bias reduction and results in a higher-order accuracy in terms of optimizing the integrated mean square error (IMSE).

In 1990 Lakshmibai and Sandyha proved a remarkable result which provides a purely combinatorial method of determining whether or not a Schubert variety is smooth. In this talk we will start by examining the combinatorial tools needed for this theorem, namely pattern containment and avoidance in permutations. We then move to the land of algebraic geometry, starting with a brief description of varieties and singular points on varieties. Finally we will construct Schubert varieties as special subsets of the complex full flag manifold and state without proof the Lakshmibai-Sandyha Theorem. In doing so we hope to show that the intersection of combinatorics and algebraic geometry is nonempty (although maybe it is only an epsilon neighborhood).

The stochastic inverse problem for determining parameter values in a physics model from observational data on the output of the model forms the core of scientific inference and engineering design. We describe a recently developed formulation and solution method for stochastic inverse problems that is based on measure theory and a generalization of a contour map. In addition to a complete analytic and numerical theory, advantages of this approach include avoiding the introduction of ad hoc statistics models, unverifiable assumptions, and alterations of the model like regularization. We present a high-dimensional application to determination of parameter fields in storm surge models. We conclude with recent work on defining a notion of condition for stochastic inverse problems and the use in designing sets of optimal observable quantities.

How does one measure area? As an example, how can one determine the area of a region on a map for the purpose of real estate appraisal? Wouldn't it be great if there were an instrument that would measure the area of a region by simply tracing its boundary? It turns out that there is such an instrument: it is called a planimeter. In this talk we will discuss a particular type of planimeter called the compensating polar planimeter. There will be a little bit of history and some analysis involving line integrals and Green's theorem. Finally, there will be a chance to see and touch actual examples of these fascinating instruments from the speaker's collection.

Free probability was introduced in the 1980’s by Voiculescu, with the aim of
studying von Neumann algebras by viewing them as non-commutative probability
spaces, and this analogy has proved quite powerful in operator algebra theory. In the
1990’s, Speicher was able to describe free independence using cumulants constructed
from the lattice of non-crossing partitions. In this talk we will give an introduction
to free probability and outline the role the non-crossing cumulants have played
in describing the theory. Time permitting we will also demonstrate some more
recent applications of combinatorics to free probability, such as in describing bi-free
probability, type B free probability, and boolean independence.

Massive data are often contaminated by outliers and heavy-tailed errors. In the presence of heavy-tailed data, finite sample properties of the least squares-based methods, typified by the sample mean, are suboptimal both theoretically and empirically. To address this challenge, we propose the adaptive Huber regression for robust estimation and inference. The key observation is that the robustification parameter should adapt to sample size, dimension and moments for optimal tradeoff between bias and robustness. For heavy-tailed data with bounded $(1+\delta)$-th moment for some $\delta>0$, we establish a sharp phase transition for robust estimation of regression parameters in both finite dimensional and high dimensional settings: when $\delta \geq 1$, the estimator achieves sub-Gaussian rate of convergence without sub-Gaussian assumptions, while only a slower rate is available in the regime $0<\delta <1$ and the transition is smooth and optimal.

In addition, non-asymptotic Bahadur representation and Wilks’ expansion for finite sample inference are derived when higher moments exist. Based on these results, we make a further step on developing uncertainty quantification methodologies, including the construction of confidence sets and large-scale simultaneous hypothesis testing. We demonstrate that the adaptive Huber regression, combined with the multiplier bootstrap procedure, provides a useful robust alternative to the method of least squares. Together, the theoretical and empirical results reveal the effectiveness of the proposed method, and highlight the importance of having statistical methods that are robust to violations of the assumptions underlying their use.

Models of information exchange that originate from economics provide interesting questions in probability. We will introduce some of these models, discuss open questions, and explain some recent results.
Joint with Wade Hann-Caruthers, Matan Harel, Vadim Martynov, Elchanan Mossel and Philipp Strack

We will discuss some geometric techniques used in proving ''arithmetic statistics'' results,
primarily using the case of Selmer groups for families of elliptic curves as a motivating example.

We consider the problem of computing a cluster of eigenvalues, and its associated eigenspace, of a (possibly unbounded) selfadjoint operator in a Hilbert space. A rational function of the operator is constructed such that the eigenspace of interest is its dominant eigenspace, and a subspace iteration procedure is used to approximate this eigenspace. The computed space is then used to obtain approximations of the eigenvalues of interest. An eigenvalue and eigenspace convergence analysis that considers both iteration error and discretization error is provided. A realization of the proposed approach for a model second-order elliptic operator is based on a discontinuous Petrov-Galerkin discretization of the resolvent, and a variety of numerical experiments illustrate its performance.

In recent years, there has been a great deal of excitement about 'big data' and about the new research problems posed by a world of vastly enlarged datasets.

In response, the field of Mathematical Statistics increasingly studies problems where the number of variables measured is comparable to or even larger than the number of observations. Numerous fascinating mathematical phenomena arise in this regime; and in particular theorists discovered that the traditional approach to covariance estimation needs to be completely rethought, by appropriately shrinking the eigenvalues of the empirical covariance matrix.

This talk briefly reviews advances by researchers in random matrix theory who in recent years solved completely the properties of eigenvalues and eigenvectors under the so-called spiked covariance model. By applying these results it is now possible to obtain the exact optimal nonlinear shrinkage of eigenvalues for certain specific measures of performance, as has been shown in the case of Frobenius loss by Nobel and Shabalin, and for many other performance measures by Donoho, Gavish, and Johnstone. We describe these results as well as results of the author and Behrooz Ghorbani on optimal shrinkage for Multi-User Covariance estimation and Multi-Task Discriminant Analysis.

Extremal set theory tries to answer questions about the maximal or minimal size of subsets of some universal set, while respecting certain imposed restrictions. In this talk we will discuss one such example, which is known as the Frankl-Wilson theorem. This theorem turns out to have applications in geometry, providing lower bounds on both the chromatic number of Euclidean space and the number of parts one needs to subdivide a bounded region in Euclidean space into smaller regions.

Harnessing disruptive innovations dynamics is theoretically presented based on Dirac-Solow-Swan model. The quantum view leads into the conclusion that hyperfine splitting of capital is occurred due to the disruption and as a consequence is the excitation of capital and labour from the old into the new industry of disruption. The bifurcation of capital dynamics occurs due to Christensen effect, after market symmetry breaking. It is shown that harnessing disruptive innovations relies on controlling expansion factor of capital accumulation on mainstream market.

$\indent$The classical congruence subgroup problem asks whether every finite
quotient of $G={\rm GL}_{n}(\mathbb{Z})$ comes from a finite quotient
of $\mathbb{Z}$. I.e. whether every finite index subgroup of $G$ contains a principal
congruence subgroup of the form
$G(m)= ker(G\to{\rm GL}_{n}(\mathbb{Z}/m\mathbb{Z})$ for some $m\in\mathbb{N}$? If the answer is affirmative we say that $G$ has the congruence subgroup property (CSP). It was already known in the $19^{\underline{th}}$ century that ${\rm GL}_{2} (\mathbb{Z})$ has many finite quotients which do
not come from congruence considerations. Quite surprising, it was proved in the sixties
that for $n\geq3, {\rm GL}_{n} (\mathbb{Z})$ does have the CSP.

Observing that ${\rm GL}_{n} (\mathbb{Z}) \cong {\rm Aut} (\mathbb{Z}^{n})$, one can generalize the congruence subgroup problem as follows: Let $\Gamma$ be a group. Does every finite index subgroup of $G = {\rm Aut}(\Gamma)$ contain a principal congruence subgroup of the form $G(M) = \ker(G\to{\rm Aut}(\Gamma/M)$ for some finite index characteristic subgroup $M\leq\Gamma$? Very few results are known when $\Gamma$ is not abelian. For example,
we do not know if ${\rm Aut} (F_{n})$ for $n\geq3$ has the CSP. But, in 2001 Asada proved, using tools from algebraic geometry, that ${\rm Aut} (F_{2})$ does have the CSP, and later, Bux-Ershov-Rapinchuk gave a group theoretic version of Asada's proof (2011).

