\indent This talk is a survey on the joint results with S.Ovsienko, V. Serganova and I. Shestakov. It is devoted to the problem of classification of indecomposable Jordan bimodules over finite dimensional Jordan algebras when squared radical is zero.

\indent Recall, that for a Jordan algebra $J$ the category $J$-bimod of $k$-finite dimensional $J$-bimodules is equivalent to the category $U$-mod of (left) finitely dimensional modules over an associative algebra $U = U(J)$, which is called the universal multiplication envelope of $J$. If $J$ has finite dimension the algebra $U$ is finite dimensional as well. In particular, in accordance with the representation type of the algebra $U$ one can define Jordan algebras of the finite, tame and wild representation types.

\indent From the other side to each Jordan algebra corresponds a Lie algera $TKK(J)$. Moreover there is a correspondence between the finite dimensional Jordan modules over $J$ and finite dimensional Lie modules over $TKK(J)$.

\indent This allows us to apply to the category $J$-bimod all the machinery developed in the representation theory of finite dimensional algebras as well as the representation theory of Lie algebras.

By utilizing machine learning techniques for privacy-preserving
classification, we consider linear programs with partitioned
constraint matrices with each partition belonging to an entity
that is unwilling to share its partition or make it public. For
vertically partitioned matrices we employ a random matrix
transformation that generates a linear program based on all the
privately held data but without revealing the data or making it
public. The component groups of the solution of the transformed
problem can be decoded and made public only by the original group
that owns the corresponding constraint matrix columns.

For a horizontally partitioned constraint matrix, we multiply
each partition by an appropriately generated and privately held
constraint matrix. This results in an equivalent linear program
that does not reveal any of the original data or make it public.
The solution vector of the transformed linear program is publicly
generated and is available to all entities.

\indent From every countable group G or measure preserving
action of G on a probability space X, one can construct a von Neumann algebra. A central theme in the theory of von Neumann algebras is understading how much of the group or group action is ``remembered'' by its von Neumann algebra. In this talk, I will survey recent results which provide the first classes of groups and group actions that can be completely recovered from their von Neumann algebras.

Recent work of Haglund, Armstrong, Rhoades
and Sagan has led to the discovery a variety of
combinatorial properties of Tesler matrices. In this talk
we focus on the connection between Tesler matrices
and Parking Functions and its consequences
in the Theory of Diagonal Harmonics.

\indent Probabilistic graphical models or Markov random fields provide a graph-based framework for capturing conditional independence relationships between random variables of a high-dimensional multivariate distribution. This interdisciplinary topic has found widespread application in a variety of areas including image processing, bioinformatics, combinatorial optimization and machine learning. Estimating the graph structure of the model from samples drawn from it forms an important task, since the structure reveals important relationships between the variables. However, structure estimation has several challenges: in general graphical models, it is NP-hard; the models are also typically in the high-dimensional regime where the number of variables is much larger than the number of samples obtained. I will address these challenges in the talk.

\indent I will talk about our recent results on learning lsing models on sparse Erdos-Renyi random graphs. We establish achievability of consistent structure learning when the model is in the so-called uniqueness regime, where there is a decay of long-range correlations. The algorithm is based on a set of conditional mutual information tests and is shown to be consistent for structure estimation with almost order-optimal sample complexity. A simpler algorithm based on correlation thresholding is also consistent, but under more stringent conditions. Thus, we establish that correlation decay is a sufficient condition for consistent structure learning in random graphs. We also prove a converse result on the number of samples required by any algorithm to consistently recover the random graph structure. I will also provide an overview on our recent results on structure learning in tree models, and more generally, in latent tree models with hidden variables.

Fusion categories are quantum analogues of finite groups. They describe certain topologically invariant 2-dimensional quantum systems, and may be relevant for building a quantum computer. Families of fusion categories can be constructed from quantum groups, subfactors or conformal field theory. Attempting to classify small examples requires techniques from analysis, combinatorics, representation theory and number theory. The classification results available so far reveal intriguing exotic examples and leave many questions.

By dynamical coupling of data with known models, we determine underlying parameters and unmeasured state variables for a variety of systems, including Lorenz, Colpitts, and Hodgkin-Huxley neurons. The dynamic coupling is mediated by use of a cost function, which is minimized by optimization software (SNOPT, IPOPT) to achieve the desired synchronization. Considering measurement and model noise, we discuss the same problem where we introduced an 'action' in order to estimate states and parameters. This action is used along with Markov Chain Monte Carlo methods in order to sample from a probability distribution.

Skew Young tableaux are simple combinatorial objects
arising in the theory of symmetric functions and the representation
theory
of the symmetric group. The number $f^\sigma$ of (standard) skew Young
tableaux of skew shape $\sigma$ has a simple determinantal formula due
to
Aitken. We will discuss some situations for which there exist other
formulas or generating functions for $f^\sigma$. For instance, for certain
skew shapes $\sigma$ the number $f^\sigma$ can be described in terms of
Euler numbers (the number of alternating permutations of $1,2,\dots,n$)
using an analytic technique introduced by Elkies and further developed
by
Baryshnikov and Romik. Certain other sequences $\sigma_n$ of skew shapes
have simple generating functions for the numbers $f^{\sigma_n}$, based
on
a well-known connection between determinants and generating functions.

