We present the Tetrahedral Compactness Theorem which states that sequences of Riemannian manifolds with a uniform upper bound on volume and diameter that satisfy a uniform tetrahedral property have a subsequence which converges in the Gromov-Hausdorff sense to a countably $\mathcal{H}^m$ rectifiable metric space of the same dimension. The tetrahedral property depends only on distances between points in spheres, yet we show it provides a lower bound on the volumes of balls. The proof is based upon intrinsic flat convergence and a new notion called the sliced filling volume of a ball.

Given a poset $P$, we are interested in determining the maximum size (denoted by $La(n,P)$) of any family of subsets of an $n$-set avoiding all extensions of $P$ as subposets. The starting point of this kind of problem is Sperner's Theorem from 1928, which can be restated as $La(n, P_2)= {n\choose \lfloor \frac{n}{2} \rfloor}$. Here $P_2$ is the chain of $2$ elements. These problems were studied by Erd\H{o}s, Katona, and others. In 2008, Griggs and Lu conjectured the limit $\pi(P):=\lim_{n\to\infty} \frac{La(n,P)} {{n\choose \lfloor \frac{n}{2} \rfloor}}$ exists and is an integer. For poset $P$ define $e(P)$ to be the maximum $k$ such that for all $n$, the union of the $k$ middle levels of subsets in the $n$-set contains no extension of $P$ as a subposet. Saks and Winkler observed $\pi(P)=e(P)$ in all known examples where $\pi(P)$ is determined. Bukh proved this conjecture holds for any tree poset $P$ (meaning its Hasse diagram is a tree). For $t\geq 2$, let crown $\O_{2t}$ be a poset of height $2$, whose Hasse diagram is cycle $C_{2t}$. De Bonis-Katona-Swanepoel proved $La(n,O_{4})= {n\choose \lfloor \frac{n}{2} \rfloor} + {n\choose \lceil \frac{n}{2} \rceil}$. Griggs and Lu proved the conjecture holds for crown $\O_{2t}$ with even $t\geq 3$. In this talk, we will prove that the conjecture holds for crown $\O_{2t}$ with odd $t\geq 7$.

Given an elliptic curve $E/\mathbb{Q}$ with good reduction at $p$, Iwasawa theory studies its arithmetic over all the fields of $p^n$-th roots of unity. For example, there are nontrivial relations among the participants in the Birch–Swinnerton-Dyer conjectures for $E$ over each layer in this tower of fields. If the reduction of $E$ mod $p$ is ordinary, then have had a satisfactory description of the scenario for quite some time. But if the reduction of $E$ mod $p$ is supersingular, the correct description has required new advances in $p$-adic Hodge theory. We will discuss the background and what is now known.

A Riemannian manifold is called geometrically formal if the wedge product of any two harmonic forms is again harmonic. We classify geometrically formal compact 4-manifolds with nonnegative sectional curvature. If the sectional curvature is strictly positive, the manifold must be homeomorphic to $S^4$ or to $CP^2$. In particular, the Hopf conjecture on $S^2 \times S^2$ holds in the class of geometrically formal manifolds. If the sectional curvature is strictly positive and we relax the condition of geometric formality to the requirement that the length of harmonic 2-forms is not too nonconstant, then the manifold must be homeomorphic to $S^4$ or to a connected sum of $CP^2$s.

Analog-to-digital (A/D) conversion is the process by which signals (e.g., bandlimited functions or finite dimensional vectors) are replaced by bit streams to allow digital storage, transmission, and processing. Typically, A/D conversion is thought of as being composed of sampling and quantization. Sampling consists of collecting inner products of the signal with appropriate (deterministic or random) vectors. Quantization consists of replacing these inner products with elements from a finite set. A good A/D scheme allows for accurate reconstruction of the original object from its quantized samples. In this talk we investigate the reconstruction error as a function of the bit-rate, of Sigma-Delta quantization, a class of quantization algorithms used in the oversampled regime. We propose an encoding of the Sigma-Delta bit-stream and prove that it yields near-optimal error rates when coupled with a suitable reconstruction algorithm. This is true both in the finite dimensional setting and for bandlimited functions. In particular, in the finite dimensional setting the near-optimality of Sigma-Delta encoding applies to measurement vectors from a large class that includes certain deterministic and sub-Gaussian random vectors. Time permitting, we discuss implications for quantization of compressed sensing measurements.

Proving that Galois representations in many situations arise from automorphic forms has been a major theme in number theory for at least two decades. However, most of the existing work concerns the situation when the mod p reduction of the Galois representation (i.e., the residual representation) is irreducible and when the number field is totally real. We will present new modularity results for n-dimensional residually reducible Galois representations over arbitrary number fields. This is joint work with T. Berger.

The Fontaine-Mazur conjecture predicts which p-adic Galois representations
arise geometrically. A few years back, Emerton and Kisin made astounding
progress in the two-dimensional case, by employing the p-adic Langlands
correspondence for GL(2) over $Q_p$, which is very well-understood by now.
A key point was the existence of ``locally algebraic" vectors.
For groups of higher rank, even a conjectural generalization remains
elusive. However, there is a conjecture of Breuil & Schneider,
which gives a weak (but precise) analogue for GL(n). Roughly it says
that a certain filtration exists if and only if a certain lattice exists.
In his thesis, Hu completely proved one direction, and produced the
expected filtration (by translating its existence into the so-called
Emerton condition). We will report on progress in the other direction,
and in many cases prove the existence of GL(n)-stable lattices in
locally algebraic representations constructed from p-adic Hodge theoretical
data. This argument is global in nature; the ultimate integral structure
comes from p-adic modular forms. We hope to also hint at a formalism,
in which an eigenvariety for U(n) parametrizes a correspondence between
semisimple Galois representations and Banach-Hecke modules with a
unitary GL(n)-action, and discuss local-global compatibility "at p"
in this context. In particular, we'll settle the Breuil-Schneider conjecture
for dR representations which ``come from an eigenvariety".

Compactified Jacobian of a singular rational curve is homeomorphic to the direct product of Jacobi factors of singularities of the curve. Therefore, to study the topology of Compactified Jacobians in this case, it suffices to study topology of Jacobi factors, which can be defined very explicitly as subvarieties in Grassmanians. J. Piontkowski showed that in certain cases Jacobi factors can be decomposed into affine cells enumerated by semimodules over the semigroup of the singularity. We proved that for quasihomogeneous plane curve singularities the cells can also be enumerated by Young diagrams inscribed in a right triangle, and dimensions of cells can be computed in a combinatorial way. The resulting combinatorial model is closely related to cell decompositions of Hilbert schemes of points on a complex plane, and to rational generalizations of Garsia-Haiman's q,t-Catalan numbers. In this talk, I will discuss our results on topology of Compactified Jacobians and, time permitting, mention connections to the theory of finite dimensional representations of rational Cherednik algebras and homological knot invariants. The talk is based on a joint work with Eugene Gorsky.

Starting from the notion of BN-pair (which is closely related to the concept of Tits building), we explain, following a classical insight, how one interprets buildings and their automorphism groups over the nonexisting ``field with one element". (As such, the symmetric groups become Chevalley groups over this ``field".) We then introduce scheme theories in ``characteristic one", and show how these are governed by a deep combinatorial (and motivic) nature.

As a vast generalization of quadratic reciprocity, class field theory
describes all abelian extensions of a number field.
Over Q, they are precisely those contained in cyclotomic fields.
However, there are a lot more non-abelian extensions, which arise naturally.
The Langlands program attempts to systematize them, by relating Galois
representations and automorphic forms; mathematical objects of rather
disparate nature. We will illustrate the basic plot for GL(2) through
the example of elliptic curves and modular forms - the context of Wiles' proof
of Fermat's Last Theorem. The main goal of the talk will be to motivate
a ``p-adic" Langlands correspondence, which is at the forefront of
contemporary number theory, but still only well-understood for GL(2) over $Q_p$.
We will discuss, in some depth, the case of semistable elliptic curves,
which provide the first non-trivial example. This leads naturally to
a result we proved recently, which shows the existence of (many) integral
structures in locally algebraic representations of ``Steinberg" type,
for any reductive group G (such as GL(n), symplectic, and orthogonal groups).
As a result, there are a host of ways to p-adically complete the Steinberg
representation (tensored with an algebraic representation). The ensuing
Banach spaces should play a role in a (yet elusive) higher-dimensional
p-adic Langlands correspondence. We hope to at least give some idea of the
proof, which goes via automorphic representations and the trace formula.

The notion of mesh patterns in permutations was introduced recently by Petter Branden and Anders Claesson to provide explicit expansions for certain permutation statistics as possibly infinite linear combinations of (classical) permutation patterns. In my talk, I will discuss eight mesh patterns of small lengths. In particular, I will link avoidance of one of the patterns to the harmonic numbers, while for three other patterns I will show their distributions on 132-avoiding permutations are given by the Catalan triangle. Also, I will show that two specific mesh patterns are Wilf equivalent meaning that for any length, the number of permutations avoiding one of the patterns equals that avoiding the other one. As a byproduct of these studies, one defines a new set of sequences counted by the Catalan numbers and provides a relation on the Catalan triangle that seems to be new. This is joint work with Jeffrey Liese.

The Chern-Simons-Schr\"odinger model in two spatial dimensions is a covariant NLS-type problem and is $L^2$ critical. We prove that

Whenever a mathematical structure is given in characteristic $p$, one can ask whether it is the reduction, in some sense, of an analogous structure in characteristic zero. If so, the structure in characteristic zero is called a ``lift'' of the structure in characteristic $p$. The most famous example is Hensel's Lemma about lifting solutions of polynomials in $\mathbb{Z}/p$ to solutions in the $p$-adic integers $\mathbb{Z}_p$. We will consider a more geometric problem: given a curve $X$ in characteristic $p$ with an action of a finite group $G$, is there a curve in characteristic zero with $G$-action that reduces to $X$? Oort conjectured that this could be done when $G$ is cyclic, and his conjecture was recently proven by the speaker, Stefan Wewers, and Florian Pop. It turns out that this question reduces to a more ``local'' question about automorphisms of power series rings in one variable. This local question will occupy most of the talk. Many examples will be given throughout.

Vesture refers to the recipe for finding nontrivial exact solutions to a nonlinear differential equation by "dressing" a known solution (which could be rather trivial) with one or more prescribed singularities. This is in particular possible if the nonlinear equation is completely integrable. We will explain what that means, and provide examples from the theory of harmonic maps, and General Relativity, to demonstrate how this procedure works. We will in particular show how to obtain the nontrivial and physically interesting hyperextreme Kerr metric, which has a naked ring singularity at its center, from the trivial solution of Einstein Vacuum Equations, namely the Minkowski metric. This is joint work with Shabnam Beheshti.

We study Galois covers of the projective line branched at three points with Galois group $G$. When such a cover is defined over a $p$-adic field, it is known to have potentially good reduction to characteristic $p$ if $p$ does not divide the order of $G$. We give a sufficient criterion for good reduction, even when $p$ does divide the order of $G$, so long as the $p$-Sylow subgroup of $G$ is cyclic and the absolute ramification index of a field of definition of the cover is small enough. This extends work of (and answers a question of) Raynaud. Our proof depends on working very explicitly with Kummer extensions of complete discrete valuation rings with imperfect residue fields.

I shall explain how a new approach via the Hodge-Laplace heat equation works in solving the Poincare-Lelong equation. This method essentially is reduced to a uniqueness theorem and some estimates concluding the preservation of the d-closedeness of the solution of the Hodge-Laplace heat equation, and circumvents the essential difficulties of the elliptic method previously adapted by many people without being able to prove the best possible result. This is a joint work with Luen-Fai Tam.

One of the simplest model problems in the kinetic theory of plasma--physics is the Vlasov-Poisson system with periodic boundary conditions. Such system describes the evolution of a plasma of charged particles (electrons and ions) under the effects of the transport and self-consistent electric field. In this talk, we present some Discontinuous Galerkin (DG) methods for the approximation of the Vlasov-Poisson system. The schemes are based on the coupling of DG approximation to the Vlasov equation (transport equation) and several finite element (conforming, non-conforming and mixed) approximations to the Poisson problem. We present the error analysis and discuss further properties of the proposed schemes. We also present numerical experiments in the 1D case that verify the theory and validate the performance of the methods in benchmark problems. If time allows, in the last part of the talk, we shall discuss the possibility of combining the proposed methods with some dimension reduction techniques, such as sparse grids. The talk is based on joint works with Saverio Castelanelli (UAB-CRM), J.A. Carrillo (Imperial College-ICREA), Soheil Hajian (Univ. Geneva) and Chi-Wang Shu (Brown University).

I will discuss the computational methods based on a continuum solvent model to calculate binding affinity for protein-ligand systems. I will also show how such models can be combined with statistical learning methods to predict binding specificity of protein recognition. An application of these methods to understand drug resistance will exemplify its usefulness in drug development.

Motivated by the quest to better understand the structure of the adèle class space of a global field in characteristic zero, Connes and Consani recently initiated a theory of hyperfield extensions of the so-called ``Krasner hyperfield'' which is one of the realizations of the field with one element. Remarkably, there appears a deep correspondence with certain group actions on certain combinatorial geometries. We will elaborate on brand new results in this theory, make wild speculations and pose several important open questions on the way.

Description trees were introduced by Cori, Jacquard and Schaeffer in 1997 to give a general framework for the recursive decompositions of several families of planar maps studied by Tutte in a series of papers in the 1960s. We are interested in two classes of planar maps which can be thought as connected planar graphs embedded in the plane or the sphere with a directed edge distinguished as the root. These classes are rooted non-separable (or, 2-connected) and bicubic planar maps, and the corresponding to them trees are called, respectively, $\beta(1,0)$-trees and $\beta(0,1)$-trees.
Using different ways to generate these trees we define two endofunctions on them that turned out to be involutions. These involutions are not only interesting in their own right, in particular, from counting fixed points point of view, but also they were used to obtain non-trivial equidistribution results on planar maps, certain pattern avoiding permutations, and objects counted by the Catalan numbers. The results to be presented in this talk are obtained in a series of papers in collaboration with several researchers.

Parallel replica dynamics was proposed by A.F. Voter as a tool for accelerating molecular dynamics simulations characterized by a sequence of infrequent, but rapid, transitions from one state to another. An example would be the migration of a defect through a crystal. Parallel replica dynamics accelerates this by simulating many replicas simultaneously, concatenating the simulation times of the realizations as though it were a single long trajectory. This motivates important questions: Is parallel replica dynamics algorithm doing what we hope? For what systems will it be useful? How do we implement it efficiently? In this talk, I will thoroughly describe the algorithm and report on progress towards rigorous justification. Open questions and related problems will also be discussed.

Given a permutation w, Stanley defined a symmetric function $F_w$ which encodes information about the reduced words of w, and showed that $F_w$ is a single Schur function exactly when w avoids the pattern 2143. We generalize this statement, showing that the Schur expansion of $F_w$ respects pattern containment in a certain sense, and that the number of Schur function terms is determined by pattern avoidance conditions on w. Along the way, we compute the cohomology of certain subvarieties of Grassmannians, resolving some cases of a conjecture of Liu. This is joint work with Sara Billey.

The sieve methods are classical methods in number theory. Inspired by the 'affine sieve method' developed by Sarnak, Bourgain, Gamburd and others, as well as by works of Rivin and Kowalsky, we develop in a systemtic way a 'sieve method' for group theory. This method is especially useful for groups with 'property tau'. Hence the recent results of Breuillard-Green-Tao, Pyber-Szabo, Varju and Salehi-Golsefidy are very useful and enables one to apply them for linear groups. We will present the method and some of its applications to linear groups and to the mapping class groups. (Joint work with Chen Meiri).