On the talk, we will give an elegant proof to the above theorem, using
basic methods of profinite groups and free groups.

Subsampling methods as well as general sampling methods appear as natural tools
to handle very large database (big data in the indivual dimension) when
traditional statistical methods or statistical learning algorithms fail to be
implemented on too large datasets. The choice of the weights of the survey
sampling sheme may reduce the loss implied by the choice of a much more smaller
sampling size (according to the problem of interest).

I will first review some asymptotic results for general survey sampling based
empirical processes indexed by class of functions (Bertail and Clemencon, 2017),
for Poisson type and conditional Poisson (rejective) survey samplings. These
results may be extended to a large class of survey sampling plans via the notion
of negative association of most survey sampling plans (Bertail and Rebecq, 2017).
Then, in the perspective to generalize some statistical learning tasks to sampled
data, we will obtain exponential bounds for the probabilities of deviation of a
sample sum from its expectation when the variables involved in the summation are
obtained by sampling in a finite population according to a rejective scheme,
generalizing sampling without replacement and using an appropriate normalization.

In contrast to Poisson sampling, classical deviation inequalities in the i.i.d.
setting do not straightforwardly apply to sample sums related to rejective schemes
due to the inherent dependence structure of the sampled points. We show here how
to overcome this difficulty by combining the formulation of rejective sampling
as Poisson sampling conditioned upon the sample size with the Escher transformation.
In particular, the Bennet/Bernstein type bounds established highlight the effect of
the asymptotic variance of the (properly standardized) sample weighted sum, and are
shown to be much more accurate than those based on the negative association property.

The numerical analysis of finite element methods in computational
electromagnetism can be developed elegantly and comprehensively if commuting
projection operators between de Rham complexes are available. Hence the
construction of such commuting projection operators is central but has been
elusive in several practical relevant settings of low regularity. In this talk
we describe how to close this gap: we construct smoothed projections over
weakly Lipschitz domains and extend the theory to mixed boundary conditions.

We study random constructions in incidence structures using a general theorem on set systems. Our main result applies to a wide variety of well-studied problems in finite geometry to give almost tight bounds on the sizes of various substructures.

In its classical setting, the Congruence
Subgroup Problem (CSP) asks whether every finite index subgroup of
$GL_{n}(\mathbb{Z})$ contains a principal congruence subgroup of the
form $\ker(GL_{n}(\mathbb{Z})\to GL_{n}(\mathbb{Z}/m\mathbb{Z}))$ for
some $m\in\mathbb{Z}$. It was known already in the 19th century that
for $n=2$ the answer is negative, and actually $GL_{2}(\mathbb{Z})$
has many finite index subgroups which do not come from congruence
considerations. On the other hand, quite surprisingly, in the sixties
it was found out by Mennicke, and separately by Bass-Lazard-Serre,
that the answer for $n>2$ is affirmative. This result was a
breakthrough that led to a rich theory which generalized the problem
to matrix groups over rings.

Viewing $GL_{n}(\mathbb{Z})\cong Aut(\mathbb{Z}^{n})$ as the automorphism
group of of $\Delta=\mathbb{Z}^{n}$, one can generalize the CSP to
automorphism groups as follows: Let $\Delta$ be a group, does every
finite index subgroup of $Aut(\Delta)$ contain a principal congruence
subgroup of the form: $\ker(Aut(\Delta)\rightarrow Aut(\Delta/M))$
for some finite index characteristic subgroup $M\leq\Delta$? Considering
this generalization, there are very few results when $\Delta$ is
non-abelian. For example, only in 2001 Asada proved, using tools from
Algebraic Geometry, that $Aut(F_{2})$ has an affirmative answer to
the CSP, when $F_{2}$ is the free group on two generators. For $Aut(F_{n})$
when $n>2$ the problem is still unsettled. On the talk, I will present
the problem from a few aspects, and introduce some recent results
for non-abelian groups. The main result will assert that while the
dichotomy in the abelian case is between $n=2$ and $n>2$, when $\Delta$
is the free metabelian group on n generators, we have a dichotomy
between $n=2,3$ and $n>3$.

A variety is called rational if it is birational to projective space. For example, the only compact, smooth, and rational Riemann surface is the Riemann sphere. A general compact Riemann surface carries two natural invariants which measure its complexity & non-rationality from both a topological and an algebraic perspective: the genus and the gonality. Both of these invariants have classically played a very important role in the study of curves. In higher dimensions there are a number generalizations of these birational invariants which measure the irrationality of an algebraic variety. I will discuss the computation of one of these invariants — the degree of irrationality — and I will pose a number of open problem about these measures of irrationality.

The conditions which determine whether or not a matrix is special orthogonal are polynomial in the matrix entries and thus give an explicit description of SO(n) as an embedded algebraic variety. We give a formula for the degree of this variety for any n which is interpretable as counting non-intersecting lattice paths. This degree also contributes to the degree of low-rank semidefinite programming. We explain how to verify the formula explicitly using numerical algebraic geometry (for $n\leq7$) and how numerical computations aid in the study of the real locus of this variety.

I will discuss the theory of theta representations for the degree n cover
of Glr and in particular those distinguished ones that have unique Whittaker
models. I will concentrate on the study of the know cuspidal distinguished
representations and possible generalizations.

Multi-strain microbial communities often exhibit complex spatial organization that emerges due to the interplay of various cooperative and competitive interaction mechanisms. One strong competitive mechanism is contact-dependent neighbor killing, such as that enabled by the type VI secretion system (T6SS). It has been previously shown that contact-dependent killing can result in bistability of bacterial mixtures, so that only one strain survives and displaces the other. However, it remains unclear whether stable coexistence is possible in such mixtures. Using a population dynamics model for a mixture of two bacterial strains, we found that coexistence can be made possible by combining contact-dependent killing with long-range growth inhibition, leading to the formation of various cellular patterns. These patterns emerge in a much broader parameter range than that required for the Turing instability, suggesting this may be a more robust mechanism for pattern formation.

I will explain how to build Koornwinder polynomials at q = t from moments of Askey-Wilson polynomials.
I will use the combinatorial theory of Viennot for orthogonal polynomials and their moments. An extension of this theory allows to build multivariate orthogonal polynomials.
The key step for this construction area Cauchy identity for Koornwinder polynomials and a Jacobi-Trudi formula for the 9th variation of Schur functions. This gives us an elegant path model for these polynomials. I will also explain a positivity conjecture for these polynomials that we can prove in several special cases. For this, we link them to the stationary distribution of an exclusion process and prove positivity by exhibiting a combinatorial model called rhombic staircase tableau. This talk is based on joint work with Olya Mandelshtam (Brown) and Lauren Williams (Berkeley).

Given a smooth projective variety over an algebraically closed field of positive characteristic, can we always dominate it by another smooth projective variety that lifts to characteristic 0? We give a negative answer to this question.

The human brain is comprised of a complex network of specific cell types. Revealing the composition of these specific cells at a given point during development, in disease versus health and in specific brain regions may provide insight into the highly specialized and regulated organization of the brain.

We show finite time blow-up for strong solutions to the 3D Euler equations on the exterior of a cone. The solutions we construct has finite energy, and velocity is axisymmetric and Lipschitz continuous before the blow up time. We achieve this by first analyzing scale-invariant (radially homogeneous) solutions, whose dynamics is governed by a 1D system. Then we make a cut-off argument to ensure finiteness of energy.