\indent The Jones polynomial can be understood in terms of the representation theory of the quantum group associated to $sl2$. This description facilitated a vast generalization of the Jones polynomial to other quantum link and tangle invariants called Reshetikhin-Turaev invariants. These invariants, which arise from representations of quantum groups associated to simple Lie algebras, subsequently led to the definition of quantum 3-manifold invariants. In this talk we categorify quantum groups using a simple diagrammatic calculus that requires no previous knowledge of quantum groups. These diagrammatically categorified quantum groups not only lead to a representation theoretic explanation of Khovanov homology but also inspired Webster's recent work categorifying all Reshetikhin-Turaev invariants of tangles.

A dual equivalence for an arbitrary collection of
combinatorial objects endowed with a descent set is a relation for which
equivalence classes group together terms according to the Schur
expansion of the corresponding generating function. After outlining the
definition of dual equivalence, we'll present three main applications:
the Schur expansion of Macdonald polynomials, Schur positivity of
k-Schur functions (joint with S. Billey), and a combinatorial rule for
the Littlewood-Richardson coefficients of the Grassmannian in the
special case of a Schubert polynomial times a Schur function (joint with
N. Bergeron and F. Sottile).

\indent I will discuss the problem of efficiently constructing
curves of genus 1 and 2 over finite fields with a
prescribed number N of points. In both cases, there are algorithms that, at least heuristically and in practice, run in time polynomial in log N. They are of complex analytic nature, using CM-techniques. Time permitting, I will also explain why these techniques are provably insufficient to efficiently construct genus-2 Jacobians with a presecribed number of points.

Let $A/K$ be an abelian variety over a global field $K$.
For each place $v$ of $K$, one associates an integer $c(v)$ called the Tamagawa
number of the place, using the reduction of the abelian variety at $v$.
Let $c$ denote the product of the $c(v)'s$. Let $t$ denote the order of the
torsion subgroup of Mordell-Weil group $A(K)$. The ratio $c/t$ is a factor in
the leading term of the L-function of $A/K$ at $s=1$ predicted by the
conjecture of Birch and Swinnerton-Dyer.
We investigate in this talk possible cancellations in the ratio $c/t$.
For elliptic curves over $Q$. the smallest ratio $c/t$ is $1/5$, obtained only
by the modular curve $X_1(11)$.

\indent Ion channels are irresistible objects for biological study because they are ‘nanovalves of life’ controlling most biological functions, much as transistors control computers. Direct simulation of channel behavior in atomic detail is difficult if not impossible. Gaps in scales of time, volume, and concentration between atoms and biological systems are each ~1012. All the gaps must be dealt with at once, because biology occurs on all the scales at once.
Simple models are surprisingly successful in dealing with ion binding in three very different (and important) channels: the sodium channel that produces the signals of nerve and muscle and two cardiac calcium channels that control contraction. Amazingly, one model with the same three parameters accounts quantitatively for qualitatively different binding in a wide range conditions for two very different calcium and sodium channels. Binding free energy is an output of the calculation, produced by crowding charged spheres into a very small space. The model does not involve any traditional chemical ‘quantum’ binding energies at all.
How can such a simple model give selectivity when crystallographic wisdom and chemical intuition says that selectivity depends on the precise structural relation of ions and side chains? The answer is that structure is a computed consequence of forces in these correlated crowded systems. Binding sites are self-organized and at their free energy minimum. Different structures form in different conditions. Binding is a consequence of the ‘induced fit’ of side chains to ions and ions to side chains.
Equilibrium is death to biology. A variational approach is obviously needed to replace our equilibrium analysis and is well under way, applying the energy variational methods of Chun Liu, used to deal with highly correlated systems like liquid crystals.

In 1913, Major Percy MacMahon showed that the major index
and inversion number statistics are equidistributed over permutations.
A bijective proof of this fact was first given in 1968 by Dominique
Foata who constructed a recursive bijection on permutations such that
the major index of the source is the inversion number of image. In
2004, Jim Haglund made a major breakthrough in the theory of Macdonald
polynomials by conjecturing a formula for Macdonald polynomials,
proved shortly thereafter with Haiman and Loehr, that involved
relatively simple generalizations of the major index and the inversion
number. In this talk, we will show how a filtration of Foata's
bijection can be used to give a simple proof of Macdonald positivity
for certain cases and outline how this approach might be generalized.
Time permitting, we will outline how similar techniques might be
useful in giving a bijective proof of the so-called q,t symmetry of
Macdonald polynomials. This talk will be accessible to first year
graduate students and contains several open problems.

\indent I will talk about the fundamental theorem of affine sieve (joint with Sarnak). The main black box in the proof of this result will be also explained. It is a theorem on a necessary and sufficient condition for a finitely generated subgroup of SL(n,Q) under which the Cayley graphs of such a group modulo square free integers form a family of expanders (joint with Varju).