Minimizing the discrepancy of a set system is a fundamental problem in combinatorics. One of the cornerstones in this area is the celebrated six standard deviations result of Spencer (AMS 1985): In any system of $n$ sets in a universe of size $n$, there always exists a coloring which achieves discrepancy $6\sqrt{n}$. The original proof of Spencer was existential in nature, and did not give an efficient algorithm to find such a coloring. Recently, a breakthrough work of Bansal (FOCS 2010) gave an efficient algorithm which finds such a coloring. His algorithm was based on an SDP relaxation of the discrepancy problem and a clever rounding procedure. In this work we give a new randomized algorithm to find a coloring as in Spencer's result based on a restricted random walk we call Edge-Walk. Our algorithm and its analysis use only basic linear algebra and is ``truly'' constructive in that it does not appeal to the existential arguments, giving a new proof of Spencer's theorem and the partial coloring lemma. Joint work with Raghu Meka

Given a compact Riemannian manifold, it is well known that the Laplace-Beltrami operator has a sequence of eigenfunctions which form an orthonormal basis for the square integrable functions. A problem of recent interest has been to prove $L^{p}$ estimates on these eigenfunctions, which bound their $L^{p}$ norm by a power of the corresponding eigenvalue, illuminating their size and concentration properties. We will investigate the validity of these bounds for manifolds equipped with rough metrics, such as those of Lipschitz regularity, which is significant in extending the theory to domains in $R^{n}$ and manifolds with boundary. In particular, we discuss a recent result regarding $L^{n}$ estimates on the restriction of these eigenfunctions to submanifolds.

The problem of sequential change-point detection is concerned with the design and analysis of fastest ways to detect a change in the statistical profi le of a random time process, given a tolerable risk of a false detection. The subject finds applications, e.g., in quality and process control, anomaly and failure detection, surveillance and security, finance, intrusion detection, boundary tracking, etc. We provide an overview of the field's state-of-the-art. The overview spans over all major formulations of the underlying optimization problem, namely, Bayesian, generalized Bayesian, and minimax. We pay particular attention to the latest advances made in each. Also, we link the generalized Bayesian problem with multi-cyclic disorder detection in a stationary regime when the change occurs at a distant time horizon. We conclude with a case study to show the field's best detection procedures at work. This is joint work with Alexander G. Tartakovsky, Department of Mathematics and Center for Applied Mathematical Sciences, University of Southern California.

The notion of a finite element system is designed to provide an alternative to Ciarlet's definition of a finite element, adapted to the needs of exterior calculus. It allows for cellular decompositions of space (rather than just simplexes or products thereof) and general functions (rather than just polynomials) yet guarantees compatibility with the exterior derivative and existence of commuting interpolation operators. We review basic definitions and properties. As an application, we show how a form of upwinding, compatible with the exterior derivative, can be carried out within this framework. References: S. H. Christiansen, H. Z. Munthe-Kaas, B. Owren. Topics in structure-preserving discretization. Acta Numer. 20 (2011), 1-119. S. H. Christiansen. Upwinding in finite element systems of differential forms. Smale lecture, Proceedings of FoCM 2011, to appear.

Broadly speaking, p-adic Hodge theory is the study of representations of Galois groups of p-adic fields on vector spaces with p-adic coefficients. One can use the theory of $(\varphi,\Gamma)$-modules to convert such Galois representations into simpler linear algebra, and one can also classify such representations in terms of how arithmetically interesting they are. In my talk, I will discuss extensions of this theory to p-adic families of Galois representations. Such families arise naturally in the contexts of Galois deformation rings and p-adic modular forms.

Consider the optimization problem of minimizing a raitonal function.
We reformulate this problem as polynomial optimization by the
technique of homogenization. These two problems are shown to be
equivalent under some generic conditions. The exact Jacobian SDP
relaxation method is used to solve the reformulated problem. We also
show that the convergence assumption of nonsingularity in Jacobian
SDP relaxation can be weakened as the finiteness of singularities.
Some numerical examples are given to show the efficiency of this
method.

In 1970, Lawson conjectured that the only embedded minimal torus in $S^3$ is the Clifford torus. Recently using a non-collapsing technique developed by Ben Andrews, Simon Brendle was able to give an affirmative answer to this question. I'll discuss the non-collapsing ideas and how they lead to Brendle's proof of the Lawson conjecture.

Incomplete knowledge of the true dynamics and its partial observations pose a notoriously difficult problem in many contemporary scientific applications which require predictions of high-dimensional dynamical systems with physical instabilities and energy fluxes across a wide range of scales. The issue of `model error' is particularly important when dealing with turbulent geophysical systems or molecular dynamics with rough energy spectra near the resolution cut-off of the numerical models. In such cases assimilation of observed data into the modeled dynamics is necessary for mitigating model error and for improving the stability and predictive skill of the imperfect models. In the talk I will give an overview of my research on a newly emerging stochastic-statistical framework which allows for information-theoretic quantification of uncertainty and mitigation of model error in imperfect statistical predictions of complex multi-scale dynamics. Two important examples used to highlight these issues will be concerned with (i) existence of `information' barriers to imperfect model improvement and (ii) real-time `stochastic superresolution' for estimation of the relevant unresolved processes in sparsely observed turbulent systems. Open mathematical and practical problems for future research will also be discussed.

In this talk, first I will formulate the general setting of
the affine sieve and then state its fundamental theorem. The outline of
its proof will be explained. In particular, we will see that the main
analytical part of the argument is to find the exact condition under
which a finitely generated subgroup of GL(n,Q) has property tau with
respect to its ``square-free principal congruence subgroups"! At the
end of the talk, some open problems will be presented.

We explore a refinement of homological mirror symmetry which relates exact symplectic topology to arithmetic algebraic geometry. We establish a derived equivalence of the Fukaya category of the 2-torus, relative to a basepoint, with the category of perfect complexes of coherent sheaves on the Tate curve over the "formal disc" Spec Z[[q]]. It specializes over the "punctured disc" Spec Z((q)), to an integral refinement of the known statement of homological mirror symmetry for the 2-torus. We will survey a general strategy of proof of homological mirror symmetry while carrying it out in the specific case of the 2-torus. In contrast to the abstract statement of our main result, the focus of the talk will be a concrete computation which we will express in more familiar terms. This is joint work with Tim Perutz.

Every number field comes with a whole array of associated invariants such as its class group, zeta function, adele ring, absolute
Galois group, etc. I will discuss to which extent such invariants characterize the number field and, in case they don't, how easily one can obtain non-isomorphic number fields for which such an invariant is
"the same".

In this talk, first I will present a kernel-free boundary integral (KFBI) method for the linearized Poisson-Boltzmann equation, which may have variable coefficients and is in general not applicable for the standard boundary integral method. Then I will present the KFBI method for the nonlinear Poisson-Boltzmann equation. The KFBI method is a Cartesian grid method. It uses Cartesian grid-based solutions to compute approximations to boundary and volume integrals involved in the solution of boundary integral equations for the Poisson-Boltzmann equation. I will also present numerical examples in both two and three space dimensions to demonstrate the efficiency and accuracy of the KFBI method.

The aim of this talk is to provide an overview of recent developments concerning the motion of a two dimensional incompressible and irrotational fluid with a free surface. The emphasis will be on the case when gravity is present, but surface tension is absent. This is joint work with John Hunter, Mihaela Ifrim and Tak Wong.

Matrix multiplication is used in a large variety of applications throughout science and engineering. It is a bottleneck for many important algorithms: for standard linear algebra problems such as solving linear systems and matrix inversion, but also for many algorithms that, on the face of it, may have nothing to do with matrices. Developing better matrix multiplication algorithms is of immense interest.

In 1969 Strassen showed that the naive algorithm for multiplying matrices is not optimal, presenting an ingenious recursive algorithm. This spawned a long line of active research on the theory of matrix multiplication algorithms. In a seminal paper from 1987, Coppersmith and Winograd designed an algorithm that can multiply two n by n matrices using $O(n^{2.376})$ arithmetic operations. This algorithm had remained the theoretically fastest approach for matrix multiplication for 24 years.

We have recently been able to design an algorithm that multiplies n by n matrices and uses at most $O(n^{2.373})$ arithmetic operations, thus improving the Coppersmith-Winograd running time. The improvement is based on a recursive application of the original Coppersmith-Winograd construction, together with a general theorem that reduces the final search for the best algorithm to solving a nonlinear constraint program. The analysis is then done by numerically solving this program. In this talk, we will give intuition for the problem and highlight the key ideas needed to obtain the improvement.

The problem of finding large average submatrices of a real-valued matrix arises in the exploratory analysis of data from a variety of disciplines, ranging from genomics to social sciences. This talk details several new results concerning the asymptotic behavior of large average submatrices of an nxn Gaussian random matrix. We begin by considering the average and joint distribution of the (globally optimal) kxk submatrix having largest average value. We then turn our attention to submatrices with dominant row and column sums, which arise as the local optima of a useful iterative search procedure for large average submatrices. Paralleling the result for global optima, we will consider the average and joint distribution of a typical (locally optimal) kxk submatrix with dominant row and column sums. The last part of the talk will be devoted to the analysis of the *number* of locally optimal kxk submatrices, $L_n(k)$, beginning with the asymptotic behavior of its mean and variance for fixed k and increasing n. Finally, we establish a central limit theorem for $L_n(k)$ that using Stein's method for normal approximation.

Joint work with Shankar Bhamidi (UNC) and Partha S. Dey (Courant)

This talk is based on joint work with Murray Schacher and Eric Rains. Given three cohomology classes satisfying a certain vanishing condition on cup products, Massey defines a triple product [a,b,c] and uses it to prove a Jacobi identity in algebraic topology.
By construction, the Massey product is a coset of a (known) subgroup
of cohomology in degree (deg a) + (deg b) + (deg c) - 1.
One interesting case is when a,b,c are degree one and the cohomology
is the Galois cohomology of a field F with GF(2) coefficients.
In this context, we can view the Massey product of three quadratic
extensions of F as a (set of) quaternion division algebras over F.
We give an interpretation of [a,b,c] as an obstruction to a lifting
problem in Galois theory. We give a new formula for [a,b,c], and use
it to show that the Massey product contains the neutral element.

In this talk we present some recent results on aggregation swarming models in two dimensions. This class of models involve pairwise interactions and an active scalar equation in the continuum limit. We show the connection between this model and the classical vorticity equation from fluid dynamics. The aggregation model can lead to a rich family of patterns. We discuss analytical and numerical methods for curve and cluster patterns.

In the seminar I would like to talk about two general problems in the theory of finite-dimensional Lie algebras:
• Finding a faithful representation of a Lie algebra (of lowest possible degree). E.g.: Can the $(2n + 1)$-dimensional Heisenberg Lie algebra $h_n$ be em-
bedded into $gl_{n+1}(C)$ Into $gl_{n+2}(C)$
• Finding properties of Lie algebras that admit a regular transformation.
E.g.: Is a Lie algebra that admits a regular periodic derivation neces- sarily abelian?
These two problems appear to be unrelated, but this is not the case. They follow naturally from some problems on the existence of affine structures on Lie algebras (or groups). I plan to briefly mention the historical background material, then formulate both problems more explicitly, illustrate which results are already known, and finally, how I hope to obtain some more (partial) answers.
For the first problem, it will be interesting to find good bounds on the minimal degree of a faithful representation. For the second problem, we would like to find good bounds on the nilpotency or solvability class of the Lie algebra.

Zero forcing (also called graph infection) on a simple,
undirected graph $G$ is based on the color-change rule: If each vertex
of $G$ is colored either white or blue, and vertex $v$ is a blue
vertex with only one white neighbor $w$, then change the color of $w$
to blue. A minimum zero forcing set is a set of blue vertices of
minimum cardinality that can color the entire graph blue using the
color change rule. In this talk will discuss the role of zero forcing
in systems control, electrical engineer, and linear algebra. Even
though various scientist have been using zero forcing, it wasn't until
recently that it was realized they were all doing the same type of
propagation. Zero forcing gets its name from the linear algebraists,
who were using the propagation to force entries of a vector to be
zero.

Sponsored by Fan Chung Graham and Jacques Verstraete

A natural question in real algebraic geometry is to ask, given polynomials $p$,
$q$, and $r$,
\begin{equation*}
\tag{Q}
\label{eq:quest}
\mbox{Does} \quad q(x) \geq 0 \quad \mbox{and} \quad r(x) = 0 \quad
\mbox{imply} \quad p(x) \geq 0\ \mbox{?}
\end{equation*}
In this talk I will focus on {\it free noncommutative} polynomials $p$, $q$,
and $r$ and substitute matrices for the variables $x_j$.
I will present an answer to a non-commutative version of (\ref{eq:quest}) which
includes a non-commutative ``real nullstellensatz'' and ``positivstellensatz''.

K . Ecker and G. Huisken have derived a priori estimate for the
curvature (second fundamental forms) when they study the mean
curvature flow of the graph of a function in Euclidean space. In this
talk, I will explain that a similar curvature estimate also exists for
higher codimensional mean curvature flow under certain natural
conditions. This is joint work with Knut Smoczyk and Mu-Tao Wang.

In this talk, I will use Kuznetsovs description of the derived category
of a smooth cubic hypersurface to give a new construction of some stable
ACM bundles on cubic threefolds and cubic fourfolds containing a plane. I
will also present a wall-crossing phenomenon that allows to relate the
compactification of the moduli space of instantons on a cubic threefold
with a moduli space of torsion sheaves a non-commutative projective plane.
This is a joint work with Emanuele Macr and Paolo Stellari.

Let X be a projective smooth variety over an algebraically closed field k.
If k has characteristic zero, then the singular (Betti) cohomology groups of X are finitely generated abelian groups and therefore all the invariants associated to them are discrete and in fact do not change under good deformations. If k has positive characteristic, then the crystalline cohomology groups of X have a much richer structure and are called F-crystals over k. In particular, one can associate to them many subtle invariants which vary a lot under good deformations and which could be of either discrete or continuous nature. We present an accessible survey of the classification of F-crystals over k via subtle invariants with an emphasis on the recent results obtain by us, by Gabber and us, and by Lau, Nicole, and us.

We first survey some results regarding the study of holomorphic functions on manifolds. We insist on Liouville theorems or, more generally, dimension estimates for the space of polynomially growing holomorphic functions. Then we present some recent joint work with Jiaping Wang on this topic. Our work is motivated by the study of Ricci solitons in the theory of Ricci flow. However, the most general results we have do not require any knowledge of curvature.

Let $k$ be an algebraically closed field of characteristic $p>0$. Let $W(k)$ be the ring of Witt vectors with coefficients in $k$.
A motivic conjecture of Milne relates, in the case of abelian schemes
over $W(k)$, the \'etale cohomology with $\Bbb Z_p$ coefficients to
the crystalline cohomology with coefficients in $W(k)$. In this talk we
report on the proof of this conjecture in the more general context of
$p$-divisible groups over $W(k)$ endowed with arbitrary families of
crystalline tensors. If $V$ is a discrete valuation ring which is a
finite extension of $W(k)$ of index of ramification $e>1$, we provide
examples which show that the conjecture is not true in general
over $V$ and we also mention some general cases in which the conjecture
does hold over $V$. Our results extend previous results of Faltings.
As a main new tool we construct global deformations of $p$-divisible
groups endowed with crystalline tensors over certain regular, formally
smooth schemes over $W(k)$ whose special fibers over $k$ have a Zariski
dense set of $k$-valued points.

Quantum secret sharing is an important multipartite cryptographic protocol in which a quantum state (secret) is shared among a set of n players. The secret is distributed in such a way that it can only be recovered by certain authorized subsets of players acting collaboratively. The recovery procedure assumes that all players are interconnected through quantum channels, or, equivalently, that the players are allowed to perform non-local quantum operations. However, for practical applications, the consumption of quantum communication resources such as entanglement or quantum channels needs to minimized.