Turbulent flows appear in many different phenomena and is of
fundamental importance in science and technology. Great part of the
conventional statistical theory of turbulence, however, is based on heuristic
arguments and empirical information, with the notion of ensemble average of
flows playing a fundamental role. The theory of statistical solutions aims
towards a rigorous foundation for the conventional statistical theory of
turbulence by rigorously defining the evolution of the probability
distributions of the velocity field within the framework of Leray-Hopf weak
solutions of three-dimensional incompressible Navier-Stokes equations. In this
talk we will review a few characteristics of turbulent flows and discuss the
concept of statistical solution. We then mention some rigorous results obtained
with this framework. If time permits, we discuss a generalization of the notion
of statistical solution to an abstract setting that easily applies to many
different equations.

Many important maps in algebraic combinatorics (the RSK bijection, the Schutzenberger involution, etc.) can be described by piecewise-linear formulas. These formulas can then be ``de-tropicalized,'' or ``lifted,'' to subtraction-free rational functions on an algebraic variety, and certain properties of the combinatorial maps become more transparent in the algebro-geometric setting. I will illustrate how this works in the case of the promotion map on semistandard tableaux of rectangular shape. I will also indicate how promotion can be viewed as the combinatorial manifestation of a symmetry coming from representation theory, and how its geometric lift fits into Berenstein and Kazhdan's theory of geometric crystals.

In this paper, we propose a new updating rule of the Levenberg–Marquardt (LM) parameter for the LM method for nonlinear equations. We show that the global complexity bound of the new LM algorithm is $O(\epsilon^{-2})$, that is, it requires at most $O(\epsilon^{-2})$ iterations to derive the norm of the gradient of the merit function below the desired accuracy $\epsilon$.

Starting in 1979, the physicist Giorgio Parisi wrote a series of ground
breaking papers introducing the idea of replica symmetry breaking, which
allowed him to predict a solution for the Sherrington-Kirkpatrick (SK)
model by breaking the symmetry of replicas infinitely many times. This
is known as full-step replica symmetry breaking (FRSB). In this talk, we
will provide a mathematical proof of Parisi's FRSB prediction at zero
temperature for the more general mixed p-spin model. More precisely, we
will show that the functional order parameter of this model is not a
step function. This talk is based on joint work with Antonio Auffinger
and Wei-Kuo Chen.

The revolution in geospatial data is transforming how we study the growth and development of cities. As improved satellite imagery becomes available, new remote-sensing methods and machine-learning approaches have been developed to convert terrestrial Earth-observation data into meaningful information about the nature and pace of change of urban landscapes and human settlements. Urban areas can be detected in satellite imagery using machine-learning approaches, which typically rely on reference ground-truth data that mark urban features, either for training or for validation. Reference data are fundamental not only for mapping and assessing cross-sectional urbanization across space, but also for classification of urbanization over time. However, because they are expensive to collect, large-scale reference datasets are scarce. We present a low-cost machine-learning approach for pixel-based image classification of built-up areas at a high-resolution and large scale. Our m
ethodology relies on data infusion of nighttime and daytime remotely sensed data for automatic collection of ground truth data, which we use for supervised pixel-based image classification of built-up land cover. We demonstrate the effectiveness of our methodology, which is implemented in Google Earth Engine, through the development of accurate 30m resolution maps that characterize built-up land cover in three diverse countries: India, Mexico, and the U.S. Our approach highlights the usefulness of data fusion techniques for studying the built environment and has broad implications for identifying the drivers of urbanization.

Classical results in ergodic theory due to Dye and Ornstein--Weiss show that, for an arbitrary countable amenable group, any two free ergodic measure-preserving actions on the standard atomless probability space are orbit equivalent, i.e. their orbit equivalence relations are isomorphic. This motivates the question of what happens for nonamenable groups. Works of Ioana and Epstein showed that, for an arbitrary countable nonamenable group, the relation of orbit equivalence of free ergodic measure-preserving actions on the standard probability space has uncountably many classes. In joint work with Gardella, we strengthen these conclusions by showing that such a relation is in fact not Borel. The proof makes essential use of techniques from operator algebras, including cocycle superrigidity results due to Popa, and answers a question of Kechris.

Let G be a smooth group scheme over the p-adic integers with reductive generic fiber.
We study the generic fiber of the universal lifting ring of a G-valued mod-p representation
of the absolute Galois group of an l-adic field. In particular, we show that it admits an
open dense regular locus, and is equidimensional of dimension dim G.
This is joint work with Stefan Patrikis.

Many eukaryotic cells chemotax, sensing and following chemical gradients. However, experiments find that even under conditions when single cells do not chemotax, small clusters may still follow a gradient. How can cell clusters sense a gradient that individual cells ignore? I will discuss possible ``collective guidance'' mechanisms underlying this motion, where individual cells measure the mean value of the attractant, but need not measure its gradient to give rise to directional motility for a cell cluster. I show that the collective guidance hypothesis can be directly tested by looking for strong orientational effects in pairs of cells chemotaxing. Collective gradient sensing also has a new wrinkle in comparison to single-cell chemotaxis: to accurately determine a gradient direction, a cluster must integrate information from cells with highly variable properties. When is cell-to-cell variation a limiting factor in sensing accuracy? I provide some initial answers, and discuss how cell clusters can sense gradients in a way that is robust to this variation. Interestingly, these strategies may depend on the cluster's mechanics: there is a fundamental bound that links the cluster's chemotactic accuracy and its rheology. This suggests that in some circumstances,
mechanical changes like fluidization can influence a cluster's sensing ability. Because of this effect, increasing the noise in a single cell's motion can actually increase the accuracy of cluster chemotaxis!

In this talk, we will pose some questions of projectivity and tensor products of modules for finite dimensional Hopf algebras. To give some answers to these questions, we will construct some examples coming from smash copropucts of Sweedler Hopf algebras. One of the fundamental tools that we use to understand the modules of these Hopf algebras is the theory of support varieties. If time allows, we will mention the definition and some of the main properties of the support varieties for these examples.

Maths anxiety is a well-defined cognitive, physiological, and psychological construct that negatively affects the maths achievement of students who suffer from it. Maths anxiety does not end when a student leaves school, and can negatively impact their adult life.

This talk will present an informal introduction into the study of maths anxiety and other related constructs that affect the learning of maths. Then a brief discussion of the use of history of mathematics in the classroom. The aim of this talk is to examine a case study investigating whether the history of mathematics can alleviate maths anxiety by developing a dialogical classroom in which success is measured not by solving equations quickly, but by engaging in discussion mathematically.

In this talk, I will consider the incompressible Navier-Stokesequations in the mild solution setting. Using this setting, I will show how to obtain analyticity of mild solutions using the Gevrey regularity technique. This regularity enables to get decay rates of weak solutions of the Navier-Stokes equations. This idea can be applied to other dissipative equations with analytic nonlinearities. I will finally consider the regularity of the flow map of mild solutions using the Log-Lipschitz regularity of the velocity field.

A mathematical analysis of Matt McDuff’s 2016 attempt to ride a world record
40 foot (12.3 meter) high vertical circular loop-de-loop he named ``The Loop of Doom''.

In this talk, I will discuss several results on determining the tower growth rate of Ramsey numbers arising in combinatorics and in geometry.

These results are joint work with David Conlon, Jacob Fox, Dhruv Mubayi, Janos Pach, and Benny Sudakov.

\indent Data science is at a crossroads. Each year, thousands of new data scientists are entering science and technology, after a broad training in a variety of fields. Modern data science is often exploratory in nature, with datasets being collected and dissected in an interactive manner. Classical guarantees that accompany many statistical methods are often invalidated by their non-standard interactive use, resulting in an underestimated risk of falsely discovering correlations or patterns. It is a pressing challenge to upgrade existing tools, or create new ones, that are robust to involving a human-in-the-loop.

In this talk, I will describe two new advances that enable some amount of interactivity while testing multiple hypotheses, and control the resulting selection bias. I will first introduce a new framework, STAR, that uses partial masking to divide the available information into two parts, one for selecting a set of potential discoveries, and the other for inference on the selected set. I will then show that it is possible to flip the traditional roles of the algorithm and the scientist, allowing the scientist to make post-hoc decisions after seeing the realization of an algorithm on the data. The theoretical basis for both advances is founded in the theory of martingales : in the first, the user defines the martingale and associated filtration interactively, and in the second, we move from optional stopping to optional spotting by proving uniform concentration bounds on relevant martingales.