\indent We consider group-invariant CR mappings from spheres to hyperquadrics. Given a finite subgroup $\Gamma \subset U(n)$, a construction of D'Angelo and Lichtblau yields a target hyperquadric $Q(\Gamma)$ and a canonical map $h_{\Gamma} : S^{2n-1}/\Gamma \to Q(\Gamma)$. For every $\Gamma \subset SU(2)$, we determine this hyperquadric $Q(\Gamma)$, that is, the numbers of positive and negative eigenvalues in its defining equation. For families of cyclic and dihedral subgroups of $U(2)$, we study these numbers asymptotically as the order of group tends to infinity. Finally, we explore connections with invariant theory and representation theory.

\indent Whereas many physical systems can be identified as Hamiltonian dynamical systems, mechanical systems under rolling and sliding constraints, even simple ones such as a rolling penny, skateboard, and sleigh, are non-Hamiltonian. This is due to the fact that those constraints are nonholonomic (non-integrable). Nonholonomic constraints destroy some nice features of Hamiltonian dynamical systems, most importantly symplecticity, while retaining some Hamiltonian properties, such as energy conservation. Many concepts and ideas in Hamiltonian dynamics have been generalized from the differential-geometric point of view to incorporate nonholonomic constraints and also to explain the "almost Hamiltonian" behavior of nonholonomic systems. In this talk, I will show how to generalize Hamilton--Jacobi theory to nonholonomic systems and its application to exactly integrating the equations of motion, touching on the basic geometric concepts of nonholonomic and Hamiltonian systems and also the tools and techniques used to reconcile the differences between them.

\indent Let $M_0$ be a closed subset of $R^n+1$ that is a smooth hypersurface except for a finite number of isolated singular points. Suppose that $M_0$ is asymptotic to a regular cone near each singular point.

Can we flow $M_0$ by mean curvature?

Theorem $(n<7)$: there exists a smooth mean curvature evolution starting at $M_0$ and defined for a short time $0<t<eps$.

Such an initial $M_0$ might arise as the limit of a smooth mean curvature evolution defined earlier than $t=0$. Thus, the result allows us to flow through singularities in some cases.

We use a monotonicity formula that complements the monotonicity formula of Huisken. The method applies to other geometric heat flows as well.

\indent Have you ever wanted to access a database in such a way that even the database can't tell what you're looking for? Better yet, don't answer that. Regardless, we will investigate how an accommodating database can allow such a query. We'll also discuss possible connections to circumventing the PATRIOT Act (don't tell the FBI!) and furthering space research.

\indent We report on recent work joint with Jason Bell. Makar-Limanov showed that the quotient division ring of the first Weyl algebra contains a copy of a free algebra on 2 generators. We prove the natural analog of this result for arbitrary iterated skew polynomial rings of fields.

\indent We study the singular set of a singular Levi-flat real-analytic
hypersurface. We prove that the singular set of such a hypersurface is Levi-flat in the appropriate sense. We also show that if the singular set is small enough, then the Levi-foliation extends to a singular codimension one holomorphic foliation of a neighborhood of the hypersurface.

We examine the problem of data assimilation in two different ways:

(1) as a special case of optimization where one attempts to minimize the parameters and state variables of a model set of equations to a time series of observations. To put this problem in context, we look at an example from neuroscience where we optimize spiking neuron models to noisy experimental data.

2) In a path integral formulation using an example from partial differential equations (the barotropic vorticity equations used as a model) as a method for obtaining means and distributions from high level integrals. This approach does not rely on optimization,but on the evaluation of a high dimensional integral.

Both approaches lend themselves to parallel implementation on GPUs using NVIDIA's CUDA C language. These algorithms vary in complexity, with some taking advantage of phenomena from nonlinear dynamics to improve their behavior. We discuss some practical limitations to parallelization due to the hardware architecture and concerns surrounding memory management.

\indent In recent years, kinetic transport models have received a lot of attention in various fields, including semiconductor device modeling, plasma physics, etc. This talk will focus on the Boltzmann equation, which is one of the most important equations in statistical physics. The Boltzmann equations describe the time evolution of the probability density functions, and are generally composed of a transport part and a collision part. Those equations have a lot of interesting structures that comes from applications and are computationally challenging to solve. In this talk, we will look into two classes of Boltzmann equations: one being the Boltzmann-Poisson systems in semiconductor device simulations, and the other being the linear Vlasov-Boltzmann transport equations. The goal is to design computationally efficient schemes that can preserve the important structures of the physical systems. I will motivate the choice of the discontinuous Galerkin (DG) finite element methods for treating those equations. The DG schemes enjoy the advantage of conservative formulation, flexibility for arbitrarily unstructured meshes, with a compact stencil, and with the ability to easily accommodate arbitrary h-p adaptivity. Numerical issues such as implementation, algorithm design and analysis for suitable applications will be addressed. Benchmark numerical tests will be provided to demonstrate the performance of the scheme compared to existing solvers such as Monte-Carlo and finite difference solvers.