We provide a novel scheme in which quantum communication is replaced by local operations and classical communication (LOCC). Our protocol is based on embedding a classical maximum distance separable (MDS) code into a quantum error correcting code and employing the properties of the latter. Our scheme is appealing for real-world scenarios where the implementation of two-qubit gates is challenging. We illustrate the results by simple examples. Our methods constitute a first step towards attacking the important problem of decoding quantum error correcting codes by LOCC.

Conformal algebras appeared initially in the theory of vertex operator
as an algebraic language describing the properties of the singular part
of the operator product expansion (OPE). On the other hand, {\em finite} conformal algebras
provide a natural approach to the study of certain infinite-dimensional algebras
in which the multiplication table is given by a finite number of algebraic functions.

In this talk, we study the natural relation between associative and Lie conformal algebras.
Every associative conformal algebra can be turned into a Lie conformal algebra in the
ordinary way. But it is still unknown whether a finite Lie conformal algebra
can be embedded into an appropriate associative conformal algebra.
A finer problem also makes sense: Whether a finite Lie conformal algebra has a faithful
finite representation?
We prove that if a finite Lie conformal algebra admits Levi decomposition then it has a
finite faithful representation.

Let $H$ be a $t$-uniform hypergraph on $k$ vertices, with $a_i\geq0$ denoting the multiplicity of the $i$-th edge, $1\leq i\leq\binom{k}{t}$. Let $\textup{\textbf{h}}=(a_1,\dotsc,a_{\binom{k}{t}})^\top$, and $N_t(H)$ the matrix whose columns are the images of $\textup{\textbf{h}}$ under the symmetric group $S_k$. We determine a diagonal form (Smith normal form) of $N_t(H)$ for a very general class of $H$.

Now, assume $H$ is simple. Let $K^{(t)}_n$ be the complete $t$-uniform hypergraph on $n$ vertices. Define $ZR_p(H)$ to be the zero-sum (mod $p$) Ramsey number of $H$, which is the minimum $n\in\mathbb{N}$ such that for every coloring $c:E\big(K^{(t)}_n\big)\to\mathbb{Z}_p$, there exists a subgraph $H'$ of $K^{(t)}_n$ isomorphic to $H$ such that $\sum_{e\in E(H')}c(e)=0$. Through finding a diagonal form of $N_t(H)$, we re-prove a theorem of Y.\ Caro in $\cite{caro}$ that gives the value $ZR_2(G)$ for any simple graph $G$. Further, we show that for a random $t$-uniform hypergraph $H$ on $k$ vertices, $ZR_2(H)=k$ asymptotically almost surely as $k\to\infty$.

Similar techniques can also be applied to determine the zero-sum (mod $2$) bipartite Ramsey numbers, $B(G,\mathbb{Z}_2)$, introduced in $\cite{caroyuster}$.

\begin{thebibliography}{99}

\bibitem{caro}
Y.\ Caro, A complete characterization of the zero-sum (mod 2) Ramsey numbers,
\textit{J.\ Combinatorial\ Th.\ Ser.\ A } \textbf{68} (1994), 205--211.

\bibitem{caroyuster}
Y.\ Caro and R.\ Yuster, The characterization of zero-sum (mod 2) bipartite Ramsey numbers,
\textit{J.\ Graph Th.\/} \textbf{29} (1998), 151--166.

\bibitem{wilsonwong}
R.\ Wilson and T.\ Wong, Diagonal forms of incidence matrices associated with $t$-uniform hypergraphs, \textit{provisionally accepted by the reviewers for Europ.\ J.\ Combinatorics}.

\end{thebibliography}

We establish a new local convex property of CAT(1)-spaces, which are metric
spaces of curvature at most 1 in the sense of Aleaxandrov, by showing that a
function which has its origin in the study of harmonic maps is convex on any
small ball of a CAT(1)-space. As an application, we also discuss barycenters
of probability measures on CAT(1)-spaces.

The diffusion tensor used in Brownian dynamics simulations is often calculated by the hydrodynamic bead models. The bead models are shown to reproduce the experimental diffusion coefficients (the diagonal tensor elements) of a single macromolecule, but the accuracy of the bead model calculation of the whole one-body or many-body diffusion tensor has not been closely investigated. Such investigation is important because Brownian dynamics simulations usually involve more than one molecule, and the whole diffusion tensor will be used instead of just the diagonal elements. As a first step, we investigated the accuracy of the bead model calculation of the diffusion tensor of two spheres. Our results show that bead models can produce fairly accurate diffusion tensors for two spheres. But its application in Brownian dynamics simulations needs extra caution, as its errors accumulate over time.

The modular j-function plays an important role in number
theory: its values at quadratic imaginary integers are called singular
moduli. Singular moduli can be interpreted as an invariant of CM
elliptic curves and play a role in explicit class field theory.
In 1985, Gross and Zagier gave an elegant formula for the factorization
of norms of differences of singular moduli associated to a pair of
imaginary quadratic discriminants d1 and d2, under the assumption that
d1 and d2 are fundamental and relatively prime. Their theorem was one
of the ingredients in the proof of the only known case of the
Birch-Swinnerton Dyer Conjecture. This talk will present a
generalization of their result to give a complete factorization for any
two fundamental discriminants which are not necessarily coprime, and
obtain at least a partial factorization for any two quadratic imaginary
discriminants. We will discuss the motivation for this generalization
arising in cryptography and give an application to proving an
intersection formula on Hilbert modular surfaces related to the work of
Bruinier and Yang. This is joint work with Bianca Viray.

$L^2$-Betti numbers were defined by Atiyah for compact manifolds X, by Cheeger and Gromov for countable groups $\Gamma$ and by Gaboriau for countable probability measure preserving equivalence relations. Henrik D. Petersen proposed a definition of $L^2$-Betti numbers for locally compact groups. I will present this definition and a recent joint work with Kyed and Petersen in which we prove that the $L^2$-Betti numbers of a locally compact group coincide with those of any associated cross section equivalence relation. As a consequence, we obtain several vanishing theorems for the reduced $L^2$-cohomology of locally compact groups.

We discuss the general problem of finding a large independent set in a hypergraph, using ordinary and probabilistic combinatorics, and summarize my research on the subject (joint with Jacques Verstraete).

The subject of this talk is at the crossroads of functional analysis, ergodic theory and group theory. Using a construction by Murray and von Neumann (1943), ergodic actions of countable groups on probability spaces give rise to algebras of operators on a Hilbert space, called von Neumann algebras. In a joint work with Sorin Popa, we proved that such crossed product von Neumann algebras by free groups or, more generally, by hyperbolic groups have a unique Cartan subalgebra. I will explain this result and its consequences for the classification of crossed products by free groups.

While the emphasis for biomedical engineering and/or bioengineering at Columbia University and UCSD are relatively new, studies on properties of such "biological" materials and events as bone, blood flow, articular cartilage date back to Galileo (1632), Harvey (1670), Borelli (1680), and Benninghoff (1900), respectively. Indeed, the word biology did not appear in the learned literature until the late 19th century. Galileo laid the foundation of mechanics in his book on the Two New Sciences in the last decade of his life while under house arrest in Pope Urban VIII's apartment in Rome. In this book the seeds of strength of material and kinematics were sown, later to be formulated by Isaac Newton after Galileo's death which occurred in 1642, which is the same year Newton was born. Today, there is no doubt of the singularly influential role played by Professor Y.C. Fung of UCSD in the development of the bioengineering discipline in America. Also, important roles were played by Professor Richard Skalak (the namesake for this memorial lecture series), and Professor Shu Chien, both formerly Columbia University professors, in enhancing the UCSD bioengineering program nationally and internationally. Professor Fung, in a stroke of genius, was able to recruit Skalak and Chien to relocate from Columbia to UCSD in 1988; this event, and the Whitaker Foundation, forever changed the landscape of bioengineering in America. Today, UCSD's bioengineering graduate research program is without doubt one of the most recognized in the world. In this lecture, I will present this short history, as well as my own role in establishing one of the best biomedical engineering departments in the world at Columbia University.

Functional magnetic resonance imaging (fMRI) is now a well established technique for studying the brain. However, in many situations, such as when data are acquired in a resting state, it is difficult to know whether the data are truly stationary or if level shifts have occurred. To this end, change-point detection in sequences of functional data is examined where the functional observations are dependent and where the distributions of change-points from multiple subjects are required. Of particular interest is the case where the change-point is
an epidemic change -- a change occurs and then the observations return to baseline at a later time. The case where the covariance can be decomposed as a tensor product is considered with particular attention to the power analysis for detection. This is of interest in the application to fMRI, where the estimation of a full covariance structure for the three-dimensional image is not computationally feasible. Using the developed methods, a large study of resting state fMRI data is conducted to determine whether the subjects undertaking the resting scan have non-stationarities present in their time courses. It is found that a sizeable proportion of the subjects studied are not stationary. The change-point distribution
for those subjects is empirically determined, as well as its theoretical properties examined.

This is joint work with John Aston (Warwick University).

Eugene Gorsky and Andrei Negut have recently put the finishing touches
to what may be viewed as the symmetric function side of the general m,n
Shuffle Conjecture. In this talk I will report of what I have learned
about the work of Tatsuyuki Hikita who constructed the combinatorial
side. This development is gravid with beautiful and challenging
Combinatorial problems. Gorsky and Negut hopefully soon will tell us
about the symmetric function side.

Symmetry is pervasive in both nature and human culture. The notion of chirality (or `handedness') is similarly pervasive, but less well understood. In this lecture, I will talk about a number of situations involving discrete objects that have maximum possible symmetry in their class, or maximum possible rotational symmetry while being chiral. Examples include geometric solids, combinatorial graphs (networks), maps on surfaces, dessins d'enfants, abstract polytopes, and even compact Riemann surfaces (from a certain perspective). I will describe some recent discoveries about such objects with maximum symmetry, illustrated by pictures as much as possible.

In a March Madness contest, players try to predict the
outcome of the NCAA Basketball Championship tournament most
accurately. If you want to win, do you choose what you think the most
likely outcome is? Or do you pick an outcome less likely so that you
have "more room" to be the most accurate? We answer this question.

We introduce semiclassical Eulerian methods that are efficient in computing high frequency waves through heterogeneous media. The method is based on the classical Liouville equation in phase space, with discontinous Hamiltonians due to the barriers or material interfaces. We provide physically relevant interface conditions consistent with the correct transmissions and reflections, and then build the interface conditions into the numerical fluxes. This method allows the resolution of high frequency waves without numerically resolving the small wave lengths, and capture the correct transmissions and reflections at the interface. This method can also be extended to deal with diffraction and quantum barriers.

We introduce the Einstein constraint equations and discuss their origin and importance. We review some of the basic tools from functional analysis, the theory of partial differential equations, and Riemannian geometry for studying the existence and uniqueness of the solutions of the constraint equations.

The original Shuffle Conjecture of Haglund et al. [2005] has a
symmetric function side and a combinatorial side. The symmetric function
side may be simply expressed as $\big\langle \nabla e_n \, , \, h_{\mu}
\big\rangle$ where $\nabla$ is the \hbox{Macdonald} polynomial
eigen-operator of Bergeron and Garsia [1999] and $h_\mu$ is the
homogeneous basis indexed by $\mu=(\mu_1,\mu_2,\ldots ,\mu_k) \vdash n$.
The combinatorial side q,t-enumerates a family of Parking Functions
whose reading word is a shuffle of $k$ successive segments of $1 2 3
\cdots n$ of respective lengths $\mu_1,\mu_2,\ldots ,\mu_k$. It can be
shown that for $t=1/q$ the symmetric function side reduces to a product
of $q$-binomial coefficients and powers of $q$. This reduction suggests
a surprising combinatorial refinement of the general Shuffle Conjecture.
Here we prove this refinement for $k=2$ and $t=1/q$. The resulting
formula gives a $q$-analogue of the well studied Narayana numbers.

This will be the first of our informal seminars on Geometric Measure Theory. We will meet each week to work through parts of one (or more) of the texts available on the topic. In this first talk, I will present the variational characterisation of smooth minimal and CMC hyper-surfaces arising from the area minimising problem and the isoperimetric problem respectively.

Given $z_0$ and $\lambda_0$
the Schwarz lemma tells us when we can find an analytic map $f:\mathbb{D}\rightarrow \mathbb{D}$ such that $f(0)=0$ and $f(z_0)=\lambda_0.$
We consider the following more general problem:
Given some data $(z_i)^{N}_{i=1}$ and $(\lambda_i)^{N}_{i=1}$ when is there an analytic $f:\mathbb{D}\rightarrow \mathbb{D}$ such that
$f(z_i)= \lambda_i.$

Since Erdos-Renyi introduced random graphs in 1959, two closely
related models for random graphs have been extensively studied. In the
$G(n,m)$ model, a graph is chosen uniformly at random from the
collection of all graphs that have n vertices and m edges. In the
$G(n,p)$ model, a graph is constructed by connecting each pair of two
vertices randomly. Each edge is included in the graph $G(n,p)$ with
probability p independently of all other edges.

Researchers have studied when the random graph $G(n,m)$ (or $G(n,p)$,
resp.) satisfies certain properties in terms of $n$ and $m$ (or $n$ and $p$,
resp.). If $G(n,m)$ (or $G(n,p)$, resp.) satisfies a property with
probability close to 1, then one may say that a ``typical graph�� with $m$
edges (or expected edge density $p$, resp.) on $n$ vertices has the
property. Random graphs and their variants are also widely used to
prove the existence of graphs with certain properties. In this talk, a
well-known problem for each of these categories will be discussed.

First, a new approach will be introduced for the problem of the
emergence of a giant component of $G(n,p)$, which was first considered
by Erd�s�Rényi in 1960. Second, a variant of the graph process
$G(n,1), G(n,2), �, G(n,m), �$ will be considered to find a tight lower
bound for Ramsey number $R(3,t)$ up to a constant factor.

No prior knowledge of graph theory is needed in this talk.

Schur functions are a basis for the ring of symmetric functions, one
with many important algebraic and combinatorial properties. k-Schur
functions are a basis for a certain subring of that ring and are turning
out to be just as interesting. Many results involving Schur functions
have analogues involving k-Schur functions. Standard strong marked
tableaux play a role for k-Schur functions similar to the role
standard Young tableaux play for Schur functions. I will discuss
results and conjectures toward an analogue of the hook length
formula. This is joint work with Matjaz Konvalinka.

I will give a general introduction to derived equivalences between smooth projective varieties. Then, I will describe recent work on the problem of determining when when two principal homogeneous spaces for an abelian variety are derived equivalent, culminating in a precise description of when two smooth projective curves over a field of characteristic zero have equivalent bounded derived categories.

In this talk, I will describe joint work with Daniele Arcara, Aaron Bertram and Jack Huizenga on running the Minimal Model Program (MMP) for the Hilbert scheme of points on $P^2$. We study
the stable base locus decomposition of the effective cone of the Hilbert scheme. We find a correspondence between the decomposition of the effective cone into the Mori chambers and the decomposition of the stability manifold into Bridgeland chambers.

F-singularities are classes of singularities defined by the behavior of Frobenius. A prominent tool for measuing these singularities is the test ideal, a characteristic $p > 0$ analog of the multiplier ideal. Recently, there has been interest in applying the methods of F-singularities to a number of geometric problems in positive characteristic. However, one gap in
the theory has been the behavior of F-singularities in families. For example generic restriction theorems for test ideals have been lacking.

In this talk, I will discuss recent joint work with Zsolt Patakfalvi and Wenliang Zhang on the behavior of F-singularities and test ideals in families. I will first define the relevant terms and explain why you these definitions and methods are useful (as a replacement for Kodaira vanishing in characterisic $p > 0$). We will then obtain generic (and non-generic) restrictions theorems for test ideals. Some global geometric consequences will also be discussed if there is sufficient time.