Bio: Aaditya Ramdas is a postdoctoral researcher in Statistics and EECS at UC Berkeley, advised by Michael Jordan and Martin Wainwright. He finished his PhD in Statistics and Machine Learning at CMU, advised by Larry Wasserman and Aarti Singh, winning the Best Thesis Award in Statistics. A lot of his research focuses on modern aspects of reproducibility in science and technology — involving statistical testing and false discovery rate control in static and dynamic settings.

This talk will feature joint work with (alphabetically) Rina Barber, Jianbo Chen, Will Fithian, Kevin Jamieson, Michael Jordan, Eugene Katsevich, Lihua Lei, Max Rabinovich, Martin Wainwright, Fanny Yang and Tijana Zrnic.

et $E$ be an elliptic curve defined over a number field $k$ and $F$ a finite cyclic extension of $k$ of $p$-power
degree for an odd prime $p$.
Under certain technical hypotheses, we describe a reinterpretation of the Equivariant Tamagawa Number Conjecture
(`ETNC') for $E$, $F/k$ and $p$ as an explicit family of
$p$-adic congruences involving values of derivatives of the Hasse-Weil $L$-functions of twists of $E$,
normalised by completely explicit twisted regulators. This
reinterpretation makes the ETNC amenable to numerical verification and furthermore leads to explicit predictions
which refine well-known conjectures of Mazur and Tate.

This is a report on joint work with Daniel Macias Castillo

We consider the problem of decomposing an even order symmetric tensor with positive eigenvalues, into a sum of rank-1 components (also knows as canonical polyadic or CP decomposition). These tensors naturally arise in many signal processing applications (such as ICA, blind source separation, source localization) when we compute higher order cumulants of the measurements. We show that when these components possess certain harmonic structures, it is possible to design clever sampling techniques to identify $O(N^{2q})$ rank-1 factors using a tensor of order $2q$ (where $q$ is an integer) and dimension $N$. This is made possible by exploiting the idea of a higher order difference set that can be associated with the cumulant tensors. For unstructured even order tensors, we show that under some mild conditions, the problem of CP decomposition is equivalent to solving a system of quadratic equations as long as the rank of the tensor is $O(N^q)$. We finally propose two algorithms, one based on convex relaxation, and the other utilizes non-convex, Jacobi iteration to solve the resulting quadratic system.

Modular forms, being some of the most
fundamental objects in number theory, have a habit of appearing in
many different contexts; such coincidences turn out to be extremely
useful for computational purposes. I'll describe three different
constructions that give the action of the Hecke operators on certain
spaces of modular forms: the classical method of modular symbols
(Manin), the``method of graphs'' based on isogenies among supersingular
elliptic curves (Mestre-Oestrele), and a less well-known method based
of reduction of quadratic forms (Birch).

In this project, we assess the impact of granting farmers title to their land on agricultural productivity in Benin. We test the hypothesis that formalizing land tenure and land use rights raises the incentives to landholders for making sustainable investments in their productive operations. We use remote sensing data to measure annual patterns of vegetation over time and apply a difference-in-differences and staggered-entry estimation strategy to identify program treatment effects. To assess patterns of annual vegetation cover across Benin, we use satellite observations of land cover from Landsat 7 to generate spectral indices that are sensitive to live vegetation and the presence of water (e.g., NDVI, SAVI, EVI, NDWI, and LSWI). To detect seasonal changes in vegetation, we analyze indices at high temporal resolution and fit sinusoids to the data. We perform the analysis at three geographical scales: the level of the village, the level of the plot, and the level of the
pixel. For the pixel-level analysis, we sample a large number of pixels within villages, and calculate and analyze temporal changes in the per-pixel spectral indices over the period 2005 to 2015. For the analysis at the village level, we aggregate the pixels spatially to the level of the village, and analyze a full time series of both means and variances at the village scale. We compare results across spatial scales (pixel, plot, village) to understand the relative importance of spatial and temporal degrees of freedom in detecting land investments.

Markov duality in spin chains and exclusion processes has found a wide variety of applications throughout probability theory. We review the duality of the asymmetric simple exclusion process (ASEP) and its underlying algebraic symmetry. We then explain how the algebraic structure leads to a wide generalization of models with duality, such as higher spin exclusion processes, zero range processes, stochastic vertex models, and their multi-species analogues.

This is a continuation of my RTG colloquium lecture on November 8. In this lecture, we study the method of Birch in more detail, to see how it can be used to compute essentially arbitrary spaces of classical modular forms. This involves relating Birch's construction to orthogonal modular forms and Clifford algebras, and applying a form of the Jacquet-Langlands correspondence. We also report on some limited computational evidence that this method can also be applied to GSp(4) Siegel modular forms. A short computer demonstration using Sage may be included if time permits. Note: this is a report on the PhD thesis of Jeffery Hein, written under John Voight at Dartmouth in consultation with Gonzalo Tornaria.

Some of the most fascinating pieces of mathematics,
such as Dirichlet's class number formula and the
celebrated Birch and Swinnerton-Dyer conjecture,
build a bridge between the distant worlds of arithmetic and analysis.
Euler systems and Iwasawa theory provide an intermediate step between the two, and both have been at the source of much of the progress to date on
BSD conjecture and its many generalizations. In this talk,
I will expand on some of these ideas, including a brief discussion of some of the recent developments in the area.

I will describe a class of modular compactifications of moduli spaces of elliptic surfaces. Time permitting, I will also discuss recent work towards connecting these compactifications with various existing compactifications of the moduli space of rational elliptic surfaces. This is joint work with Dori Bejleri.

We consider the problem of finding an elevated submatrix inside a large, noisy matrix. We are interested in both the detection problem (detecting the existence of the elevated submatrix) and the localization problem (find out the row and column index sets of the submatrix). Treating the elevated mean as the signal strength, we illustrate the fundamental signal boundaries of detection and localization, and propose estimators that reach the boundaries. The relationship of detection and localization problems will also be addressed.

I will describe some interesting selection problems for fully nonlinear, degenerate elliptic PDEs. In particular, I will focus on the vanishing discount procedure and show the convergence result via a new variational technique.

Abelian covers of finite volume hyperbolic manifolds are ubiquitous. We will discuss ergodic properties of the geodesic flow/ frame flow on such spaces. In particular, we will discuss the local mixing property of the geodesic flow/ frame flow, which we introduce to substitute the well-known strong mixing property in infinite volume setting. We will also discuss applications to measure classification problems and to counting and equidistribution problems. Part of the talk is based on the joint work with Hee Oh.

How complicated is a rational number? Its size is not a very good indicator for this. For instance, 1987985792837/1987985792836 is approximately 1, but so much more complicated than 1. We'll explain how to measure the complexity of a rational number using various notions of height. We'll then see how heights are used to prove some basic finiteness theorems in number theory.

We construct ``large'' Cantor sets whose complements resemble the unit disk arbitrarily well from the point of view of the squeezing function, and we construct ``large'' Cantor sets whose complements do not resemble the unit disk from the point of view of the squeezing function.

In this talk, we introduce the LRHB algorithm, which is an extension of the reduced-Hessian method of Gill and Leonard for unconstrained problems to problems with simple bound constraints. Numerical results for LRHB will be presented. We will also consider computational and practical issues with methods for nonlinear optimization and present results on a large test collection of problems indicating the reliability and efficiency of sequential quadratic programming methods and interior-point methods on certain classes of problems. This is joint work with Michael Ferry and Philip E. Gill.

Continuous-time Markov chain models are often used to describe the stochastic dynamics of networks of reacting chemical species, especially in the growing field of systems biology. Discrete-event stochastic simulation of these models rapidly becomes computationally intensive. Consequently, more tractable diffusion approximations are commonly used in numerical computation, even for modest-sized networks. However, existing approximations (e.g., linear noise and Langevin), do not respect the constraint that chemical concentrations are never negative.