\indent Spectral Collocation methods are a powerful set of methods used to solve partial differential equations and ordinary differential equations numerically. This talk will present some of the basic theory behind Spectral Collocation methods, and provide several examples of how they can be utilized. Along the way, we will encounter approximation theory, orders of convergence, wave and soliton equations, and even some nice pictures. No prior knowledge of numerical analysis is assumed; if you are curious about numerical methods but don't have much background, this talk should provide a gentle introduction.

\indent Covariance matrix is of fundamental importance in many aspects of statistics. Recently, there is a surge of interest on regularized covariance matrix estimation using banding, tapering and thresholding methods, in high dimensional statistical inference, where multiple iid copies of the random vector from the underlying multivariate distribution are required.

\indent In the context of time series analysis, however, it is typical that only one realization is available, so the current results are not applicable. In this talk, we shall exploit the connection between covariance matrices and spectral density functions using the idea in Toeplitz~(1911) and Grenander and Szeg\"o~(1958)

\indent In this talk we will prove the following obstruction result to
functors extending the Zariski spectrum: every contravariant functor from rings to sets whose restriction to the full subcategory of commutative rings is isomorphic to Spec must assign the empty set to complex matrix algebras of order at least 3. The proof relies crucially on the Kochen-Specker "no-hidden-variables" theorem of quantum mechanics. We will also discuss a (very) recent generalization of the result due to van den Berg and Heunen.

The talk will give an introduction
to this new branch of compressed sensing.
One has an affine subspace intersect the positive semideinite (PSD)
matrices and wishes to find a smallest rank
matrix therein. This is a highly nonconvex
problem. Minimizing the trace is a convex problem
which often gives the correct answer. There is an
elegant probablistic analysis which applies in some
situations. The talk gives an exposition of this.

\indent Improvements of the obvious lower bounds on the independence number of (hyper)graphs have had impact on problems in discrete geometry, coding theory, number theory and combinatorics. One of the most famous examples is the result of Komlos-Pintz-Szemeredi (1982) on the independence number of 3-uniform hypergraphs which made important progress on the decades old Heilbronn problem. We give a sharp upper bound on the chromatic number of simple k-uniform hypergraphs that implies the above result as well as more general theorems due to Ajtai-Komlos-Pintz-Spencer-Szemeredi, and Duke-Lefmann-Rodl. Our proof technique is inspired by work of Johansson on graph coloring and uses the semi-random or nibble method. This is joint work with Alan Frieze.

\indent We usually only justify time series estimators using asymptotic theory, but the sample size for time series, say those yearly macro series, is usually limited, not more than 100. Additionally, high dimensionality and serial dependence makes the asymptotics harder to be a good approximation for a finite sample. My works in high dimensional time series modeling tries to solve these problems, i.e. to quantify the interplay and strike a balance between degree of time dependence, high dimensionality and moderate sample size (relative to dimensionality). In this talk, I will talk about generalized dynamic factor models (briefly), large vector autoregressions for modeling expectation (in detail), and also dynamic volatility matrix estimation (briefly) if time permits.

\indent A multivariate real polynomial $p$ is nonnegative if $p(x) \geq 0$ for all $x \in R^n$. I will review the history and motivation behind the problem of representing nonnegative polynomials as sums of squares. Such representations are of interest for both theoretical and practical computational reasons and I will discuss several applications. I will then discuss some theoretical aspects of sums of squares representations of nonnegative polynomials, in particular, some underlying fundamental reasons that there exist nonnegative polynomials that are not sums of squares.

Tutte founded the enumeration theory of planar maps in a
series of papers in the 1960s. We are interested in rooted planar maps
which can be thought as connected planar graphs embedded in the sphere
with a directed edge distinguished as the root. A planar map is
non-separable if it has no loops and no cut-vertices. Non-separable
planar maps are also called 2-connected maps. Another class of maps of
our interest is bicubic maps, which after removing the root orientation
are connected regular bipartite graphs with vertex degree 3.
Cori, Jacquard and Schaeffer introduced description trees in 1997, to
give a general framework for the recursive decompositions of several
families of planar maps studied by Tutte. These trees are not only
interesting in their own right, but also they proved to be a useful tool
in obtaining non-trivial equidistribution results on planar maps,
certain pattern avoiding permutations, and objects counted by Catalan
numbers.

In this talk, I will provide an overview of several recent results and
research trends related to planar maps and description trees. Most of
the results are ``work in progress'' of several research teams.

The famous Carleson corona theorem asserts that the open
unit disk is dense in the maximal ideal space of the algebra of
bounded holomorphic functions on it (denoted $H^\infty$). Similar
statements for the algebra of bounded holomorphic functions on a
polydisk and for slice algebras for $H^\infty$ remain the major open
problems of multivariate complex analysis. For instance, the answer to
the last problem would be obtained if one were able to show that
$H^\infty$ has the Grothendieck approximation property. This problem
posed by Lindenstrauss in the early 1970th is also unsolved. The
strongest result in this direction was proved by Bourgain and Reinov
in 1983 and asserts that $H^\infty$ has the approximation property up to
logarithm. In the talk I will present a proof of the corona theorem
for slice algebras for $H^\infty$, describe topological structure of the
maximal ideal space of $H^\infty$ and as a corollary present some
results on $Sz$. Nagy operator-valued corona problem for $H^\infty$.