A semisimple (more generally, reductive) group scheme is called isotropic, if it contains a proper parabolic subgroup. In particular, the special orthogonal group of a non-degenerate quadratic form is isotropic if and only if the form itself is isotropic. To any isotropic reductive group G over a commutative ring R one associates a group-valued functor $K_1^G$ on the category of commutative R-algebras, which is an analog of the non-stable $K_1-functor GL_n/E_n$. When R is a field, $K_1^G$ coincides on smooth algebras with the group of $A^1$-connected components of G in the sense of Morel-Voevodsky. We will discuss various properties of these functors, and connections with classification of principal G-bundles.

We study the stability in the $H^1$-seminorm of the $L_2$-projection
onto finite element spaces in the case of nonuniform but shape regular
meshes in two and three dimensions and prove, in particular, stability
for conforming triangular elements up to order twelve and conforming
tetrahedral elements up to order seven.

Parity games are two player games which are played by moving a token around a finite graph. Parity games have important applications in automata theory, logic, and verification - for example, the model checking problem for the modal mu-calculus is polynomial time equivalent to solving parity games. It is a long-standing open problem whether the winner in a parity game can be decided in polynomial time.

In the talk, we will relate this problem to weak automatizability of resolution. A propositional proof system is weakly automatizable if there is a polynomial time algorithm which separates satisfiable formulas from formulas which have a short refutation in the system, with respect to a given length bound.
We explain that if the resolution proof system is weakly automatizable, then parity games can be decided in polynomial time.

This is joint work with Pavel Pudlak and Neil Thapen.

I will present the connection between two topics which seem completely
unrelated, other than by a fortuitous common name. The original shuffle
conjecture equates two symmetric polynomials $Phi_n[X;q,t]$ and
$Pi_n[X;q,t]$. The first was conjectured by Garsia-Haiman in the early
1990's to produce the bigraded Frobenius characters of diagonal harmonic polynomials. The second was defined by Haglund et al around 2002 as a weighted enumeration of parking functions in the $n \times n$ lattice square. In a recent paper, Hikita constructs a new polynomial
$Pi_{m,n}[X;q,t]$ for any pair of relatively prime natural numbers $m$ and
$n$, by extending the notion of parking function to the case of an $m
\times n$ lattice rectangle. It follows from Hikita's construction that
$Pi_n[X;q,t] = Pi_{n+1,n}[X;q,t]$.

The shuffle algebra is a representation-theoretic object constructed by
Feigin and Odesskii, which acts on the K-theory of the Hilbert scheme of
points in the plane. The K-theory is isomorphic to the ring of symmetric
functions in infinitely many variables, so the shuffle algebra action
produces many interesting symmetric functions. In particular we are able
to construct a new polynomial $Phi_{m,n}[X;q,t]$ which reduces to
$Phi_n[X;q,t] for m=n+1$, and conjecture that $Phi_{m,n}[X;q,t] =
Pi_{m,n}[X;q,t]$ for general coprime $m,n$. In this talk, we will explain the ample connection that led to this extension of the Shuffle Conjecture. This is joint work with Eugene Gorsky.

This will be the second of our informal seminars on Geometric Measure Theory. This week I will finish the variational characterisation of smooth minimal and CMC hyper-surfaces arising from the area minimising problem and the isoperimetric problem respectively.

A subfactor has infinite depth if its standard representation generates infinitely many irreducibles. Such subfactors are hard to construct and only very few methods are known to produce examples with non-integer Jones index. We will explain one such procedure, which involves a notion of free product for the associated planar algebras (joint work with Vaughan Jones).

A signal molecule, such as a hormone, interacts with receptor molecules
triggering a cascade of downstream events which we call its pathway.
Xenobiotic chemicals can alter normal biological functions by interacting
with receptors in ways that mimic natural signaling molecules. A new
way to test xenobiotic chemicals for interference along these pathways
is to use simple cell-based assays. These tests are considerably
cheaper and far less time-consuming than traditional animal-based tests
(a very important consideration when we think of the tens of thousands
of chemicals to which we are exposed). However, the assays are themselves complex biological processes that can respond to chemical perturbations in multiple ways, so determining the levels of interference of the chemicals from the data is non-trivial.
In this talk I discuss a mathematical model that can be used to
integrate data from a large number of assays and derive a measure of the
probability that activity in the assay is due to activity in the intended pathway or due to activity in a number of assay interference pathways. The approach is employed to analyze data from a battery of 20 estrogen receptor assays in which 1800 chemicals were tested. The results are compared with expected data from a set of reference chemicals.
The work was initiated at a one-week workshop titled ``Modeling Problems Related to our Environment,'' at the American Institute of Mathematics. The nature of the collaboration is itself of interest and I will devote some time to discussing that.

Using results of Drinfeld and Taguchi, we establish an equivalence of categories between the category of ``Drinfeld displays'' (objects to be introduced) and the category of $\pi$-divisible modules. We define tensor products of $\pi$-divisible modules and using the above equivalence, we prove that the tensor products of $\pi$-divisible modules over locally Noetherian base schemes exist and commute with base change. If time permits, we will show how this will provide tensor products of Lubin-Tate groups and formal Drinfeld modules.

We will look how to model cells (or more precisely vesicles) with energy functionals. We will also look at the sharp interface limit of one such phase field model constructed by Shao, Rappel, and Levine. Finally we will look at some numerical simulations based on Shao-Rappel-Levine model and other models.

This framework is based on the causal set approach to discrete quantum gravity. We first construct a causal growth process (CGP). We �quantize� the CGP by forming Hilbert spaces of sets of finite paths. A quantum dynamics is generated by a sequence of probability operators on these Hilbert spaces which then gives a quantum sequential growth process. We also obtain a quantum measure space using suitable subsets of the path space. We define a covariant bidifference operator on the Hilbert space of causal sets. Employing this operator we define a discrete curvature operator and discrete Einstein equations. We comment on the relationship of this to classical general relativity.

We continue with our seminars on GMT. Lei Ni will talk about some of the analysis preliminaries we require, such as compactness theorems for Radon measures and BV functions.

Come play America's favorite game show! This round has all new questions.

A knot homology theory associates to a knot or link a chain complex whose graded Euler characteristic is a classical knot polynomial. This type of knot invariant has been increasingly influential in low-dimensional topology since the first one was defined in 1999. This primarily expository talk will introduce some knot homology theories with an emphasis on the commonalities in their constructions and their relationships to other areas in mathematics (symplectic geometry, representation theory, etc.). Towards the end, we will encounter some areas of current research. No significant knowledge of low-dimensional topology will be assumed. The talk should be accessible to beginning graduate students, but hopefully still interesting to those who are further along!

Any smooth affine hypersurface Z of complex dimension n deformation retracts to a cell complex of real dimension n. Starting from the Newton polytope of the defining equation of Z, I will give an explicit combinatorial construction of a compact space S, comprised of n-dimensional components, which embeds into Z as a deformation retract. In particular, Z is homotopy equivalent to S. The construction uses toric degenerations, Nakayama-Ogus's work in log geometry and the Kato-Nakayama space. It is motivated by the homological mirror symmetry program. If time permits, I will explain the connections. This work is joint with Nicola Sibilla, David Treumann and Eric Zaslow.

This talk is prompted by the long standing question of whether it is
possible for the universal enveloping algebra of an in finite
dimensional Lie algebra to be noetherian. To address this problem,
we answer a 23-year-old question of Carolyn Dean and Lance Small;
namely, we prove that the universal enveloping algebra of the Witt
(or centerless Virasoro) algebra is not noetherian. We employ
algebro-geometric techniques from Sue Sierra's classification of
(noncommutative) birationally commutative projective surfaces to
accomplish this task.

As a consequence of our main result, we show that the enveloping
algebras of many other infinite dimensional Lie algebras also are not
noetherian. These Lie algebras include the Virasoro algebra and
all Z-graded simple Lie algebras of polynomial growth.

This is joint work with Sue Sierra.

Genetically encoded biosensors based on fluorescence resonance energy transfer (FRET) have been widely applied to visualize the molecular activity in live cells with high spatiotemporal resolution. The enormous amount of video images and the complex dynamics of signaling events presented tremendous challenges for data analysis and demand the development of intelligent and automated imaging analysis methods specifically designed for the studies of live cell imaging. We have developed advanced and automated computational imaging analysis methods for quantifying and simulating the motion of biosensors, reconstructing the de facto molecular activities, and tracking and analyzing the spatiotemporal molecular interactions in a single live cell with high-throughput power. Based on fluorescence recovery after photobleaching (FRAP) experiments, we have developed a finite element (FE) method to analyze, simulate, and subtract the diffusion effect of mobile biosensors. The results indicate that the Src biosensor located in the cytoplasm moves 48 folds faster ($0.9360.06 mm^2/sec$) than those anchored on different compartments in plasma membrane (at lipid raft: $0.1160 mm^2/sec$ and outside: $0.1860.02 mm^2/sec$). The mobility of biosensor at lipid rafts is slower than that outside of lipid rafts and is dominated by two-dimensional diffusion. Furthermore, we have developed a general correlative FRET imaging method (CFIM) to quantify the subcellular coupling between an enzymatic activity and a phenotypic response in live cells, e.g. at focal adhesions (FAs). CFIM quantitatively evaluated the cause-effect relationship between Src kinase activation and FA dynamics monitored in single cells. CFIM showed that the growth factor-stimulated FA disassembly at cell periphery was linearly dependent on the local Src activation with a time delay. The FA disassembly per unit of Src activation (coupling capacity), as well as the time delay, was regulated by cell-matrix interaction via different integrin receptors. The results revealed a tight enzyme-phenotype coupling in FA populations mediated by integrin a[v]b[3], but not in those by integrin a[5]b[1]. Therefore, our computational analysis methods can allow the high-throughput quantification of molecular motions and interactions at subcellular levels in single live cells. The results should advance our systems understanding of the hierarchical interactions of signaling molecular network at subcellular microdomains.

We continue with our seminars on GMT. Lei Ni will talk some more about the compactness theorem for Radon measures.

For a proper smooth variety over a perfect field of characteristic p, crystalline cohomology is a good integral model for rigid cohomology and crystalline Chern classes are integral classes which are rationally compatible with the rigid ones. The overconvergent de Rham-Witt complex introduced by Davis, Langer and Zink provides an integral p-adic cohomology theory for smooth varieties designed to be compatible with rigid cohomology in the quasi-projective case. The goal of this talk is to describe the construction of integral Chern classes for smooth varieties rationally compatible with rigid Chern classes using the overconvergent complex.

Building on the work which will appear in my thesis and
certain formalism introduced in F. Pellarin's paper \textit{Values of certain $L$-series in positive characteristic},
I will introduce a family arising from the Carlitz module in the ring of twisted power series over the ring $K$ of rational
functions in one indeterminate with coefficients in a finite field.
I will describe how this family gives rise to both the
explicit calculation of the rational functions appearing
(inexplicitly) in Pellarin's main result in the paper above
and also to certain $K$-relations between Thakur's
multizeta values originally introduced in his
book \textit{Function Field Arithmetic}.
This is a joint work with F. Pellarin.

I will discuss an explicit Iwasawa theoretic construction of Tate (exact) sequences and give some of its applications to various conjectures on special values of Artin L-functions. This is based on joint work with Greither and Banaszak.

A subgraph of an edge colored graph is called rainbow if each edge has a distinct color. Brualdi and Hollingsworth proved that in any edge coloring of a complete graph by perfect matchings, there are two edge disjoint rainbow spanning trees and conjectured that a full partition into rainbow spanning trees is possible (for $K_{2n}$ , where $n ? 3$). Kaneko, Kano, and Suzuki proved that one may always find three edge disjoint rainbow spanning trees. In this talk, we show that in this situation a constant fraction of the graph can be covered by edge disjoint rainbow spanning trees. We'll discuss some of the main challenges and ideas in the proof, and an extension of the conjecture by Kaneko, Kano, and Suzuki that considers all proper edge colorings and not just those by perfect matchings.

Having a fundamental spacetime view of physics, appears more justified in attempting to construct a quantum theory of gravity. In this talk, for this reason, we adopt a path integral approach, and we attempt to construct a self-consistent realistic histories formulation of quantum theory, namely the "coevents formulation". After revising the histories viewpoint of classical physics, we attempt to apply this to the quantum case. However, the nature of the probabilities arising from the path integral, leads us to alter the classical picture. The new ontology is that of a coevent (or a coarse grained history), and we analyse the consequences of this (relatively) novel proposal.

Adopting a path integral approach to quantum theory leads to the definition of quantum measure. One of its main properties, is that one can cover the full sample space with sets of quantum measure zero. The physical consequence of this, is that one cannot maintain the one-history-realised viewpoint. This leads to the proposed coevents formulation. In this talk, after a very brief introduction to the formulation, we examine certain important issues that arise. Firstly, how the Kochen Specker theorem and the existence of zero covers is evaded. At this point, a brief comparison of the coevents formulation with the consistent histories will be done. Secondly, how requiring deductive reasoning for the quantum world (and the Modus Ponens inference rule) necessarily leads to the multiplicative coevents defined earlier.

Recently it has been observed that the numbers of 2-crossings and 2-nestings have a symmetric joint distribution over many combinatorial structures, including permutations, matchings, set partitions, and linked partitions. These results have been put by Kasraoui in the larger context of enumeration of northeast and southeast chains of size 2 in fillings of moon polyominoes.
In this talk we present much stronger symmetric properties of northeast and
southeast chains.
Our results are obtained by generalizing the polyomino model in two ways:
(1) The polyomino is equipped with a charge function, and
(2) We relax the constraints on the polyominoes to allow a full action of the symmetric group.
This is a joint work with Andrew Wang and Jean Yeh.

In this talk we show how analyzing basic Depth First Search algorithm leads to a simple proof of the two most classical results about random graphs:

1. The result of Erdos-Renyi that the random graph $G(n,p)$ experiences sharp phase transition around $p=1/n$. For $p=(1-\epsilon)/n$, all connected components of $G(n,p)$ are typically of size $O(\log n)$, while for $p=(1+\epsilon)/n$, with high probability there exists a connected component of size linear in n.

2. The result of Ajtai-Komlos-Szemeredi that in the supercritical regime $p=(1+\epsilon)/n$, random graph typically contains a path of linear length.

Joint work with M. Krivelelvich

Pick functions are the analytic maps of the upper half plane into itself. Their boundary behavior was studied classically by many household names such as Caratheodory, Julia and Nevanlinna. For example, Nevanlinna showed that asymptotic expansions of a certain type of Pick function at infinity parametrize the solutions to the Hamburger moment problem. We propose a program for studying interpolation and approximation multivariable Pick functions on the boundary using techniques from functional analysis which have roots in systems engineering.

We describe joint work (with David Ben-Zvi and David Nadler) that
constructs an equivalence between the derived category of smooth
representations of $\mathrm{GL}_n(\mathbb{Q}_p)$ and a certain category
of coherent sheaves on the moduli stack of Langlands parameters for
$\mathrm{GL}_n$. The proof of this equivalence is essentially a
reinterpretation of $K$-theoretic results of
Kazhdan and Lusztig via derived algebraic geometry. We will also discuss
(conjectural) extensions of this work to other quasi-split groups, and
to the modular representation theory of $\mathrm{GL}_n$.