In this talk, we propose an approximation for such Markov chains, via reflected diffusion processes, that respects the fact that concentrations of chemical species are non-negative. This fixes a difficulty with Langevin approximations that they are frequently only valid until the boundary of the positive orthant is reached. Our approximation has the added advantage that it can be written down immediately from the chemical reactions. This contrasts with the linear noise approximation, which involves a two-stage procedure --- first solving a deterministic ordinary differential equation, followed by a stochastic differential equation for fluctuations around those solutions.

Under mild assumptions, we first prove that our proposed approximation is well defined for all time. Then we prove that it can be obtained as the weak limit of a sequence of jump-diffusion processes that behave like the Langevin approximation in the interior of the positive orthant and like a rescaled version of the Markov chain on the boundary of the orthant. For this limit theorem, we adapt an invariance principle for reflected diffusions, due to Kang and Williams, and modify a result on pathwise uniqueness for reflected diffusions, due to Dupuis and Ishii. Some numerical examples illustrate the advantages of our approximation over direct simulation of the Markov chain or use of the linear noise approximation.

Joint work with Saul Leite (Federal University of Juiz de Fora, Brazil), David Anderson (U. Wisconsin-Madison) and Des Higham (U. Strathclyde).

In the pre-talk for graduate students, we will define discrete series representations and
give examples for $SL(2,\mathbb{R})$ and $GL(2,F)$, where $F$ is a local non-archimedean
field of characteristic $0$ with residue field of order not divisible by $2$. In the main talk,
first, we will review how the multiplicities of discrete series representations of
$SL(2,\mathbb{R})$ in $L^2(\Gamma \backslash SL(2,\mathbb{R}))$ are given by
dimensions of spaces of holomorphic cusp forms for $\Gamma$; we will take a look at
what happens if we try to use the formation of Poincar\'e series as an intertwiner; and we
will summarize some of the methods available for guaranteeing occurrence of discrete
series representations in $L^2(\Gamma \backslash G)$ when $G$ is a semisimple Lie
group other than $SL(2,\mathbb{R})$. Second, we will compute the product of the formal
dimension of two particular discrete series representations of $PGL(2,F)$ and the covolume
of a torsion-free lattice $\Gamma$ in $PGL(2,F)$ by dealing carefully with Haar measure
and applying standard facts from $\mathfrak{p}$-adic representation theory, thereby giving
the first explicit computation of multiplicities of those two discrete series representations in
$L^2(\Gamma \backslash PGL(2,F) )$; and we will say how the local Jacquet-Langlands
correspondence and the work of Corwin, Moy, and Sally could be used to carry out similar
calculations. (This material is part of our dissertation on representations of von Neumann
algebras coming from lattices in $SL(2,\mathbb{R})$ and $PGL(2,F)$.)

\def\R{ {\mathbb{R}} }
\def\bbS{ {\mathbb S}}

Let $M_n (\bbS)^g$ denote $g$-tuples of $n \times n$ complex self-adjoint matrices. Given tuples $X=(X_1, \dots, X_g) \in M_{n_1} (\bbS^g)$ and $Y=(Y_1, \dots, Y_g) \in M_{n_2}(\bbS)^g$, a matrix convex combination of $X$ and $Y$ is a sum of the form
\[
V_1^* XV_1+V_2^* Y V_2 \quad \quad \quad V_1^* V_1+V_2^* V_2=I_n
\]
where $V_1:M_{n} (\R) \to M_{n_1}$ and $V_2:M_n (\R) \to M_{n_2}$ are contractions. Matrix convex sets are sets which are closed under matrix convex combinations.

While in the classical setting there is only one good notion of an extreme point, there are three natural notions of extreme points for matrix convex sets: Euclidean, matrix, and absolute extreme points. A central goal in the theory of matrix convex sets is to determine if one of these notions of extreme points for matrix convex sets is minimal with respect to spanning.

Matrix extreme points are the most restricted type of extreme point known to span matrix convex sets; however, they are not necessarily the smallest set which does so. Absolute extreme points, a more restricted type of extreme points that are closely related to Arveson's boundary, enjoy a strong notion of minimality should they span. However, until recently it has been unknown if general matrix convex sets are spanned by their absolute extreme points.

This talk will give a class of closed bounded matrix convex sets which do not have absolute extreme points. The sets considered are noncommutative sets, $K_X$, formed by taking matrix convex combinations of a single tuple $X$. In the case that $X$ is a tuple of compact operators with no nontrivial finite dimensional reducing subspaces, $K_X$ is a closed bounded matrix convex set with no absolute extreme points.

Markoff triples are integer solutions of the equation $x^2+y^2+z^2=3xyz$ which arose in Markoff's spectacular and fundamental work (1879) on diophantine approximation and has been henceforth ubiquitous in a tremendous variety of different fields in mathematics and beyond. After reviewing some of these, we will discuss joint work with Bourgain and Sarnak on the connectedness of the set of solutions of the Markoff equation modulo primes under the action of the group generated by Vieta involutions, showing, in particular, that for almost all primes the induced graph is connected. Similar results for composite moduli enable us to establish certain new arithmetical properties of Markoff numbers, for instance the fact that almost all of them are composite. \\
Time permitting, we will also discuss recent joint work with Magee and Ronan on the asymptotic formula for integer points on Markoff-Hurwitz surfaces $x_1^2+x_2^2 + \dots + x_n^2 = x_1 x_2 \dots x_n$, giving an interpretation for the exponent of growth in terms of certain conformal measure on the projective space.

The Ran space Ran(X) is the space of finite
subsets of X, topologized so that points can collide. Ran spaces have
been studied in diverse works from Borsuk-Ulam and Bott, to
Beilinson-Drinfeld, Gaitsgory-Lurie and others. The alpha form of
factorization homology takes as input a manifold or variety X together
with a suitable algebraic coefficient system A, and it outputs the
sheaf homology of Ran(X) with coefficients defined by A.
Factorization homology simultaneously generalizes singular homology,
Hochschild homology, and conformal blocks or observables in conformal
field theory. I'll discuss applications of this alpha form of
factorization homology in the study of mapping spaces in algebraic
topology, bundles on algebraic curves, and perturbative quantum field
theory. I'll also describe a beta form of factorization homology,
where one replaces Ran(X) with a moduli space of stratifications of X,
designed to overcome certain strict limitations of the alpha form. One
such application is to proving the Cobordism Hypothesis, after
Baez-Dolan, Costello, Hopkins-Lurie, and Lurie. This is joint work
with David Ayala.

A central theme of large-scale convex optimization is the search for ``optimal methods.’’ These are the algorithms whose convergence guarantees match complexity theoretic lower bounds for a given problem class. Standard optimal methods are notoriously unintuitive. I will begin by describing a new transparent optimal method for minimizing smooth convex functions that is rooted in elementary cutting plane ideas.

Despite the successes of convex techniques, recent years have seen a resurgence of interest in nonconvex and nonsmooth optimization. In such settings, it is essential to exploit problem structure to make progress. One typical example of favorable structure occurs when minimizing a composition of a finite convex function with a smooth map. In the second part of the talk, I will discuss various aspects of this problem class, focusing on both worst-case and average case guarantees. The phase retrieval problem will illustrate the algorithms and theory.

This is joint work work with D. Davis (Cornell), M. Fazel (Washington), A.S. Lewis (Cornell), C. Paquette (Lehigh), and S. Roy (Washington).

We present several results on the size of the nodal (zero) set for solutions of partial differential equations of elliptic and parabolic type. In particular, we show a sharp upper bound for the $(n-1)$-dimensional Hausdorff measure of the nodal sets of the eigenfunctions of regular analytic elliptic problems in ${\mathbb R}^n$. We also show certain more recent results concerning semilinear equations (e.g. Navier-Stokes equations) and equations with non-analytic coefficients. The results on the size of nodal sets are connected to quantitative unique continuation, i.e., on the estimate of the order of vanishing of solutions of PDEs at a point. The results on unique continuation are joint with Ignatova and Camliyurt.