\indent Function theory of several complex variables is much less well understood than function theory of functions of one variable. One approach to attempting to bridge this divide is to study an analytic function $f$ on $D^n$ as follows. Fix a 1-dimensional algebraic variety $V\subset C^n$ and let $F$ denote the restriction of $f$ to $V$. Since $V$ is 1-dimensional, $F$ behaves somewhat like a function of one complex variable and we could potentially apply the theory of functions of one variable to understanding $F$. If we can use this approach to prove facts about $F$, then we could potentially extend some of these results to $f$. In particular, we take this approach to generalize to $D^n$ the classic Schwarz Lemma on the disc $D$. Familiarity with the definition of an analytic function of one variable is the only thing that will be assumed.

Function theory of several complex variables is much less well understood than function theory of functions of one variable. One approach to attempting to bridge this divide is to study an analytic function f on the polydisc as follows. Fix a $1$-dimensional algebraic variety $V$ in $C^n$ and let $F$ denote the restriction of $f$ to $V$. Since $V$ is $1$-dimensional, $F$ behaves somewhat like a function of one complex variable and we apply the theory of functions of one variable to $F$. We use this approach to prove facts about $F$ and then we extend certain results about $F$ to results about $f$. In particular, we take this approach to generalize to $D^n$ the classic Schwarz Lemma on the disc $D$ and give sufficient conditions for a bounded analytic function on $D^n$ to be uniquely determined by its values on a finite set of points. In terms of the Pick problem on $D^n$, we give sufficient conditions for a Pick problem to have a unique solution.

\indent The problem of the densest packing of space by congruent regular tetrahedra has a long history, starting with Aristotle's assertion that regular tetrahedra fill space, and continuing through its appearance in Hilbert's 18th Problem. This talk describes its history and many recent results obtained on this problem, including contributions from physicists, chemists and materials scientists. The current record for packing density is held by my former graduate student Elizabeth Chen, jointly with Michael Engel and Sharon Glotzer.

If A is an abelian variety over a p-adic field K then A has good
reduction if and only if the p-adic Tate module of A is a crystalline
representation of the absolute Galois group of K. As there are examples of
curves over K with bad reduction whose Jacobian has good reduction, the
Galois action on the p-adic etale cohomology of the curve does not
determine its reduction. We will discuss these issues and point
to a p-adic criterion of good reduction for curves.

\indent The talk will review semidefinite programming (SDP) relaxations for polynomial optimization and show how to solve them. We propose regularization type methods to solve such large scale SDP problems. Significantly bigger problems would be solved, which is not possible by using prior existing methods like interior-point algorithms. Numerical examples will also be shown.

We consider the following dynamic Boolean model introduced by
van den Berg, Meester and White (1997). At time $0$, let the nodes of the
graph be a Poisson point process in $R^d$ with constant intensity and let
each node move independently according to Brownian motion. At any time $t$,
we put an edge between every pair of nodes if their distance is at most $r$.
We study two features in this model: detection (the time until a target
point--fixed or moving--is within distance $r$ from some node of the graph),
coverage (the time until all points inside a finite box are detected by
the graph) and percolation (the time until a given node belongs to the
infinite connected component of the graph). We obtain asymptotics for
these features by combining ideas from stochastic geometry, coupling and
multi-scale analysis. This is joint work with Yuval Peres, Alistair
Sinclair and Alexandre Stauffer.

Given a Riemannian manifold $M$, when can it be isometrically
embedded in Euclidean space? When can a local isometry be found, and
when can a global isometry be found? What is the minimum dimension of
the target Euclidean space if $M$ has dimension $n$? These questions have
been extensively studied during the last century, with perhaps the
best known result being the famous Nash embedding theorem from the
1950's. The aim of this talk is to introduce the problem and some of
the well known results. The talk is intended to be more of a history
lesson rather than technical, so there will be minimal discussion of
proofs, and no background required.

\indent This talk will introduce a new class of models which can lead to efficient estimation in multivariate analysis. Some members in the class include the basic envelope model, partial envelope model, inner envelope model, scaled envelope model, and heteroscedastic envelope model. They have the common word ``envelope'' in their names because they are all constructed by enveloping: use reducing subspaces to connect the mean function and the covariance function, so that the number of parameters can be reduced. The application of enveloping is very broad and can be used in many contexts to control parameterization. Theoretical results, simulations and a large number of data examples show that the efficiency gains obtained by enveloping can be substantial.

\indent The eigenvalues of the Laplacian (or Laplacian with potential) on a smoothly bounded domain domain are very natural quantities, arising as the fundamental tones of a drum, the rates of decay of diffusions, and the energy levels of quantum systems. I will discuss some of the history relating to inequalities for low eigenvalues, leading up to the proof of a conjecture of Yau and van den Berg for the `fundamental gap' or excitation energy of a convex domain.

\indent In this talk, we will discuss about Li-Yau-Hamilton type
differential Harnack inequalities, heat kernel estimates and their
applications to study type I ancient solutions of the Ricci flow.