G-protein coupled receptors (GPCRs) mediate cellular responses to various hormones and neurotransmitters and are important targets for treating a wide spectrum of diseases. While significant advances have been made in structural studies of GPCRs, details of their activation mechanism remain unclear. The X-ray crystal structure of the M2 muscarinic receptor, a key GPCR that regulates human heart rate and contractile forces of cardiomyocytes, was determined recently in an inactive antagonist-bound state. Here, activation of the M2 receptor is directly observed via accelerated molecular dynamics (aMD) simulation, in contrast to previous microsecond-timescale conventional MD (cMD) simulations in which the receptor remained inactive. Receptor activation is characterized by formation of a $Tyr206^{5.58}$-$Tyr440^{7.53}$ hydrogen bond and ~6 $\AA$ outward tilting of the cytoplasmic end of TM6, preceded by relocation of $Trp400^{6.48}$ towards $Phe195^{5.47}$ and $Val199^{5.51}$ and flipping of $Tyr430^{7.43}$ away from the ligandbinding cavity. Network analysis reveals that communication in the intracellular domains is greatly weakened during activation of the receptor. Together with the finding that residue motions in the ligand-binding and G-protein coupling sites of the apo receptor are correlated, this highlights a dynamic network for allosteric regulation of the M2 receptor activation.

I will report on an ongoing joint project with David Helm and Yichao Tian. Let $p$ be a prime unramified in a totally real field $F$. The Goren-Oort strata are defined by the vanishing locus of the partial Hasse invariants; they furnish an analog of the stratification of modular curves mod $p$ by the ordinary locus and the supersingular locus. We give an explicit global description of the Goren-Oort stratification of the special fiber of the Hilbert modular variety for $F$. An interesting application of this result is that, when $p$ is inert of even degree, certain generalizations of the strata considered by Goren-Oort contribute non-trivially as Tate cycles to the cohomology of the special fiber of the Hilbert modular varieties. Under some mild conditions, they generate all the Tate cycles.

Extremal Combinatorics is one of the central branches of discrete mathematics which deals with the problem of estimating the maximum possible size of a combinatorial structure which satisfies certain restrictions. Often, such problems have also applications to other areas including Theoretical Computer Science, Additive Number Theory and Information Theory. In this talk we will illustrate this fact by several closely related examples focusing on a recent work with Alon and Moitra.

Causal set theory is a promising candidate of a discrete theory of quantum gravity.
Yet even measuring the dimension of a causal set and deciding if it is at all manifold-like are hard.
We propose a new observable, the abundance of intervals, which can help with both these tasks.
It is at once characteristic for the dimension of a causal set and gives a strong indicator if the space time is manifold like.

We will discuss joint work with D. Lafountain on a proof of the Jones Conjecture. Recently, Dynnikov and Prasolov provided a proof of this conjecture. We present here an alternative approach.

I will explain how to prove the Breuil-Mezard conjecture for split (non-scalar) residual representations by local methods. Combined with the cases previously proved by Kisin and Paskunas, this completes the proof of the conjecture for $\mathrm{GL}_2(\mathbb{Q}_p)$. As an application, we can prove the Fontaine-Mazur conjecture in the cases that the global residual representation restricts to the decomposition group at $p$ as an extension of the trivial character by the mod $p$ cyclotomic character. These are the cases complementary to Kisin's. This is a joint work with Yongquan Hu.

The ``Shuffle Conjecture'' states that the bigraded Frobeneus characteristic of the space of diagonal harmonics (equal to \(\nabla e_n\)) can be computed as the weighted sum of combinatorial objects called parking functions. In a 2010 paper Haglund, Morse, and Zabrocki studied the family of polynomials \(\nabla C_{p_1}\dots C_{p_k}1\), where $p=(p_1,\ldots ,p_k)$ is a composition and the $C_a$ are certain rescaled Hall-Littlewood vertex operators. They conjecture that these polynomials enumerate a composition indexed family of parking functions weighted by the same statistics. This refinement of the nearly decade old ``Shuffle Conjecture,'' when combined with properties of the Hall-Littlewood polynomials implies the existence of certain bijections between these families of parking functions. The existence of these bijections then follows from some relatively simple properties of a certain recursively constructed family of polynomials. This work introduces those polynomials, explains their connection to the conjecture of Haglund, Morse, and Zabrocki, and explores some of their surprising properties, both proven and conjectured. The result is an intriguing new approach to the Shuffle Conjecture and a deeper understanding of some classical parking function statistics.

The Strauss conjecture for the Minkowski spacetime in three dimensions
states that the semilinear equation
\[
\Box u = |u|^p, \quad u(0)=\epsilon f, \quad \partial_t u(0)=\epsilon g
\]
has a global solution for all $f$ and $g$ smooth, compactly supported and
$\epsilon$ small enough if $p > 1+\sqrt 2$. We prove a similar result in the
context on Schwarzschild and Kerr with small angular momentum spacetimes. This is
joint work with H. Lindblad, J. Metcalfe, C. Sogge, and C. Wang

Let $Z$ be a variety and $A$ an elliptic curve over the function field of $Z$. I. Dolgachev and M. Gross define the \emph{geometric Shafarevich-Tate group} of $A$ over $Z$ to classify the set of isomorphism classes of principal homogeneous spaces for $A$ which are locally trivial in the \'etale topology. In joint work with Chad Schoen, we describe how to compute the Shafarevich-Tate group when $A$ is the generic fiber of a class of elliptic threefolds and $Z$ is the base. We also obtain results on the Brauer groups of such threefolds.

Equipped with the $L^2$-distortion distance, the space "$X$" of all metric measure spaces $(X,d,m)$ is proven to have nonnegative curvature in the sense of Alexandrov. Geodesics and tangent spaces are characterized in detail. Moreover, classes of semiconvex functionals and their gradient flows on "$X$" are presented.

Tractable characterizations of polynomials which are nonnegative on a set K, is a topic of independent interest in Mathematics, but is also of primary importance in many important applications, and notably in global optimization. We will review two kinds of tractable characterizations for nonnegative polynomials through linear matrix inequalities (LMIs) and semidefinite programs (SDPs).

The first characterization of nonnegativity is based on the defining polynomials of the set K, via sums of squares (SOS)-weighted representation. For instance, in global optimization this allows to define a hierarchy of semidefinite relaxations which yields a monotone sequence of lower bounds converging to the global optimum (and in fact, finite convergence is generic).

The second (dual) characterization of nonnegativity is based on moments of a measure whose support is K. In this approach, checking nonnegativity is reduced to solving a sequence of generalized eigenvalue problems. When applied in global optimization over K, this results in a monotone sequence of upper bounds converging to the global minimum, which complements the previous sequence of lower bounds.

These two (dual) characterizations provide convex inner (resp. outer) approximations (by spectrahedra) for the convex cone of polynomials that are nonnegative on K.

Deterministic dynamical system models with delayed feedback and state constraints arise in a variety of applications in science and engineering. Under certain conditions oscillatory behavior has been observed and it is of interest to know when there are periodic solutions. Here we consider a prototype for such models --- a one-dimensional delay differential equation with non-negativity constraints. We obtain sufficient conditions for the existence of slowly oscillating periodic solutions of such equations when the delay/lag interval is long. Under further restrictions, including longer delay intervals, we prove uniqueness and uniform exponential asymptotic stability of such solutions.

Uncertainty relations are a distinctive characteristic of quantum theory that imposes intrinsic limitations on the precision with which physical
properties can be simultaneously determined. The modern work on
uncertainty relations employs entropic measures to quantify the lack of
knowledge associated with measuring non-commuting observables. However, I will show here that there is no fundamental reason for using entropies as quantifiers; in fact, any functional relation that characterizes the uncertainty of the measurement outcomes can be used to define an uncertainty relation. Starting from a simple assumption that any measure of uncertainty is non-decreasing under mere relabeling of the measurement outcomes, I will show that Schur-concave functions are the most general uncertainty quantifiers. I will then introduce a novel fine-grained uncertainty relation written in terms of a majorization relation, which generates an infinite family of distinct scalar uncertainty relations via the application of arbitrary measures of uncertainty. This infinite family of uncertainty relations includes all the known entropic uncertainty relations, but is not limited to them. In this sense, the relation is universally valid and captures the essence of the uncertainty principle in quantum theory.

We will show that every group admitting an unbounded
quasi-cocycle into a mixing non-amenable representation is not inner
amenable. We will discuss how our approach generalizes a recent result of
Dahmani, Guirardel, and Osin on the non-inner amenability of so-called
acylindrically hyperbolic groups. Other applications such as to central
sequences in
group-measure space algebras as well as some open problems will also be
touched upon. This is based on joint work with Ionut Chifan and Bogdan
Udrea.

Circulants are Cayley graphs for cyclic groups, and admit dihedral symmetries.
This talk is a summary of some work done with a research student (from Spain)
on the genus spectrum of embeddings of circulants on orientable surfaces.
First, we derived a formula for the {\em maximum genus\/} of such embeddings.
Then we turned to the question of finding the {\em minimum\/} genus of
such embeddings, for various classes of circulants. In doing this, we found
several counter-examples to a claimed theorem by Costa, Strapasson, Alves
and Carlos (2010) on embeddings of circulants on the torus (genus 1), and
then we went on to determine all circulants that have minimum genus 1 or 2.

This work involved a combination of mathematics and computer experimentation,
some of which will be described, with illustrations.

A bilevel program is a sequence of two optimization problems where the constraint region of the upper level problem is determined implicitly by the solution set to the lower level problem. The classical approach to solve a bilevel program is to replace the lower level program by its first order optimality condition. This approach, however, is not valid for the case where the lower level program is nonconvex. The reformulation based on the value function of the lower level program, on the other hand, is completely equivalent to the original bilevel program but may not have the optimal solution of the bilevel program as a stationary point. The combined program with both the value function constraint and the first order condition is much more likely to have the optimal solution as a stationary point. Since the value function is in general a nonsmooth function, we propose to solve the problem by smoothing techniques. Using smoothing technique, we approximate each nonsmooth Lipschitz continuous function by a family of smoothing functions. We use certain penalty based methods to solve the smooth problem and drive the smoothing parameter to infinity. Based on the sequence of iteration points and the family of smoothing functions we introduce the weak extended generalized Mangasarian-Fromovitz constraint qualification (WEGMFCQ). We show that if the WEGMFCQ holds at the accumulation point of the iteration points, then the accumulation point is a stationary point of the nonsmooth optimization problem. Numerical experiments show that while the EGMFCQ never hold for bilevel programs, the WEGMFCQ may hold for bilevel programs easily.

The term "Gray code" usually refers to an ordering of the n-bit binary
strings in which successive strings differ in a single bit. For
example, 000, 001, 011, 010, 110, 111, 101, 100 suffices for $n=3$. The
term "de Bruijn sequence" usually refers to a circular string of $2^n$
bits in which each n-bit binary string appears once as a substring. For
example, 0000100110101111 suffices for $n=4$ since its substrings are
0000, 0001, 0010, ..., 1110, 1100, 1000 (where the last three substrings
wrap-around).

These concepts have natural generalizations to other combinatorial
objects. For example, 123, 132, 312, 321, 231, 213 is a Gray code for
the permutations of {1,2,3} in which successive permutations differ by
an adjacent-transposition. Simiarly, 321312 is a universal cycle for
the 2-permutations of {1,2,3}. (The term "universal cycle" is often
used for non-binary de Bruijn sequences.)

Gray codes and universal cycles are usually constructed 'globally',
meaning that the overall order is easy to describe. In this talk, we
instead consider 'local' constructions, where it is easy to describe the
next object in the order. With this change in focus, we are able to
unify a number of previous results, and solve several well-known open
problems. The talk will begin with an overview of the research area and
its applications, and should be of interest to computer scientists and
discrete mathematicians.

In addition to frequent single-nucleotide mutations, mammalian and
many other genomes undergo rare and dramatic changes called genome
rearrangements. These include inversions, fissions, fusions, and
translocations. Although analysis of genome rearrangements was
pioneered by Dobzhansky and Sturtevant in 1938, we still know very
little about the rearrangement events that produced the existing
varieties of genomic architectures. Recovery of mammalian
rearrangement history is a difficult combinatorial problem that I will
cover in this talk. Our data sets have included sequenced genomes
(human, mouse, rat, and others), as well as radiation hybrid maps of
additional mammals.

Kronecker coefficients count the multiplicities of
irreducible representations in the tensor product of two irreducible
representations of the symmetric group. While their study was initiated
almost 75 years, very little is still known about them, and one of the
major problems of algebraic combinatorics is to find a positive
combinatorial interpretation for the Kronecker coefficients. Recently
this problem found a new meaning in the field of Geometric Complexity
Theory, initiated by Mulmuley and Sohoni, where certain conjectures on
the complexity of computing and deciding positivity of Kronecker
coefficients are part of a program to prove the ``P vs NP'' Millennium
problem.

In this talk we will describe several problems and some results on
different aspects of the Kronecker coefficients. A conjecture of Jan
Saxl states that the tensor square $S_\rho$ $ \otimes$ $S_\rho$, where $\rho$ is
the staircase partition, contains every irreducible representation of
$S_n$. We will present results towards this conjecture, as well as a tool
for determining the positivity of certain Kronecker coefficients. We
will also explore the combinatorial aspect of the problem and show how
to prove certain unimodality results using Kronecker coefficients,
including Sylvester's theorem on the unimodality of $q$-binomial
coefficients (as polynomials in $q$) and some new extensions thereof, like
strict unimodality. The results are based on joint work with Igor Pak
and Ernesto Vallejo.

Shrinking Ricci solitons are generalizations of positive Einstein
manifolds which arise naturally in the analysis of singularities of
the Ricci flow. At present, all known complete noncompact examples
either split locally as products or possess conical structures at
infinity. I will describe recent joint work with Lu Wang in which we
prove that such conical structures admit little flexibility: if two
shrinking solitons are asymptotic along some end of each to the same
regular cone, then the solitons must actually be isometric on some
neighborhoods of infinity of these ends. As an application, we prove
that the only complete connected shrinking soliton asymptotic to a
rotationally symmetric cone is the Gaussian soliton.

We consider critical branching Brownian motion with absorption, in which
there is initially a single particle at $x > 0$, particles move according to
independent one-dimensional Brownian motions with the critical drift of
negative the square root of 2, and particles are absorbed when they reach
zero. Kesten (1978) showed that almost surely this process eventually
dies out. Here we obtain upper and lower bounds on the probability that
the process survives until some large time t. These bounds improve upon
results of Kesten (1978), and partially confirm nonrigorous predictions of
Derrida and Simon (2007). We will also discuss results concerning the
behavior of the process before the extinction time, as x tends to
infinity. We estimate the number of particles in the system at a given
time and the position of the right-most particle, and we obtain asymptotic
results for the configuration of particles at a typical time. This is
based on joint work with Julien Berestycki and Nathanael Berestycki.

Technological innovations have revolutionized the process of scienti�c research and knowledge discovery. Nowadays, massive abundance of data is not uncommon in many scientific areas ranging from genomics to economy. They give rise to many novel opportunities in knowledge discovery but come coupled with many unforeseen computational and statistical challenges, including scalability and storage bottleneck, noise accumulation, spurious correlation, incidental endogeneity, and measurement errors. These challenges are distinguished and require both new computational and new statistical paradigms. My current research lies at the frontier of such big data paradigm and focuses mostly on only one task of variable selection and feature extraction.
In this talk I will expose the challenges that arise in this one task and show how optimization, probability and matrix analysis interplay in the new statistical theory.

Let $\chi$ be a totally odd character of a totally real number
field. In 1981, B. Gross formulated a p-adic analogue of a conjecture of
Stark which expresses the leading term at s=0 of the p-adic L-function
attached to $\chi\omega$ as a product of a regulator and an algebraic
number. Recently, Dasgupta-Darmon-Pollack proved Gross' conjecture in the
rank one case under two assumptions: that Leopoldt's conjecture holds for F
and p, and a certain technical condition when there is a unique prime above
p in F. After giving some background and outlining their proof, I will
explain how to remove both conditions, thus giving an unconditional proof
of the conjecture. If there is extra time I will explain an application to
the Iwasawa Main Conjecture which comes out of the proof, and make a few
remarks on the higher rank case.