We study the existence of perfect matchings and Hamiltonian cycles in the preferential attachment model. In this model, vertices are added to the graph one by one, and each time a new vertex is created it establishes a connection with $m$ random vertices selected with probabilities proportional to their current degrees. (Constant $m$ is the only parameter of the model.) We prove that if $m \ge 1{,}260$, then asymptotically almost surely there exists a perfect matching. Moreover, we show that there exists a Hamiltonian cycle asymptotically almost surely, provided that $m \ge 29{,}500$. One difficulty in the analysis comes from the fact that vertices establish connections only with vertices that are ``older'' (i.e.~are created earlier in the process). However, the main obstacle arises from the fact that edges in the preferential attachment model are not generated independently. In view of that, we also consider a simpler setting---sometimes called uniform attachment---in which vertices are added one by one and each vertex connects to $m$ older vertices selected uniformly at random and independently of all other choices. We first investigate the existence of perfect matchings and Hamiltonian cycles in the uniform attachment model, and then extend the argument to the preferential attachment version.

The mirror symmetry conjecture (inspired by physics) has spurred a
lot of development in symplectic geometry. In the last few years, a
wave of modern homotopy theory has also entered the symplectic
landscape, and begun to present new questions about the structure of
symplectic manifolds. In this talk, we’ll explain a basic invariant
in symplectic geometry (the Fukaya category) and, as time allows,
give a survey of new inroads being opened through Lagrangian
cobordisms, derived geometry, and deformation theory.

Symplectic reductions from compact Hamiltonian Lie group actions on symplectic manifolds are important examples in the study of symplectic topology and mirror symmetry. In late 90s, Givental introduced an equivariant Gromov-Witten theory and used it to prove the mirror conjecture under the semi-positive assumption. During the past ten years, several groups of people have been working hard in generalizing the theory using symplectic vortex equations, but unfortunately, the corresponding moduli spaces suffer serious defect in compactness for higher genus case. In my talk, I will explain my ongoing project with Bohui Chen and Bai-Ling Wang in defining a new Gromov-Witten type of invariants for the equivariant cohomology of the ambient space. Using it, we also construct a quantum Kirwan morphism for a symplectic reduction.

Have you ever looked at two matrices and thought to yourself ``Man, I wonder if there's a polynomial of the first matrix equal to a polynomial of the second matrix?'' If yes, then boy is this the perfect talk for your highly specific interests. For everyone else, I hope to convince you that asking such a question can be a surprisingly interesting and fun process.
Specifically, we're going to look at this question when our two matrices come from some graph G. When our matrices satisfy a certain relation, we'll be able to use this relation to translate from eigenvalues of one matrix to eigenvalues of the other, and using spectral graph theory we'll be able to conclude various properties about our original graph from this.

We study the complexity of the classification problem for Cartan subalgebras in von Neumann algebras. We will discuss a construction that leads to a family of II$_1$ factors whose Cartan subalgebras, up to unitary conjugacy, are not classifiable by countable structures. We do this via establishing a strong dichotomy, depending if the action is strongly ergodic or not, on the complexity of the space of homomorphisms from a given equivalence relation to $E_0$. We will start with some of the necessary preliminaries, and then outline the proofs of the aforementioned results.

A classical result of Siegel asserts that the (2,3,7)-triangle group attains the smallest covolume among lattices of $\rm{SL}_2(\mathbb{R})$. In general, given a semisimple Lie group $G$ over some local field $F$, one may ask which lattices in $G$ attain the smallest covolume. A complete answer to this question seems out of reach at the moment; nevertheless, many steps have been made in the last decades. Inspired by Siegel's result, Lubotzky determined that a lattice of minimal covolume in $\mathrm{SL}_2(F)$ with $F=\mathbb{F}_q(\!(t)\!)\) is given by the so-called the characteristic $p$ modular group $\mathrm{SL}_2(\mathbb{F}_q[1/t])$. He noted that, in contrast with Siegel's lattice, the quotient by $\mathrm{SL}_2(\mathbb{F}_q[1/t])$ was not compact, and asked what the typical situation should be: for a semisimple Lie group over a local field, is a lattice of minimal covolume a cocompact or nonuniform lattice?

In the talk, we will review some of the known results, and then discuss the case of $\mathrm{SL}_n(\mathbb{R})$ for $n > 2$. It turns out that, up to automorphism, the unique lattice of minimal covolume in $\mathrm{SL}_n(\mathbb{R})$ is $\mathrm{SL}_n(\mathbb{Z})$. In particular, it is not uniform, giving an answer to Lubotzky's question in this case.

The most fundamental objects in number theory are number fields, field extensions of the rational numbers that are finite dimensional as vector spaces over Q. Their arithmetic is governed heavily by certain invariants such as the discriminant, Artin conductors, and the class group; for example, the ring of integers inside a number field has unique prime factorization if and only if its class group is trivial. The behavior of these invariants is truly mysterious: it is not known how many number fields there are having a given discriminant or conductor, and it is an open conjecture dating back to Gauss as to how many quadratic fields have trivial class group.

Nonetheless, one may hope for statistical information regarding these invariants of number fields, the most basic such question being “How are such invariants distributed amongst number fields of degree d?” To obtain more refined asymptotics, one may fix the Galois structure of the number fields in question. There are many foundational conjectures that predict the statistical behavior of these invariants in such families; however, only a handful of unconditional results are known. In this talk, I will describe a combination of algebraic, analytic, and geometric methods to prove many new instances of these conjectures, including some joint results with Altug, Bhargava, Ho, Shankar, and Wilson.

A major open problem in the periodic homogenization theory of Hamilton-Jacobi equations is to understand ``deep'' properties of the effective equation, in particular, how the effective Hamiltonian depends on the original Hamiltonian. In this talk, I will present some recent progress in both the convex and non-convex settings.

The enumerative geometry of curves on K3 surfaces is governed by modular forms. I will discuss a parallel connection between the enumerative geometry of hyper-Kaehler varieties and Jacobi forms. The case of genus 1 curves is particularly interested and leads to the Igusa cusp form conjecture. In the last part I will explain recent work with Junliang Shen and Aaron Pixton which yields a proof of this conjecture.

Physics predicts that the Gromov-Witten theory of Calabi-Yau threefolds satisfies two fundamental properties: Finite generation and a holomorphic anomaly equation. I will explain a recent conjecture with Pixton that extends these conjectures to all elliptic fibrations, and indicate how to prove it in several basic cases. If time permits, we will also discuss holomorphic anomaly equations for hyper-Kaehler varieties.

The real rank two locus of an algebraic variety is the closure of the union of all secant lines spanned by real points. We seek a semi-algebraic description of this set. Its algebraic boundary consists of the tangential variety and the edge variety. Our study of Segre and Veronese varieties yields a characterization of tensors of real rank two.

In an attempt to generalize beyond solvable methods of analysis for last passage percolation, recently Eric Cator (Radboud University, Nijmegen) and I have started analyzing the piecewise linearity of the last passage model. The tools we use to this point are purely geometric, but have the potential advantage that they can be used for very general choices of random inputs. I will describe the very pretty geometry of the last passage model, our work in progress to use it to produce probabilistic information, and some connections to algebraic geometry.

I will discuss some recent ongoing work with Paul Skoufranis to create a non-microstates bi-free entropy. I will propose a definition of bi-free conjugate variables and bi-free Fisher information, which admit desirable properties such as additivity in the presence of bi-free independence and versions of Cramer-Rao and Stam inequalities. I will also discuss the analogue of the free difference quotient, and some of the quirks present in the bi-free setting.