\indent We explore the question of whether there are nontrivial
solutions to the two-valued minimal surface (2MSE) equation defined over the punctured plane. The 2MSE is a non-uniformly elliptic PDE, degenerate at the origin, originally introduced by N.Wickramasekera and L.Simon to produce examples of stable branched minimal immersions.

The groups $F \leq T \leq V$ were defined by Richard Thompson in 1965 and used to construct finitely presented groups with unsolvable word problems. $T$ and $V$ were also the first examples of infinite, finitely presented simple groups. Since then, these groups have been studied extensively using a rich interplay of algebraic, topological, and dynamical approaches. I will discuss recent work regarding the higher dimensional analogues of Thompson groups, $nV$, including the fact that $mV$ is not isomorphic to $nV$ for $n \neq m$, and that for every $n$ the group $nV$ is finitely presented and simple. The only background required for this talk is basic group theory.

\indent Since 2005 a class of algebras, the 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 insight regarding the ideas which prompted the initial description of these structures.

\indent I'll briefly describe some results of the expected form, namely, results of the form: $E$ has property $P$ if and only if $L_K(E)$ has property $P'$.

\indent However, the main goal of the talk will be to show how Leavitt path algebras have been used to answer various questions outside the subject per se. For example, results about von Neumann regular rings; about prime or primitive algebras; about $C*-$algebras; and about Lie algebras have been gleaned from these structures.

\indent A discretization scheme for space-time domains and two second order solvers for convection diffusion reaction partial differential equations.

\indent Categorical quantum mechanics seeks to distill quantum mechanics to minimal assumptions, based on categories with tensor products. We address the question of how to usefully represent observables in this setting. Orthonormal bases in the category of finite-dimensional Hilbert spaces turn out to correspond to Frobenius algebras. We show that for arbitrary dimensions one needs H*-algebras instead, which can be defined in any monoidal category. Finally we compare the notion of H*-algebra to that of nonunital Frobenius algebra in various categories

Non-uniform circuit lower bounds are among the strongest
impossibility results attainable in complexity theory, but they are
also among the most difficult to prove. The circuit class ACC consists
of circuit families with constant depth over unbounded fan-in AND, OR,
NOT, and MODm gates, where $m > 1$ is an arbitrary constant. Despite the
apparent simplicity of such circuits, the power of MODm has been very
hard to reason about. For instance, it was not known whether a
complexity class as large as $EXP^{NP}$ (the class of languages
recognized in $2^{O(n^k)}$ time with an NP oracle) could be simulated
with depth-$3$ polynomial size circuits made out of only MOD6 gates.

We prove:
- There are functions computable in Nondeterministic Exponential Time
that cannot be simulated with non-uniform ACC circuits of polynomial
size. The size lower bound can be slightly strengthened to
quasi-polynomials and other less natural functions.
- There are functions in $EXP^{NP}$ that cannot be simulated with
non-uniform ACC circuits of $2^{n^{o(1)}}$ size. The lower bound gives
an exponential size-depth tradeoff: for every $d$ and $m$ there is a $b > 0$
such that the relevant function doesn't have depth-$d$ ACC circuits of
size $2^{n^b}$ with MODm gates.

The proofs are more interesting than the results. The high-level
strategy is to design faster algorithms for the circuit satisfiability
problem over ACC circuits, then show how such algorithms can be
applied to yield lower bound proofs against ACC circuits, via a more
general "algorithm-lower bound" connection. This connection provides a
new direction for further progress in circuit complexity.

Given any complex power series in n-variables one can always
ask what is the largest negative power which is integrable. This
number is the log canonical threshold (its reciprocal is called the
Arnold multiplicity) of the underlying hypersurface. It is a measure
of the complexity of the singularity at the origin, which carries more
information than the multiplicity.

I describe some recent work with Hacon and Xu, where we prove some
conjectures of Kollár and Shokurov, which state that the set of log
canonical thresholds satisfies the ascending chain condition and which
identifies the accumulation points.

Given a curve (Riemann surface), one can construct an abelian variety: its Jacobian. Abelian varieties are quotients of vector spaces by lattices. The classical Torelli theorem states that the Jacobian determines the curve. We discuss some generalizations of this and their history.

\indent Given a fixed graph $G$ on $n$ vertices, we can create a random coloring of $G$ in the following way: randomly pick an edge, then randomly pick a color, and then color both endpoints of that edge that color. We can continue this process on a graph that is already colored by simply overwriting any vertices that have already been assigned a color. This gives rise to a random walk on the $2^n$ colorings of $G$, and it is this random walk that we will investigate. The eigenvalues of the transition matrix are known and have a simple form. We discuss these and other quantities as well as several related problems.

\indent We give a survey of what is known about how many symmetries an algebraic variety can possess. We start with some classical results, including those of Hurwitz, Noether and Riemann, to do with the automorphism group of the plane and the automorphism group of curves (or equivalently Riemann surfaces), and we end with some more recent results to do with the automorphism group of threefolds of low degree and varieties with finite automorphism group.

Perhaps the most studied varieties in algebraic geometry are
the moduli spaces of smooth and stable curves. A stable curve is a
nodal curve whose canonical divisor is ample. Adding stable curves
gives a geometrically meaning compactification to the space of smooth
curves.