Many biochemical processes at the sub-cellular level involve a small number of molecules. The local numbers of these molecules vary in space and time, and exhibit random fluctuations that can only be captured with stochastic simulations. We present a novel stochastic operator-splitting algorithm to model such reaction-diffusion phenomena. The reaction and diffusion steps employ stochastic simulation algorithms and Brownian dynamics, respectively. Through theoretical analysis, we have developed an algorithm to identify if the system is reaction-controlled, diffusion-controlled or is in an intermediate regime. The time-step size is chosen accordingly at each step of the simulation. We have used three examples to demonstrate the accuracy and robustness of the proposed algorithm. The first example deals with diffusion of two chemical species undergoing an irreversible bimolecular reaction. It is used to validate our algorithm by comparing its results with the solution obtained from a corresponding deterministic partial differential equation at low and high number of molecules. In this example, we also compare the results from our method to those obtained using a Gillespie multi-particle (GMP) method. The second example, which models simplified RNA synthesis, is used to study the performance of our algorithm in reaction- and diffusion-controlled regimes and to investigate the effects of local inhomogeneity. The third example models reaction-diffusion of CheY molecules through the cytoplasm of Escherichia coli during chemotaxis. It is used to compare the algorithm’s performance against the GMP method. Our analysis demonstrates that the proposed algorithm enables accurate simulation of the kinetics of complex and spatially heterogeneous systems. It is also computationally more efficient than commonly used alternatives, such as the GMP method.

Since 2005 a class of algebras, the {\it Leavitt path algebras} $L_K(E)$ (for $K$ any field and $E$ any directed graph), has been a focus of investigation by both algebraists and C$^*$-analysts. In this talk I'll define these algebras, and give some of their general properties. Then I'll describe some of the current lines of investigation in the area. In particular, I'll show a connection between ideas from symbolic dynamics (``flow equivalence") and the Grothendieck group $K_0(L_K(E))$. With that connection in mind, I'll explain one of the most compelling open problems in Leavitt path algebras, the {\it Algebraic Kirchberg Phillips Question}, which can be paraphrased as: can we recover $L_K(E)$ from $K_0(L_K(E))$? While the answer to the corresponding question for graph C$^*$-algebras is {\it Yes}, there remains a barrier to a complete answer on the algebra side.

If $n$ is a nonnegative integer, a partition of $n$ is a sequence of weakly decreasing positive integers which sum to $n$. Partitions arise in the study of the symmetric group of permutations of the set $\{1, 2, \ldots, n\}$, the geometry of the Grassmannian of $k$-dimensional subspaces of an $n$-dimensional vector space, and in numerous combinatorial areas such as finite field theory. I will show how a visualization of partitions obtained by shoving boxes into a corner can be used to define a polynomial refinement of the binomial coefficients called the Gaussian polynomials and will discuss various properties of this polynomial refinement (and present at least one open problem).

It has been a long history of utilizing interactions in regression
analysis to investigate interactive effects of covariates on response
variables. In this paper we aim to address two kinds of new challenges
resulted from the inclusion of such high-order effects in the regression
model for complex data. The first kind arises from a situation where
interaction effects of individual covariates are weak but those of
combined covariates are strong, and the other kind pertains to the
presence of nonlinear interactive effects. Generalizing the single index
coefficient regression model, we propose a new class of semiparametric
models with varying index coefficients, which enables us to model and
assess nonlinear interaction effects between grouped covariates on the
response variable. As a result, most of the existing semiparametric
regression models are special cases of our proposed models. We develop a
numerically stable and computationally fast estimation procedure utilizing
both profile least squares method and local fitting. We establish both
estimation consistency and asymptotic normality for the proposed
estimators of index coefficients as well as the oracle property for
thenonparametric function estimator. In addition, a generalized likelihood
ratio test is provided to test for the existence of interaction effects or
the existence of nonlinear interaction effects. Our models and estimation
methods are illustrated by both simulation studies and an analysis of body
fat dataset.

Recent work of Eugene Gorsky and Andrei Negut have put the finishing touches of what may be viewed as the symmetric function side of the Extended Shuffle conjectures. Their work, together with Hikita's construction of the combinatorial side combine into a truly remarkable family of conjectures which beautifully extend the classical Shuffle conjecture. These developments have opened up a vast research area of Algebraic Combinatorics considerably enlarging classical symmetric function theory as well as the Theory of Parking functions created by Computer Scientists. This talk is an elementary introduction to this new research area.

It is well-known that adding extra capacity to queues in networks where individuals choose their own route can sometimes severely degrade performance, rather than improving it. We will discuss some simple examples of queueing networks where this is the case under probabilistic routing, but where under state-dependent routing the worst case performance is no longer seen. This raises the question of whether giving arrivals more information about the state of the network leads to better performance more generally.

This is joint work with Heti Afimeimounga, Lisa Chen, Mark Holmes, Bill Solomon, and, latterly, Niffe Hermansson and Elena Yudovina.
(Jointly sponsored by the Mathematics and ECE Departments.)

In this talk I will try to give some idea of what originally got me interested in the theory of noncommutative rings. The Weyl algebras are fundamental examples in noncommutative ring theory. They have many interesting properties which illustrate the many ways that noncommutative rings are more complicated (and intriguing) than commutative ones. After introducing these examples, I will discuss how Weyl algebras have served as a starting point for one avenue in my recent research.

In this work, we study singular solutions and pattern formation in aggregation swarming models in two dimensions. This class of models involve pairwise interactions and an active scalar equation in the continuum limit. We show the connection between this model and the classical vorticity equation from fluid dynamics. The aggregation model can lead to a rich family of patterns. We discuss the stability of the singular patterns formed with this model.

Geometry and physics have been, in recent times, a great inspiration for mathematical problems. It is therefore useful to consider numerical methods that pay special attention to various invariant geometric structures, and minimize dependence on choices of coordinate systems. Attention to this aspect can lead to better stability and qualitative behavior. One important tool we have in capturing geometric structure is differential forms; many common differential equations find their most natural expressions in terms of forms. The Finite Element Exterior Calculus (FEEC) provides a framework for discretizing differential forms as finite elements. We present examples of how FEEC recasts problems into a more geometric form, and describe generalization to hyperbolic problems, by, specifically, application of FEEC to solving Maxwell's equations. We describe a choice of discretization (Whitney Forms) and possible generalizations and their issues.

We consider the problem of finding a hidden clique in a random graph. This problem was studied by Alon, Krivelevich and Sudakov in 1998. Using Matlab, we wrote two algorithms that are designed to find a hidden clique. One of the algorithms was suggested by the work of Alon, Krivelevich and Sudakov. The other algorithm is a slight variation that seems to perform better in our experiments.

For a graph $G$, the bipartition number $\tau(G)$ is the minimum number of complete bipartite subgraphs whose edge sets partition the edge set of $G$. In 1971, Graham and Pollak proved that $\tau(K_n)=n-1$. Since then, there have been a number of short proofs for Graham-Pollak theorem by using linear algebra or by using matrix enumeration. We present a purely combinatorial proof for Graham-Pollak theorem. For a graph $G$ with $n$ vertices, one can show $\tau(G) \leq n- \alpha(G)$ easily, where $\alpha(G)$ is the independence number of $G$. Erd\H{o}s conjectured that almost all graphs $G$ satisfy $\tau(G)=n-\alpha(G)$. In this talk, we prove upper and lower bounds for $G(n,p)$ which gives support for Erd\H{o}s' conjecture. Joint work with Fan Chung.

Cryptography has a long history, dating back to the first century BC.
The twentieth century saw huge advances in cryptography as systems for
rapid long-distance communications developed, to the point where now
cryptographic practices are used on a daily basis for commonplace
activities such as checking email. In this talk I will give a brief
survey of mathematics that go into the modern practice of cryptography,
in both designing and attacking cryptographic systems.

Understanding solid liquid phase transition is of great importance in many applications starting from growth of nano-crystals in solutions for solar cells to making the perfect ice cream/slushy. This forms the basis for understanding more complex phenomena such as solvation and self assembly in active matter systems.The element of interest in this talk is to understand the role and the effect of the fluid flow on the phase transition. The same effect that prevents the ice cream or slushy from freezing into a solid block.This talk will consider a system of particles that interact through a pair potential and choose a revised Enskog kinetic theory to describe the time evolution. Using a generalized Chapman-Enskog procedure non-local hydrodynamics that take the form of a density functional theory will be derived. A numerical study on the effect of melt flow on the freezing transition will be discussed in detail. Some potential applications to self assembly in active matter systems and solvation will also be outlined.
This work was done in collaboration with John Lowengrub and Aparna Baskaran.

I will talk on some fundamental problems in the representation theory of Lie superalgebras, more particularly about so-called exceptional Lie superalgebras. I will give basic definitions, some history, and explain details.

A Path to the Numerical Discretization of General Relativity

Gauge symmetries play a central role in every physical theory, from fluid dynamics to general relativity (GR). Unfortunately, their role both within theoretical and numerical applications is often overlooked, as one must pick a gauge in order to proceed with a well-defined problem in hand. As a result, most discretization schemes choose a gauge and hence fail to preserve the full symmetry structure of the Lagrangian. This problem is perhaps the most egregious within the framework of GR, as the gauge symmetry of the theory is the group of diffeomorphisms of spacetime; in other words, spacetime coordinate transformations lead to physically indistinguishable dynamics.

In this talk, I will showcase the first baby steps that have been made towards a geometric, symmetry-preserving discretization scheme for GR. I will present an extension of finite element exterior calculus (FEEC) to flat, pseudo-Riemannian manifolds of arbitrary dimension. In particular, I will apply this framework to electromagnetism (EM) in Minkowski spacetime, and highlight the automatic imposition of a self-consistent gauge condition at first order. Furthermore, using the homothetic subdivision and refinement scheme given by Rapetti and Bossavit, I will also demonstrate that this scheme can be extended to arbitrary order. Within the context of EM, this leads to a much richer approximation to the full gauge group.

A quiver is a directed graph, that is, a collection of points and arrows pointing from some points to others. A representation of the quiver is a way of assigning a vector space to each point and a linear transformation to each arrow. We discuss what it means for two representations to be isomorphic. We then show how the theory of representations of quivers captures many of the basic ideas of linear algebra such as similarity and equivalence of matrices. If time permits, we will discuss Gabriel's theorem, one of the most fundamental results in the subject, which classifies those quivers of finite representation type.

I'll present a general theory of "tropical schemes", or skeleta, which is based on reversing the correspondence that associates to a "tropical object" its semiring of convex piecewise-affine functions; in this way, we can understand, and generalise, certain classical constructions using semiring theory. For example, we can recover the dual intersection complexes of degenerations of varieties as skeleta. This construction is naturally intertwined with the theory of non-Archimedean geometry.

Recently Ikavi has studied the evolution of a the polar dual of a convex body evolving by curvature flows. The technique has been used by Stancu and Ivaki previously in studying curvature flows are their relation to affine inequalities. I will describe how Ivaki shows the evolution of the polar dual is very similar to the evolution of the original body, evolving by an expanding Gauss curvature flow. Using techniques that go back to Tso and then later Andrews, an upper curvature bound is obtained for the original body. The novel part of this paper is that essentially the same technique yields an upper curvature bound for the dual body, which by duality corresponds to a lower curvature bound for the original body. Convergence results are then obtained in a fairly standard manner.

Tropical food will be provided.

We consider the so-called Drinfeld setting, a function field analogue of some aspects of the theory of modular forms, modular curves and elliptic curves. In this setting Drinfeld constructed families of modular curves defined over a complete, algebraically closed field of characteristic $p.$ We are interested in studying their Weierstrass points, a finite set of points of geometric interest. In this talk we will present some tools from the theory of Drinfeld modular forms that were developed to further this study, some geometric and analytic considerations, and some partial results towards computing the image of these points modulo a prime ideal of the base ring.

In this talk, we give a mathematical derivation of a pathway- based mean-field model for E. coli chemotaxis based on the moment closure in kinetic theory. The pathway based model incorporate the most recent intracellular chemical dynamics. The derived moment system, under some assumptions, gets to the chemotaxis model pro- posed in [G. Si, T. Wu, Q. Quyang and Y. Tu, Phys. Rev. Lett., 109 (2012), 048101], especially an important physical assumption made in which can be understood explicitly in this new moment system. We obtain the Keller-Segal limit by considering the moment system in the regime of long time and strong tumbling rate. Numerical experiments are presented to show the agreement of the moment system with (individual based) signaling pathway- based E. coli chemotaxis simulator ([L. Jiang, Q. Ouyang and Y. Tu, PLoS Comput. Biol., 6 (2010), e1000735]).

Abstract: Brownian motion is continuous random motion, discovered by early 19th Century botanist Robert Brown, studied by Albert Einstein in one of the three 1905 papers that led to his Nobel prize, and finally put on firm mathematical footing by Norbert Wiener in the 1920s. It is intimately tied to local and global geometry, and is an important tool in studying heat flow on more general manifolds.
In this talk, I will give an overview of some results on Brownian motion on classical Lie groups, focusing on unitary groups $U_N$ and general linear groups $GL_N$. I will discuss my recent work on the large-N limit of Brownian motions on these groups, their fluctuations, and applications to random matrix theory and operator algebras.

In this talk we discuss the proof of a conjecture made by Ron Evans on the relationship between the eighth symmetric moment of the classical Kloosterman sum and the Fourier coefficients of the newform of level 6 and weight 6. Rather than delve deeply into any particular technical part of the proof, we will emphasize the fascinating way in which ideas from analysis, geometry, Galois representations, automorphic forms, and combinatorics interact in the course of the proof.

Traditionally, research focusing on the design of scheduling and
staffing policies for service systems has modeled servers as having
fixed (possibly heterogeneous) service rates. However, service systems
are often staffed by people. Then, the rate a server chooses to work may
be impacted by the scheduling and staffing policies used by the system.
We present a model for such "strategic servers" that choose their service
rate in order to maximize a trade-off between an "effort cost", which
captures the idea that servers exert more effort when working at a faster rate,
and a "value of idleness", which assumes that servers prefer to be idle as much as possible. In this strategic server framework, we re-visit classic scheduling and staffing questions in many-server systems, and, in particular, we investigate the performance of the common square-root safety staffing rule.

Based on joint work of
S. Doroudi, R. Gopalakrishnan, A. R. Ward (Presenter), and A. Wierman

The harmonic oscillator problem in quantum mechanics is to find
operators $a$ and $b$ acting on a Hilbert space satisfying the relation
$ab-ba=1$. This is one of the physical motivations behind studying the
Weylalgebra and the enveloping algebra of the Heisenberg Lie algebra.
In this talk, I will present a two-parameter version of this problem and discuss some of the subtleties in looking for simple, primitive factor rings in quantum enveloping algebras.

We discuss how randomization and rank structures can be used in the direct solution of some large dense or sparse linear systems with nearly $O(n)$ complexity and storage. Randomized sampling and related adaptive strategies help significantly improve both the efficiency and the flexibility of structured solution. We also demonstrate how these can be extended to the development of matrix-free direct solvers based on matrix-vector products only. This is especially interesting for problems with few varying parameters (e.g., frequency or shift). We show that such structured solvers also have significantly better stability than classical LU factorizations. They are then very suitable for ill-conditioned problems and can provide solutions with controllable accuracies. Applications to some difficult situations will be demonstrated and the effectiveness will be justified: 1. preconditioning certain indefinite problems where only matrix-vector products are available; 2. solving ill-conditioned Toeplitz least squares problems; 3. finding good initial estimates for iterative (e.g., Newton) eigenvalue solutions.

I will talk about the problem of separating multiple signals from each other when we only have access to a few linear (or non-linear) combinations of them. An example of this type of problem is at a cocktail party when you are trying to have a conversation with a friend but there are several conversations happening around you. Your ears provide you with a superposition of all the voices, and your brain does remarkably well at focusing on your friend's voice and drowning out all the others. We will talk about one computer algorithm (or time permitting, more) that does such a task (reasonably) successfully. Along the way, we will talk about important tools in mathematical signal processing, including the Fourier transform and sparsity.