The cell membrane is the first interface that separates the inside of a cell from its surrounding medium. It serves not only as a protective mechanical barrier, but also as a platform for cells to exchange material with their environment. In this talk we will illustrate both of these essential membrane functions through two examples where the biophysics of lipid bilayers determine the response of the cell membrane.
First, we will examine the out-of-equilibrium response of cell-sized lipid vesicles exposed to solute imbalance. Based on experimental observations that giant vesicles in hypotonic condition exhibit a non-intuitive pulsatile behavior characterized by swell-burst cycles, we will present a theoretical description of the system in the form of coupled stochastic differential equations. We will show how thermal fluctuations enable stochastic pore nucleation, leading to a dependence of the lytic strain on the load rates, and unravel scaling relationships between the pulsatile dynamics and the vesicles properties. We will then demonstrate how vesicles encapsulating polymer solutions - mimicking the crowded cytoplasm of a cell - undergo swell-burst cycles even in the absence of a concentration imbalance.

Then, we will investigate how membrane necks, a necessary step to produce trafficking membrane vesicles, are generated by curvature-inducing proteins. Based on an augmented Helfrich model for lipid bilayers to include membrane-protein interaction, we will show how the spontaneous curvature field induced by proteins can be computed based on the knowledge of the neck geometry. We will apply this methodology to catenoid-shaped necks, for which the shape equation reduces to a variable coefficient Helmholtz equation for spontaneous curvature, where the source term is proportional to the Gaussian curvature. We will finally present numerical results showing how boundary conditions and geometric asymmetries determine an energetic landscape constraining the geometry of catenoid-shaped membrane necks.

Resolvent degree is an invariant of a branched cover which quantifies how ``hard'' is it to specify a point in the cover given a point under it in the base. Historically, this was applied to the branched cover $\mathbb{P}^n/S_{n-1}\to \mathbb{P}^n/S_n$, from the moduli of degree n polynomials with a specified root to the moduli of degree n polynomials. Classical enumerative problems and congruence subgroups provide two additional sources of branched covers to which this invariant applies. In ongoing joint work with Benson Farb, we develop the theory of resolvent degree as an extension of Buhler and Reichstein's theory of essential dimension. We apply this theory to systematize an array of classical results and to relate the complexity of seemingly different problems such as finding roots of polynomials, lines on cubic surfaces, and level structures on intermediate Jacobians.

Donaldson and Chen introduced the J-functional in '99, and explained its importance in the existence problem for constant scalar curvature metrics on compact Kahler manifolds. An important open problem
is to find algebro-geometric conditions under which the J-functional has a critical point. The critical points of the J-functional are described by a fully-nonlinear PDE called the J-equation. I will discuss some recent progress on this problem, and indicate the role of algebraic geometry in proving estimates for the J-equation.

Representation stability is an exciting new area that combines
ideas from commutative algebra and representation theory. The meta-idea
is to combine a sequence of objects together using some newly defined
algebraic structure, and then to translate abstract properties about
this structure to concrete properties about the original object of
study. Finite generation is a particularly important property, which
translates to the existence of bounds on algebraic invariants, or some
predictable behavior. I'll discuss some examples coming from topology
(configuration spaces) and algebraic geometry (secant varieties).

We will discuss how one can tile the hyperbolic plane with various polygons. We will focus on the tiling with the smallest possible tile, the (2,3,7)-triangle. The reflexion group associated to it turns out to have fundamental importance in the theory of Hurwitz surfaces.

Consider the tilings of an oriented surface by triangles, or squares, or hexagons, up to combinatorial equivalence. The combinatorial curvature of a vertex is 6, 4, or 3 minus the number of adjacent polygons, respectively. Tilings are naturally stratified into all such having the same set of non-zero curvatures. We outline a proof that for squares and hexagons, the generating function for the number of tilings in a fixed stratum lies in a ring of quasi-modular forms of specified level and weight. First, we rephrase the problem in terms of Hurwitz theory of an elliptic orbifold---a quotient of the plane by an orientation-preserving wallpaper group. In turn, we produce a formula for the number of tilings in terms of characters of the symmetric group. Generalizing techniques pioneered by Eskin and Okounkov, who studied the pillowcase orbifold, we express the generating function for a stratum in terms of the q-trace of an operator acting on Fock space. The key step is to compute the trace in a different basis to express it as an infinite product, and apply the Jacobi triple product formula to conclude quasi-modularity.

The Torelli subgroups of mapping class groups are fundamental objects in low-dimensional topology, through some basic questions about their structure remain open. In this talk I will describe these groups, and how to use tools from representation theory to establish patterns their homology. This project is joint with Jeremy Miller and Peter Patzt. These ``representation stability'' results are an application of advances in a general algebraic framework for studying sequences of group representations.

The moduli space of Riemann surfaces of fixed genus is one of the hubs of modern mathematics and physics. We will tell the story of how simple sounding problems about polygons, some of which arose as toy models in physics, became intertwined with problems about the geometry of moduli space, and how the study of these problems in Teichmuller dynamics lead to connections with homogeneous spaces, algebraic geometry, dynamics, and other areas. The talk will mention joint works with Alex Eskin, Simion Filip, Curtis McMullen, Maryam Mirzakhani, and Ronen Mukamel.

The AdS instability conjecture is a conjecture about the initial value problem for Einstein vacuum equations with a negative cosmological constant. Proposed by Dafermos and Holzegel in 2006, the conjecture states that generic, arbitrarily small perturbations to the initial data of the AdS spacetime, under evolution by the vacuum Einstein equations with reflecting boundary conditions on conformal infinity, lead to the formation of black holes.Following the work of Bizon and Rostworowski in 2011, a vast amount of numerical and heuristic works have been dedicated to the study of this conjecture, focusing mainly on the simpler setting of the spherically symmetric Einstein-scalar field system. In this talk, we will provide the first rigorous proof of the AdS instability conjecture in the simplest possible setting, namely for the Einstein-null dust system, allowing for both ingoing and outgoing dust. This system is a singular reduction of the spherically symmetric Einstein-massless Vlasov system, in the case when the Vlasov field is supported only on radial geodesics. In order to overcome the 'trivial' break down occurring once the null dust reaches the center $r=0$, we will study the evolution of the system in the exterior of an inner mirror with positive radius $r_0$ and prove the conjecture in this setting. After presenting our proof, we will briefly explain how the main ideas can be extended to more general matter fields, including the regular Einstein-massless Vlasov system.

The stochastic inverse problem for determining parameter values in a physics model from observational data on the output of the model forms the core of scientific inference and engineering design. We describe a recently developed formulation and solution method for stochastic inverse problems that is based on measure theory and a generalization of a contour map. In addition to a complete analytic and numerical theory, advantages of this approach include avoiding the introduction of ad hoc statistics models, unverifiable assumptions, and alterations of the model like regularization. We present a high-dimensional application to determination of parameter fields in storm surge models. We conclude with recent work on defining a notion of condition for stochastic inverse problems and the use in designing sets of optimal observable quantities.

When we choose a metric on a manifold we determine the spectrum of the Laplace operator. Thus an eigenvalue may be considered as a functional on the space of metrics. For example the first eigenvalue would be the fundamental vibrational frequency. In some cases the normalized eigenvalues are bounded independent of the metric. In such cases it makes sense to attempt to find critical points in the space of metrics. For surfaces, the critical metrics turn out to be the induced metrics on certain special classes of minimal (mean curvature zero) surfaces in spheres and Euclidean balls. The eigenvalue extremal problem is thus related to other questions arising in the theory of minimal surfaces. In this talk we will give an overview of progress that has been made for surfaces with boundary, and discuss some recent results in higher dimensions. This is joint work with R. Schoen.

The combinatorics of permutations in the symmetric group $S_n$ has deep connections to algebraic properties of the {\em coinvariant ring} (through work of Artin, Chevalley, Lusztig-Stanley, and others) and geometric properties of the {\em flag variety} whose points are complete flags in $\mathbb{C}^n$ (through work of Ehresmann, Borel, and others). We will discuss new generalizations of the coinvariant ring and flag variety indexed by two positive integers $k \leq n$. The algebraic and geometric properties of these objects are controlled by ordered set partitions of $[n]$ with $k$ blocks. There are connections between these objects and the Delta Conjecture in the theory of Macdonald polynomials. Joint with Jim Haglund, Brendan Pawlowski, and Mark Shimozono.