A motivating problem in higher dimensional geometry is to construct
the moduli space of varieties of general type, in any dimension. Just
as with curves, we need to consider non-normal varieties, possibly
with more than one component. Unlike the case of curves, if we fix
the degree, it is not at all clear how to bound the number of
components.

Motivated by recent work of Mok and others on local holomorphic
isometric embeddings, we consider local conformal maps from an irreducible
Hermitian symmetric space of compact type into the products of such
manifolds. We allow the conformal factors to be arbitrary real numbers and
derive a necessary and sufficient condition for the global rigidity to
hold. This is a joint work with Yuan Yuan.

We show that there is a countable structure M such that for any set X, X computes a copy o M if and only if X is not hyperarithmetic. This gives a strong generalisation of the Slaman-Wehner theorem to the hyperdegrees. On the other hand, the generalization of the Slaman-Wehner theorem to the degrees of constructibility is false.
This is related to work of Kalimullin and his co-authors on structures whose degree spectrum has full measure. We show, for example, that there are only countably many such structures.
We also touch on the possible form of structures M as above. For example, they can be linear orderings, but not have uncountably categorical theories. Many open questions remain. Among them: can we similarly capture the nonarithmetic degrees?
Joint work with Antonio Montalban and Ted Slaman.

Hermitian symmetric spaces have played a distinguished role in the
history of representation theory and they continue to be a source of
beautiful and often surprising results. In this talk, I will use a natural
generalization of Young diagrams for Hermitian symmetric spaces to give a
concrete and uniform approach to a wide variety of interconnected topics
including non-compact roots and canonical reduced expressions, abelian
ideals in a Borel subalgebra, rational smoothness of Schubert varieties in
co-minuscule flag varieties, equivalences of categories of highest weight
modules, BGG resolutions, and syzygies of determinantal ideals.

A $(v,k,\lambda,\mu)$-strongly regular graph (SRG for short) is a finite undirected graph without loops or multiple edges such that (i) it has $v$ vertices, (ii) it is regular of degree $k$, (iii) each edge is in $\lambda$ triangles, (iv) any two nonadjacent points are joined by $\mu$ paths of length 2. The connectivity of a graph is the minimum number of vertices one has to remove in order to make it disconnected (or empty).
In 1985, Brouwer and Mesner used Seidel's characterization of strongly regular graphs with eigenvalues at least $-2$ to prove that the vertex-connectivity of any $(v,k,\lambda,\mu)$-SRG equals its degree $k$. Also, they proved that the only disconnecting sets of size $k$ are the neighborhoods $N(x)$ of a vertex $x$ of the graph.
A natural question is what the minimum number of vertices whose removal will disconnect a $(v,k,\lambda,\mu)$-SRG into non-singleton components. In 1996, Brouwer conjectured that this number is $2k-\lambda-2$. In this talk, I will report some progress on this problem.
This is joint work with Kijung Kim and Jack Koolen (POSTECH, South Korea).

The origins of multilinear measure theory (also known as multidimensional measure theory) can be traced back to the work of Fr$\mathrm{\acute{e}}$chet in 1915. Fr$\mathrm{\acute{e}}$chet characterized
the bounded bilinear functionals on $C[0, 1]$, generalizing the characterization
of bounded linear functionals given by Riesz. It was not until much later that
these bounded bilinear functionals came to be identified with set functions called
bimeasures. Since that time, multilinear measure theory has developed, and contains
many interesting and deep results. Despite the progress, however, several key
measure-theoretic results have eluded satisfactory generalization. In this talk, we
will use results in operator theory to provide a generalization of the Radon-Nikod$\mathrm{\acute{y}}$m
theorem, and then use it to prove a bounded convergence theorem for bimeasures.

This talk will focus on the beautiful interplay between probability theory and harmonic functions. This connection is provided through the notion of random walks. A gentle introduction to this concept will be given, along with a realization of solutions to a discrete Laplace's equation in terms of random walks. Time permitting, we will also discuss the continuous analog of these results, along with some related areas of study at the intersection of probability and classical analysis. No knowledge beyond linear algebra will be assumed, and all results will be discussed on a heuristic level, so even the most stochastic-phobic among us is encouraged to attend.

Sincerely,
Mr. Thought \\

Please note different location and time than usual.

Abstract: The central axis of the famous DNA double helix is often
constrained or even circular. The topology of this axis can influence
which proteins interact with the underlying DNA. Subsequently, in all
cells there are proteins whose primary function is to change the DNA
axis topology -- for example converting a torus link into an unknot.
Additionally, there are several protein families that change the axis
topology as a by-product of their interaction with DNA. This talk will
describe some typical DNA conformations, and the families of proteins
that change these conformations. I'll present a few examples
illustrating how 3-manifold topology has been useful in understanding
certain DNA-protein interactions, and discuss the most common
techniques used to attack these problems.