A shortest remaining processing time (SRPT) queue is a single server queue in
which the server dedicates its effort to the job that has the least remaining
processing time among all jobs in the system. This is done with preemption
so that when the total processing time of an arriving job is less than that
remaining for the job in service, service of the job in service is paused and
the newly arriving job enters service. Shortest remaining processing time is of interest because it is performance optimal in the sense that over all non-idling service disciplines it minimizes queue length. However, this may come at the expense of long delays for jobs with large total processing times. From a mathematical point of view, SRPT is challenging to analyze, in part because the preemptive nature of SRPT leads to an infinite dimensional state space. By formulating and analyzing a probabilistic model for SRPT that employs measure-valued processes, we investigate this performance trade off. The talk will begin with a discussion of a fluid limit (first order approximation) for the stochastic model. This will yield some interesting insights about the anticipated performance trade-off. Next work in progress on a diffusion limit (second order approximation) will be discussed. Various parts of this work are joint with D. Down (McMaster University), H. C. Gromoll (U Virginia), and L. Kruk (Maria Curie-Sklodowska University)

General Relativity (GR) suffers, in several ways -- from source modeling to
data analysis--, from the "curse of dimensionality", by which it is here
roughly meant that the complexity of the system grows beyond practical
control as more physical parameters of interest are taken into account.

This is a very concerning, practical bottleneck for the upcoming generation
of advanced gravitational wave detectors, worth billions of dollars, which
is expected to detect within a few years gravitational waves in a direct
way for the first time in history and reach unexplored portions of the
universe. This is the most anticipated era of general relativity since the
study of Hulse and Taylor which lead to their Nobel prize in 1993.

Due to the low signal-to-noise ratio of any expected detection, matched
filtering and catalogs of templates are needed, both for detection and
parameter estimation of the source of any trigger.

Numerical relativity simulations of the Einstein equations typically take
hundreds of thousands of hours, making a survey of the full parameter space
intractable with standard search methods. Parameter estimation algorithms
are just impractical, would take years of computing time.

I will first summarize previous work I have involved in and the
state-of-the-art of Einstein's equations as an initial-boundary value
problem, both at the continuum and discrete level. Next I will discuss my
current research program, in collaboration with many colleagues, to tackle
the curse of dimensionality in GR, existing results, and plans for the
future. The effort is essentially about dealing with parametrized systems
using reduced order models (ROM), and the techniques are generic and
applicable to many areas.

In our driving field, general relativity, we typically obtain several (from
3 to 11) orders of magnitude of computational speedup compared to direct
approaches, both on the modeling and data analysis sides. As an example, we
can now substitute a typical months-long supercomputer simulation of
colliding black holes with a surrogate model that can run on a smartphone
within a tenth of a millisecond without loss of accuracy. Real time
calculations on mobile devices not only provide a huge opportunity of
outreach about complicated systems such as colliding black holes, but also
application-specific ones in many areas, such as on-site or remote design
and control.

The effort is a combination of theoretical physics, analytical and
numerical methods for partial differential equations, scientific computing,
large scale computations, reduced order modeling, approximation theory,
sparse representations and signal processing.

It is well know that water-mediated interactions play critical roles in biomolecular recognition processes. Decades of theoretical and experimental studies showed us a very complicated picture. No single reliable model can provide simple and consistent descriptions to its role in kinetics, thermodynamic, and structural characterizations yet.

However, a joint explicit solvent molecular dynamics (MD) simulations and the variational implicit-solvent model (VISM) may still provide semi-quantitative insight into these complicated heterogeneous hydration, the solute-solvent interface, and individual interesting water molecules around proteins.

In this study, we used this combination approach to study the hydration properties of the biologically important p53/MDM2 complex. Unlike simple model solutes, in such a realistic and heterogeneous solute-solvent system with both geometrical and chemical complexity, it occurs that the local water distribution sensitively depends on nearby amino acid properties and the geometric shape of the protein. We show that the VISM can accurately describe the locations of high and low density solvation shells identified by the MD simulations, and can explain them by a local coupling balance of solvent-solute interaction potentials and curvature. In particular, capillary transitions between local dry and wet hydration states in the binding pocket are captured for inter-domain distance between 4 to 6 � right at the onset of binding. The underlying physical connection between geometry and polarity are illustrated and quantified. Our study offers a microscopic and physical insight into the heterogeneous hydration behavior of the biologically highly relevant p53/MDM2 system and demonstrates the fundamental importance of hydrophobic effects for biological binding processes. We hope this study may help to establish new design rules for drugs and medical substances.

A character on a group is a class function of positive type. For finite groups, the classification of characters is directly connected to the representation theory of the group and plays a key role in the classification of finite simple groups. Based on the rigidity results of Mostow, Margulis, and Zimmer, it was conjectured by Connes that for lattices in higher rank Lie groups the space of characters should be completely determined by the finite dimensional representations of the lattice. In this talk, I will give an introduction to this conjecture (which has now been solved in a number of cases), and I will discuss its relationship to ergodic theory, abstract harmonic analysis, invariant random subgroups, and von Neumann algebras.

Extracting knowledge and providing insights into the complex
mechanisms underlying noisy high-dimensional data sets is of utmost
importance in many scientific domains. Networks are an example of
simple, yet powerful tools for capturing relationships among entities
over time. For example, in social media, networks represent
connections between different individuals and the type of interaction
that two individuals have. In systems biology, networks can represent
the complex regulatory circuitry that controls cell behavior.
Unfortunately the relationships between entities are not always
observable and need to be inferred from nodal measurements.

I will present a line of work that deals with the estimation of
high-dimensional dynamic networks from limited amounts of data. The
framework of probabilistic graphical models is used to develop
semiparametric models that are flexible enough to capture the dynamics
of network changes while, at the same time, are as interpretable as
parametric models. In this framework, estimating the structure of the
graphical model results in a deep understanding of the underlying
network as it evolves over time. I will present a few computationally
efficient estimation procedures tailored to different situations and
provide statistical guarantees about the procedures. Finally, I will
demonstrate how dynamic networks can be used to explore real world
systems.

An infinite dimensional Lie algebra is locally finite if every
finitely generated subalgebra is finite dimensional. On one extreme are the
simple, locally finite Lie algebras. We provide structure theorems which
describe such algebras over fields of positive characteristic. On the other
extreme are the maximal, locally solvable Lie algebras, which are Borel
subalgebras. We provide a theorem which shows that such Lie algebras are
stabilizers of maximal, generalized flags, which is a generalization of
Lie's theorem. We will finish by describing some new directions in the
study of these Lie algebras.

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.

We consider a random model for directed graphs whereby an arc is placed from one vertex to another with a prescribed probability
$p_{ij}$ which may vary from arc to arc. Using perturbation bounds as well as Chernoff inequalities, we show that the stationary
distribution of a Markov process on a random graph is concentratednear that of the “expected” process under mild conditions. These
conditions involve the ratio between the minimum and maximum in- and out-degrees, the ratio of the minimum and maximum entry in the
stationary distribution, and the smallest singular value of the transition matrix. Lastly, we give examples of applications of our
results to known models of directed graphs.

I will talk about work in progress joint with Yuguang Zhang,
trying to relate Boucksom et al's solution to the non-archimedean Monge-Ampere
equation on K3 surfaces to the real Monge-Ampere equation on the skeleton.

In this talk, we will first introduce some basic fact on the Maximal
slice and its brief history. we will also try to review some existence
and non-existence result in Minkowski space. Then, we will show that
the Bernstein Theorem in ADS space fails.

I give a bad math talk. Over the past several years, I have kept a list of all of the bad things speakers have done during their talks, and I will incorporate them into a single horrible talk and discuss each of the bad aspects afterwards. In the end, it will be entertaining and informative; and hence, not such a bad math talk after all.

The integers do not admit any nontrivial derivations. We will explain
how the operation $x \to$ ${x-x^p \over p}$ can be thought of as a replacement for
the derivative operator on the integers. After the introduction we hope
to explain some recent work on the meaning of "linearity" in this theory.

I will begin with an introduction to the subfactor
classification program, which has two main focuses: restricting the
list of possible principal graphs, and constructing examples when the
graphs survive known obstructions. I will discuss recent joint work
with Liu and Morrison which classifies 1-supertransitive subfactors
without intermediates with index in $(3+\sqrt{5},6.2)$. We show there
are exactly 3 examples corresponding to a BMW algebra and two
"twisted" variations.

Fano varieties are in some sense the simplest type of
algebraic varieties. They are the algebraic analogue of manifolds
with positive curvature, such as spheres. In low dimensions one can
classify Fano varieties (where for an algebraic geometer low means up
to dimension three) and as the dimension increases, they form bounded
families, so that one can in principle classify Fano varieties in all
dimensions.

In this talk we explain some of the known and conjectured results,
both for an explicit classification, and for some of the boundedness
results.

Gauge functions significantly generalize the notion of a norm, and gauge optimization is the class of problems for finding the element of a convex set that is minimal with respect to a gauge. These conceptually simple problems appear in a remarkable array of applications. Their gauge structure allows for a special kind of duality framework that may lead to new algorithmic approaches. I will illustrate these ideas with applications in sparse signal recovery.

Consider the question of how a derivatives security (e.g., a call option) should be priced. Due to their potential utility for hedging and speculation, economists started becoming interested in this question in the early 1900s. In the late 1960s, this emerged as an important concern for finance markets as well, due to a relaxation in regulations that allowed insurance companies and banks to invest in derivatives. After decades of growing interest and research efforts, Fisher Black and Myron Scholes developed a mathematically based pricing strategy in the early 1970s that was later simplified and expanded on by Robert Merton. The resulting formulas revolutionized trading practices worldwide. In 1997, Merton and Scholes received the Nobel prize for this body of work (Black had passed away and therefore was ineligible).

In the talk, we will examine an extremely simplified version of this body of work. In particular, we will analyze the single-period Cox-Ross-Rubenstein (CRR) model using some of the key principles and methodologies that Black, Scholes, and Merton developed to construct their derivatives pricing theory. Despite the fact that this model is very simple, the analysis will illustrate certain essential aspects of their Nobel prize winning work such as dynamic hedging and the risk neutral probability measure. This will give a flavor of the mathematical content of Math 194, being offered in the winter quarter.

We exhibit a numerical method to compute a three-point branched covers
of the complex projective line. We develop algorithms for working
explicitly with Fuchsian triangle groups and their finite index
subgroups, and we use these algorithms to compute power series
expansions of modular forms on these groups. As one application, we
find an explicit rational function of degree 50 which regularly
realizes the group $PSU_3(5)$ as a Galois group over the rationals.
This is joint work with Michael Klug, Michael Musty, and Sam
Schiavone.

In this talk, we summarize the compact coupled interface method and introduce modifications for handling a wider class of Poisson-type problems, as well as comparisons to the coupled interface method and immersed interface method. We pay special attention to the Poisson-Boltzmann equation and its role in electrostatic effects specifically for implicit solvation.

The Langlands-Shahidi method provides us with a constructive
way of studying automorphic L-functions. For the classical groups over
function fields we will present recent results that allow us to obtain
applications towards global Langlands functoriality. This is done via
the Converse Theorem of Piatetski-Shapiro, which we can apply since our
automorphic L-functions have meromorphic continuation to rational
functions and satisfy a functional equation. We lift globally generic
cuspidal automorphic representations of a classical group to an
appropriate general linear group. Then, we express the image of
functoriality as an isobaric sum of cuspidal automorphic representations
of general linear groups, where the symmetric and exterior square
automorphic L-functions play a technical role. As a consequence, we can
use the exact Ramanujan bounds of Laurent Lafforgue for GL(N) to prove
the Ramanujan conjecture for the classical groups. Our results are
currently complete for the split classical groups under the assumption
that characteristic p is different than two.

I will give a brief introduction to Hodge theory and discuss Hodge-theoretic problems involving abelian varieties.

For a quiver with potential, we can associate a vanishing cycle to each representation space. If there is a nice torus action on the potential, the vanishing cycles can be expressed in terms of truncated Jacobian algebras. We study how these vanishing cycles change under the mutation of Derksen-Weyman-Zelevinsky. We use Ringel-Hall algebras as the main organizing tools. The wall-crossing formula leads to a categorification of quantum cluster algebras under the assumption of existence of certain potential. This is a special case of A. Efimov's result, but our approach is more concrete and down-to-earth. We also obtain a formula relating the representation Grassmannians under sink-source reflections. In this talk, I will start with basic definitions of mutations of quivers with potentials, vanishing cycles, and Hall algebras.

We study the problem of finding best rank-1 approximations for both
symmetric and nonsymmetric tensors. For symmetric tensors, this is
equivalent to optimizing homogeneous polynomials over unit spheres; for
nonsymmetric tensors, this is equivalent to optimizing multi-quadratic
forms over multi-spheres. We propose semidefinite relaxations, based on
sum of squares representations, to solve these polynomial optimization
problems. Some numerical experiments are presented to show that this
approach is practical in getting best rank-1 approximations.

\def\RR{{\mathbb R}}

Where do polynomial inequalities come from? We have polynomials $p$ and $q$ acting on tuples $x$ of numbers in $\RR^n$. Suppose $p(x) > 0$ for all $x$ in $\RR^n$ making $q(x) > 0$. Is there some algebraic relationship between $p,q$ equivalent to this? That is a lot to hope for, but an ``algebraic certificate'' equivalent to an inequality often exists and this is the substance of much of real algebraic geometry (RAG). It is a subject which bloomed in the last 50 years.

Now consider $n$ tuples $X:= \{ X_1, \dots, X_n \}$ of symmetric matrices $X_j$ and polynomials $p$ and $q$ acting on such tuples, for example, $n=2$ and $$p(X):= X_1 {X_2}^3 + {X_2}^3 X_1 + {X_1}^5.$$ The polynomial yields a value $p(X)$ that is a symmetric matrix, and we can consider the same issues as in classical RAG. We have polynomials $p$ and $q$. Suppose $p(X)$ is positive definite for all $X$, making $q(X)$ a positive definite matrix. Recall a positive definite matrix is one whose eigenvalues are all $>0$. A theory parallel to RAG for converting such inequalities to algebra formulas has emerged in the last ten years. The motivation came from problems in systems engineering but spread out from this in many directions.

The talk will give a taste of selections from this smorgasboard.

We'll discuss various constraints on rational cuspidal curves in the projective plane, and consider the question of whether such curves are necessarily equivalent to a line, under a birational automorphism of the plane.

This is the first in a series of seven Food for Thought talks on the Millennium Prize Problems. In this talk we give an introduction to the Clay Mathematics Institute and its Millennium Prize Problems program. We then focus our attention on the question of whether \textbf{P}, the class of problems with polynomial-time algorithms, equals \textbf{NP}, the class of problems with polynomial-time nondeterministic algorithms. We will define \textbf{P} and \textbf{NP} in terms of Turing machines. We will then move on to some diverse contemporary approaches to proving either \textbf{P} = \textbf{NP} or \textbf{P} $\neq$ \textbf{NP}, and discuss the implications that such a theorem would have. Along the way, we will see some famous results that have appeared out of the struggle to solve this hallmark riddle of theoretical computer science.

Biological processes are carried out through molecular conformational transitions, ranging from the structural changes within biomolecules to the formation of macromolecular complexes and the associations between the complexes themselves. These transitions cover a vast range of timescales and are governed by a tangled network of molecular interactions. The resulting hierarchy of interactions, in turn, becomes encoded in the experimentally measurable “mechanical fingerprints” of the biomolecules, their force–extension curves. However, how can we decode these fingerprints so that they reveal the kinetic barriers and the associated timescales of a biological process? Here, we show that this can be accomplished with a simple, model-free transformation that is general enough to be applicable to molecular interactions involving an arbitrarily large number of kinetic barriers. Specifically, the transformation converts the mechanical fingerprints of the system directly into a map of force-dependent rate constants. This map reveals the kinetics of the multitude of rate processes in the system beyond what is typically accessible to direct measurements. With the contributions from individual barriers to the interaction network now “untangled”, the map is straightforward to analyze in terms of the prominent barriers and timescales. Practical implementation of the transformation is illustrated with simulated biomolecular interactions that comprise different patterns of complexity—from a cascade of activation barriers to competing dissociation pathways.