Many important maps in algebraic combinatorics (the RSK bijection, the Schutzenberger involution, etc.) can be described by piecewise-linear formulas. These formulas can then be ``de-tropicalized,'' or ``lifted,'' to subtraction-free rational functions on an algebraic variety, and certain properties of the combinatorial maps become more transparent in the algebro-geometric setting. I will illustrate how this works in the case of the promotion map on semistandard tableaux of rectangular shape. I will also indicate how promotion can be viewed as the combinatorial manifestation of a symmetry coming from representation theory, and how its geometric lift fits into Berenstein and Kazhdan's theory of geometric crystals.

Free boundary minimal surfaces in the ball are proper branched minimal immersions of a surface into the ball that meet the boundary of the ball orthogonally. Such surfaces have been extensively studied, and they arise as extremals of the area functional for relative cycles in the ball. They also arise as extremals of an eigenvalue problem on surfaces with boundary. In this talk we will discuss existence and uniqueness theorems for such surfaces, focusing on a uniqueness result for free boundary minimal annuli. This is joint work with M. Li and R. Schoen.

We construct a new quasi-local mass in space-time and show that this mass is non-decreasing along any null flow of doubly convex 2-spheres. As a result, we prove the Penrose conjecture for conical null slices, or null cones, under fairly generic conditions.

Grothendieck’s standard conjectures, Voevodsky’s nilpotence conjecture, and Tate’s conjecture, play a key central role in algebraic geometry. Notwithstanding the effort of several generations of mathematicians, the proofs of these celebrated
conjectures remain ellusive. The aim of this talk, prepared for a broad audience, is to give an overview of a recent topological/noncommutative approach which has led to the proof of the aforementioned important conjectures in several new cases.

The alternating direction method of multipliers (ADMM) has been successfully used in solving problems arising from different fields such as machine learning, image processing, statistics and so on. However, most existing works on analyzing the convergence and complexity of ADMM are for convex problems. In this talk, we discuss several recent results on convergence behavior of ADMM for solving nonconvex problems. We consider two nonconvex models. The first model allows the objective function to be nonconvex and nonsmooth, but the constraints are convex. The second model allows the constraints to be Riemannian manifolds. For both models, we propose ADMM variants for solving them and analyze their iteration complexities for obtaining an $\epsilon$-stationary solution. Numerical results on tensor robust PCA, maximum bisection problem and community detection problem are reported to demonstrate the efficiency of the proposed methods.

I will begin by introducing the Gaussian, Laguerre and Jacobi ensembles and their corresponding eigenvalue densities. The moments of these eigenvalue densities are generated by the corresponding resolvent, R(x). When investigating large matrices of size N, it is natural to expand R(x) as a series in 1/N, as N tends to infinity. The loop equation formalism enables one to compute R(x) to any desired order in 1/N via a triangular recursive system. This formalism is equivalent to the topological recursion, the Schwinger-Dyson equations and the Virasoro constraints, among other things. The loop equations provide a relatively accessible entry-point to these topics and my derivation will rely on nothing more than integration by parts, as Aomoto applied to the Selberg integral. Time permitting, I may also explore links to the topological recursion and/or some combinatorics.

All original results will be from joint work with Peter Forrester and Nicholas Witte.

In this talk, I will present several works related to tensor data analysis. Firstly, hypergraph matching (HGM) is a popular tool in establishing corresponding relationship between two sets of points, which becomes a central problem in computer vision. We reformulate HGM as a sparse constrained model, and show its relaxation problem can also recover the global optimizer. A quadratic penalty method is presented to solve the relaxation model. Secondly, the analytic connectivity (AC) is an important quantity in spectral hypergraph theory. The definition of AC involves a series of polynomial optimization problem (POP). The number of POPs can be reduced by the structure of hypergraphs. Further, we proposed a simplex constrained model, a equality constrained model and a sparse constrained model for computing AC under different situations. Thirdly, identifying new indications for known drugs, i.e., drug repositioning (DR), attracts a lot of attentions in bioinformatics. We develop a novel method for DR based on projection onto convex sets.

We will discuss two different approaches to computing the L-functions of
Shimura varieties associated with GU(2,1). Both approaches employ the
comparison of the Grothendieck-Lefschetz formula with the Arthur-Selberg
trace formula. The first approach, carried out by the author, takes as its
starting point the recent work of Laumon and Morel. The second approach is
due to Flicker. In both approaches, the principal challenge is that the Shimura
varieties in question are non-compact, and one must use cohomology with
compact supports. Time permitting, we will discuss the prospects for extending
these approaches to the non-compact Shimura varieties associated with
higher-rank unitary groups.

Many real-world examples of fluid-structure interaction, including the transit of red blood cells through the narrow slits in the spleen, involve the near-contact of elastic structures separated by thin layers of fluid. Motivated by such problems, we introduce an immersed boundary method that uses elements of lubrication theory to resolve thin fluid layers between immersed boundaries. We apply this method to two-dimensional flows of increasing complexity, including eccentric rotating cylinders and elastic vesicles near walls in shear flow, to show its increased accuracy compared to the classical immersed boundary method.

Associated to a finite volume hyperbolic 3-manifold is a
number field and a quaternion algebra over that number field. Closed
hyperbolic 3-manifolds arising from Dehn surgeries on a hyperbolic
knot complement provide a family of number fields and quaternion
algebras that can be viewed as ``varying'' over a
certain curve component (the so-called canonical component) of the
$SL(2,C)$-character variety of the knot group. This talk will give
examples of different behavior and survey recent work on how the
varying behavior can be explained using the language of Azumaya
algebras over the canonical curve.

We will discuss recent work with Z. Zhang on Torelli theorems for bidouble covers of a smooth quintic curve and 2 lines in the plane, and cubic 4-folds arising from a cubic 3-fold and a hyperplane intersecting transversely in $P^4$.
The talk for graduate students will be, ``Abelian Varieties and the Torelli Theorem''. I will explain what an Abelian variety is, and discuss the Torelli theorem for curves.

Our capacity to produce and store large sets of data has increased exponentially over the course of the last two decades; the development of algorithms for sifting through it efficiently is somewhat lagging behind. Randomization is being recognized as a powerful tool, whether in constructing models on which algorithms can be tested, or in sampling the data reliably, or in speeding up and optimizing existing algorithms. In particular, basic results from random matrix and random graph theory are being employed at the forefront of scientific computing; often, assembling algorithms and producing theoretical guarantees for them requires a blend of probability, combinatorics, graph theory, numerical analysis, and optimization.
I will speak about two results, one in which randomization is used to achieve a communication-minimizing non-symmetric eigenvalue solver, and one establishing a spectral gap in bipartite biregular graphs, with applications in areas as varied as community detection, matrix completion, and error-correcting codes. This is joint work with Jim Demmel and Grey Ballard, respectively, Gerandy Brito and Kameron Harris.

In this talk we discuss an algorithm to overcome the curse of dimensionality, in possibly non-convex/time/state-dependent Hamilton-Jacobi partial differential equations. They may arise from optimal control and differential game problems, and are generally difficult to solve numerically in high dimensions.

A major contribution of our works is to consider an optimization problem over a single vector of the same dimension as the dimension of the HJ PDE instead. To do so, we consider Hopf-type formulas. The sub-problems are now independent and they can be implemented in an embarrassingly parallel fashion. That is ideal for perfect scaling in parallel computing.

The algorithm is proposed to overcome the curse of dimensionality when solving high dimensional HJ PDE. Our method is expected to have application in control theory, differential game problems, and elsewhere. A similar technique can be extended to the computational of a Hamilton-Jacobi partial differential equations in the Wasserstein space, and this is also expected to have applications in mean field control problems and mean field games.