Some central limit results on stochastic processes in a compact connected 2-point homogeneous space $E(d)$ of growing dimension $d$ are reformulated within the theory of polynomial convolution structures. This approach stresses the algebraic-topological relationship between those structures and the asymptotic properties of the stochastic processes under consideration, in particular of random walks and Gaussian processes on $E(d)$ with $d\to\infty$.

Superconformal algebras are superextensions of the Virasoro algebra. They play an important role in the string theory and conformal field theory.

Central extensions of contact superalgebras $K(2)$ and $K'(4)$ are known to physicists as the $N = 2$ and the big $N = 4$ superconformal algebras. A remarkable property of $\widehat{K}'(4)$ and the exceptional superconformal algebra $CK_6$ is that they admit embeddings into the Lie superalgebras of pseudodifferential symbols on the circle, extended by
$N = 2, 3$ odd variables. Associated to these embeddings, there are ``small" irreducible representations of these superalgebras and their realizations in matrices of size $2^N$ over a Weyl algebra. The general construction of such matrix realizations is connected with the spin
representation of $\mathfrak{o}(2N + 1, \mathbb{C})$.

We also obtain a realization of the family of simple exceptional finite-dimensional Lie superalgebras $D(2; 1; \alpha)$, related to $K(4)$ in matrices over a Weyl algebra.

In 1643, Ren\'{e} Descartes discovered a formula relating curvatures of circles in Apollonian circle packings, constructed by Apollonius of Perga in 200 BC. This formula has recently led to a connection between the construction of Apollonius and orbits of a certain so-called \emph{thin} subgroup $\Gamma$ of $\textrm{GL}_4(\mathbb Z)$. This connection is key in recent results on the arithmetic of Apollonian packings, which I will describe in this talk. A crucial ingredient in the proofs is the spectral gap coming from families of expander graphs
associated to $\Gamma$ -- this gap is far less understood in the case of thin groups than that of non-thin groups. Motivated by this problem, I will then discuss the ubiquity of thin groups and present results on thinness of monodromy groups of hypergeometric equations in the case where these groups act on hyperbolic space.

The Finite Element Method is a powerful tool for approximating the solutions of a large class of PDE's. Traditional FEM requires the function spaces to be linear. We discuss an extension of FEM, Geodesic Finite Elements, which allows the functions being approximated to have range in a non-linear Riemannian Manifold. We realize the set of pseudo Riemannian metrics as a symmetric space in order to pull back the matrix exponential onto it, thus endowing it with a Riemannian metric. Once done, we are able to discretize a class of pseudo-Riemannian manifolds. This has possible applications in non-linear hyperbolic PDE's.

In recent years, it has become interesting from a number-theoretic point of view to be able to determine whether a finitely generated subgroup of $GL_n(\mathbb Z)$ is a so-called thin group. In general, little is known as to how to approach this question. In this talk we discuss this question in the case of hypergeometric monodromy groups, which were studied in detail by Beukers and Heckman in 1989. We will convey what is known, explain some of the difficulties in answering the thinness question, and show how one can successfully answer it in many cases where the group in question acts on hyperbolic space. This work is joint with Meiri and Sarnak.

The Global Positioning System involves about 24 satellites
orbiting the earth each sending a unique signal. Your GPS receiver
receives the signal from at least 4 of them and from these signals it
tells you where you are within 10 meters (if your GPS can handle
WAAS the accuracy is 3 meters). The calculations done by your
receiver involve general relativity, error correction and geometry.

Abstract: In game theory, an Evolutionarily Stable Set (ES set) is a set of Nash Equilibrium (NE) strategies that give the same payoffs. Similar to an Evolutionarily Stable Strategy (ES strategy), an ES set is also astrict NE. This work investigates the evolutionary stability of classical and quantum strategies in the quantum penny flip games. In particular, we developed an evolutionary game theory model to conduct a series of simulations where a population of mixed classical strategies from the ES set of the game were invaded by quantum strategies. We found that when only one of the two players� mixed classical strategies were invaded, the results were different. In one case, due to the interference phenomenon of superposition, quantum strategies provided more payoff, hence successfully replaced the mixed classical strategies in the ES set. In the other case, the mixed classical strategies were able to sustain the invasion of quantum strategies and remained in the ES set. Moreover, when both players� mixed classical strategies were invaded by quantum strategies, a new quantum ES set was emerged. The strategies in the quantum ES set give both players payoff 0, which is the same as the payoff of the strategies in the mixed classical ES set of this game.

Mirror symmetry, notably in the work of Mark Gross and Berndt Siebert, identifies involutory "mirror pairs" (X,Y) of (degenerating, polarized)
compact Calabi-Yau manifolds and makes predictions relating the symplectic
geometry of X with the algebraic geometry of Y. Those predictions range
from those of the "open string sector", where homological mirror symmetry
(HMS) relates the Lagrangian submanifolds of X to algebraic vector bundles
on Y, to those of the "closed string sector" where, for example, counts of
holomorphic spheres in X are predicted to equal certain period integrals on Y.
I'll report recent work, joint with Nick Sheridan, which says the following: if one can prove a certain fragment of HMS (a fragment which
we expect to fit neatly into the Gross-Siebert program) then, without
knowing anything more about the geometry of X and Y, one can deduce (i) the
full statement of HMS; and (ii) certain algebraic and enumerative claims
from closed-string mirror symmetry.

The product replacement graph (PRG) of a group G is the set of generating k-tuples of G, with edges corresponding to Nielsen moves. It is conjectured that PRGs of infinite groups are nonamenable. We verify that PRGs have exponential growth when G has polynomial growth or exponential growth, and show that this also holds for a group of intermediate growth: the Grigorchuk group. We also provide some sufficient conditions for nonamenability of the PRG, which cover elementary amenable groups, linear groups, and hyperbolic groups.

***Integral equation methods for singular problems and application to the evaluation of Laplace eigenvalues***

Boundary integral equation methods represent a powerful tool for the numerical solution of a variety of problems in acoustics, electromagnetics, fluid mechanics etc. Together with Oscar Bruno at Caltech, we developed a numerical method for the solution of scattering problems with mixed Dirichlet/Neumann boundary conditions. We investigated and obtained the exact form of the solution singularities -- which arise at transition points where Dirichlet and Neumann boundary conditions meet. These singularities are incorporated in the numerical approach and are resolved via Fourier Continuation technique. The resulting method exhibits spectral convergence. Additionally, jointly with Nilima Nigam at SFU, we applied the mixed boundary value solver to Laplace eigenvalue problems. The challenge is in eigenvalue search as a result of the properties of the objective function, which requires scanning through a range of frequencies. We introduce an improved search algorithm, that allows to locate the eigenvalues using standard root-finding methods. We also apply another integral equation method for domains with corners for a mode matching problem in electromagnetics. Jointly with Ahmed Akgiray at Caltech, we calculated TE and TM modes for domains of specific geometry. Those are further used for antenna design.

A Sidon set is a set of integers such that no difference of two
distinct numbers in the set equals the difference of any other two
numbers in the set. Constructions of large Sidon sets have
interesting connections to geometry and number theory. I will show
how to construct Sidon sets and discuss these connections.

Genus one curves, defined over the rationals, need not have rational points.
The set of all such curves, whose Jacobian is a fixed elliptic curve E, form
a group, called the Weil-Chatelet group. It has an important subgroup, the
Tate-Shafarevich group, formed by those curves which have points over all
completions of the rationals.

This talk will address two aspects of the arithmetic of genus one curves:
(1) (with J. Stix) the divisibility of the Tate-Shafarevich group inside
the Weil-Chatelet group; (2) (with A. Wiles) work in progress on the existence
of points defined over number fields with solvable Galois group over the
rationals. Earlier work, also with Wiles, proved existence when the curve
represents an element of the Tate-Shafarevich group; we now aim to extend
this to the whole Weil-Chatelet group.

More than 25 years ago, Kenig, Ruiz, and Sogge established uniform $L^p$ resolvent estimates for the Laplacian in the Euclidean space. Taking their remarkable estimate as a starting point, we shall describe more recent developments concerned with the problem of controlling the resolvent of elliptic self-adjoint operators in $L^p$ spaces in the context of a compact Riemannian manifold. Here some new interesting difficulties arise, related to the distribution of eigenvalues of such operators. Applications to inverse boundary problems and to the absolute continuity of spectra for periodic Schr\"odinger operators will be presented as well."

In a report published in 1937, Doeblin - who is regarded the father of the "coupling method" - introduced an ergodicity coefficient that provided the first necessary and sufficient condition for the weak ergodicity of non-homogeneous Markov chains over finite state spaces.

In today's jargon, Doeblin's coefficient corresponds to the maximal coupling of the probability transition kernels associated with a Markov chain, and the Monte Carlo literature has (often implicitly) used it to draw perfectly from the stationary distribution of a homogeneous Markov chain over a Polish state space.

In this talk, I will show how Doeblin's coefficient can be used to approximate the distribution of additive functionals of homogeneous Markov chains, particularly sojourn-times, instead of characterizing asymptotic objects such as stationary distributions. The methodology leads to easy to compute and explicit error bounds in total variation distance, and gives access to approximations in Markov chains that are too long for exact calculation but also too short to rely on Normal approximations or stationary assumptions underlying Poisson approximations.

``Schubert calculus'' is the (somewhat misleading) name given to the classical branch of enumerative geometry which counts intersections of certain simple varieties like points, lines, and planes. Rather than actually doing any geometry, we'll find these intersection numbers with a combinatorial rule known as the Littlewood-Richardson rule. In particular, we'll show why the problem of counting the number of lines in $\mathbb{C}^3$ that intersect four generic lines is now considered to be ``just combinatorics.''

This is a report on joint work with Ana Caraiani, Matthew Emerton, Toby Gee, David Geraghty, and Vytautas Paskunas. We will explain some ideas around the global construction of new representation-theoretic objects called "patched modules" by a variant of the Taylor-Wiles-Kisin method. They are in many ways better suited than p-adically completed cohomology for a global attempt to understand the p-adic local Langlands correspondence. As an application, we obtain new cases of the Breuil-Schneider conjecture.

We extend a model for the morphology and dynamics of a crawling eukaryotic cell to describe cells on micropatterned substrates. This model couples cell morphology, adhesion, and cytoskeletal flow in response to active stresses induced by actin and myosin. We propose that protrusive stresses are only generated where the cell adheres, leading to the cell's effective confinement to the pattern. Consistent with experimental results, simulated cells exhibit a broad range of behaviors, including steady motion, turning, bipedal motion, and periodic migration, in which the cell crawls persistently in one direction before reversing periodically. We show that periodic motion emerges naturally from the coupling of cell polarization to cell shape by reducing the model to a simplified one-dimensional form that can be understood analytically. Additionally, we will discuss a turning instability arising from our model applying onto a free moving cell without interaction with the micropatterned substrates. Some attempts have made to test how the instability depends on the parameters in the model numerically. For a much simplified model, we do find that surface tension is a key factor to stabilize the cell turning.

Algorithms have two costs: arithmetic and communication, i.e. moving data between levels of a memory hierarchy or processors over a network. Communication costs (measured in time or energy per operation) already greatly exceed arithmetic costs, and the gap is growing over time following technological trends. Thus our goal is to design algorithms that minimize communication. We present algorithms that attain provable lower bounds on communication, and show large speedups compared to their conventional counterparts. These algorithms are for direct and iterative linear algebra, for dense and sparse matrices, as well as direct n-body simulations. Several of these algorithms exhibit perfect strong scaling, in both time and energy: run time (resp. energy) for a fixed problem size drops proportionally to the number of processors p (resp. is independent of p). Finally, we describe extensions to algorithms involving arbitrary loop nests and array accesses, assuming only that array subscripts are affine functions of the loop indices.

Suppose that G is a connected reductive p-adic group. We will
describe the classification of irreducible admissible smooth mod p
representations of G in terms of supercuspidal representations. This is
joint work with N. Abe, G. Henniart, and M.-F. Vigneras.

The study of mean curvature flow not only is fundamental in geometry, topology and analysis, but also has important applications in applied mathematics, for instance, image processing. One of the most important problems in mean curvature flow is to understand the possible singularities of the flow and self-shrinkers, i.e., self-shrinking solutions of the flow, provide the singularity models.

In this talk, I will describe the rigidity of asymptotic structures of self-shrinkers. First, I show the uniqueness of properly embedded self-shrinkers asymptotic to any given regular cone. Next, I give a partial affirmative answer to a conjecture of Ilmanen under an infinite order asymptotic assumption, which asserts that the only two-dimensional properly embedded self-shrinker asymptotic to a cylinder along some end is itself the cylinder. The feature of our results is that no completeness of self-shrinkers is required.

The key ingredients in the proof are a novel reduction of unique continuation for elliptic operators to backwards uniqueness for parabolic operators and the Carleman type techniques. If time permits, I will discuss some applications of our approach to shrinking solitons of Ricci flow.

In this talk I will survey current results on the long-time existence and behavior of Ricci flows in dimensions 2, 3 and higher. Moreover, I will point out analogies with construction techniques for Einstein metrics.
In dimension 3, the Ricci flow together with a certain surgery process has been used by Perelman, amongst many others, to establish the Poincaré and Geometrization Conjectures. Despite the depth of this result, a precise description of the long-time behavior of this flow has remained unknown. For example, it was only conjectured by Perelman that it suffices to carry out a finite number of surgeries and that the geometric decomposition of the manifold is exhibited by the flow as $t \to \infty$. Recently I was able to confirm Perelman's first conjecture and I partially answered his second one.

I will first give a brief overview of Ricci flows with surgery and explain the finite surgery theorem. Next, I will present long-time existence results in dimensions 4 and higher and describe possible further directions in this field.

Since the foundational results of Chung-Graham-Wilson on quasirandom graphs over 20 years ago, here has been a lot of effort by many researchers to extend the theory to hypergraphs. I will present some of this history, and then describe our recent results that provide such a generalization in some cases. One key new aspect in the theory is a systematic study of hypergraph eigenvalues first introduced by Friedman and Wigderson. This leads to the study of various extremal questions
on hypergraphs, for example, spectral Tur an problems and spectral packing
problems. This is joint work with Peter Keevash and John Lenz.

Although the Ricci flow with surgery has been used by Perelman to solve the Poincar�© and Geometrization Conjectures, some of its basic properties are still unknown. For example it has been an open question whether the surgeries eventually stop to occur (i.e. whether there are finitely many surgeries) and whether the full geometric decomposition of the underlying manifold is exhibited by the flow as $t \to \infty$.

In this talk I will show that the number of surgeries is indeed finite and that the curvature is globally bounded by $C t^{-1}$ for large $t$. This confirms a conjecture of Perelman. Using the new curvature bound, it is possible to give a more precise geometric picture of the long-time behavior of the flow.

We consider three distinct topics of independent interest;
one in enumerative combinatorics, one in symmetric function theory, and
one in algebraic geometry. The topic in enumerative combinatorics
concerns a q-analog of a generalization of the Eulerian numbers, the one
in symmetric function theory deals with a refinement of Stanley's
chromatic symmetric functions, and the one in algebraic geometry deals
with a representation of the symmetric group on the cohomology of the
regular semisimple Hessenberg variety of type A. Our purpose is to
explore some connections between these topics and consequences of these
connections. This talk is based on joint work with John Shareshian.