The blob complex is a chain complex which can be associated to any pair ($n$-dimensional manifold, $n$-category). It has very nice properties under cutting and pasting manifolds along lower-dimensional pieces (with ``corners" of arbitrary dimension) and so amounts to an $n$-dimensional topological quantum field theory which makes sense ``all the way down to the point".

It is a generalisation and unification of several ideas. Well-known TQFTs such as Turaev-Viro theory and Chern-Simons theory can be recovered from the cases $n=2$ and $n=3$, with appropriate kinds of $2$-category (spherical tensor category) and $3$-category (modular category) respectively.

What is novel is that the construction is a homotopy-invariant or``derived" construction, which allows for much more general input categories, and hence new kinds of TQFT. In the simplest case - when $n=1, M$ is the circle, and $C$ is any associative algebra (viewed as a $1$-category with just one object) - it is equivalent to the Hochschild chain complex of $C$, which can be thought of as the ``derived cocentre" of $C$ (this example is very closely related to Costello's work on topological conformal field theories.) It seems very likely that the blob complex formulation is general enough to allow $4$-dimensional gauge theories, with their exact triangles and other homological baggage, to fall into place as TQFTs which can be encoded using algebra and combinatorics.

A large part of the idea consists of giving a suitable definition of n-category. The approach here is very natural in this geometric context, and seems to be relatively easy to understand in comparison with the other current approach (via multisimplicial sets) by Lurie, whose ``topological chiral homology" is presumably just a different way of saying the same thing.

\indent Many data-related applications utilize principal component analysis and/or data dimension reduction techniques that require efficiently computing dominant part of singular value decompositions (SVD) of very large matrices which are also very dense. In our talk, we introduce a limited memory block krylov subspace optimization method which remarkablely accelerate the traditional simultaneous iteration scheme. We present extensive numerical results comparing the algorithm with some state-of-the-art SVD solvers. Our tests indicate that the proposed method can provide better performance over a range of dense problem classes under the MATLAB environment. We also present some convergence properties of our algorithm.

\indent Central to many factoring algorithms in use today is the following random process: generate random numbers in the interval [1,N] until some subset has a product which is a square. Naive probabilistic models for the distribution of prime factors suggest that this stopping time has a sharp threshold. Based on more sophisticated probabilistic models, we find a rigorous upper bound that is within a factor of 4/pi of a proven lower bound, and conjecture that our upper bound is in fact asymptotically sharp. This is joint work with Andrew Granville, Ernie Croot and Prasad Tetali.

\indent This is an expository talk. I will discuss the recent paper
of Tian and Zhu on the convergence of the Kahler-Ricci flow on Fano manifolds.

\indent Many questions in extremal graph theory can be phrased like
this: what is the maximum of a certain linear combination of densities
of given graphs in an arbitrary graph? Using the theory of graph limit
objects (called graphons), it is now possible to pose and in some
cases answer some rather general questions about extremal graphs.

- Which linear inequalities between subgraph densities are valid?
Hatami and Norine very recently proved that this question is
undecidable. On the other hand, it follows from results of Lovasz and
Szegedy that if we allow an arbitrarily small ``slack'', then it
becomes decidable.

- Can all valid inequalities between subgraph densities be proved
using just Cauchy-Schwarz? Using the notions of graphons and graph
algebras one can give an exact formulation of this question, which
turns out to be analogous to Hilbert's 17th Problem about
representing nonnegative polynomials as sums of squares. Hatami and
Norine showed that the answer is negative, but Lovasz and Szegedy
proved that it becomes positive if we allow an arbitrarily small
error.

- Is there always an extremal graph? One can prove that there is
always an extremal graphon, which then gives a ``template'' for
asymptotically extremal graphs.

- Which graphs are extremal? In other words, what are the possible
"templates" of extremal graphs? There are nontrivial conditions and
quite
interesting families, but a complete charaterization remains an
exciting but difficult open problem.

Classical statistical approaches for multiclass probability estimation are typically based on regression techniques such as multiple logistic
regression, or density estimation approaches such as linear discriminant analysis (LDA) and quadratic discriminant analysis (QDA). These
methods often make certain assumptions on the form of probability functions or on the underlying distributions of subclasses. In this article, we develop a model-free procedure to estimate multiclass probabilities based on large-margin classifiers. In particular, the new estimation
scheme is employed by solving a series of weighted large-margin classifiers and then systematically extracting the probability information
from these multiple classification rules. A main advantage of the proposed probability estimation technique is that it does not impose any
strong parametric assumption on the underlying distribution and can be applied for a wide range of large-margin classification methods.
A general computational algorithm is developed for class probability estimation. Furthermore, we establish asymptotic consistency of the
probability estimates. Both simulated and real data examples are presented to illustrate competitive performance of the new approach and
compare it with several other existing methods.

This is a joint work with Hao Helen Zhang and Yufeng Liu.

We survey several commonly-used quasi-Newton methods
and line-search algorithms for unconstrained and box-constrained optimization and consider
their underlying strategies. By taking advantage of an implicit similarity in two existing algorithms, we develop a method for box-constrained optimization that includes a new way to compute a search direction and a new line-search
algorithm. On a collection of standardized problems, this method is over $35\%$ faster than the leading comparable alternative.

I will discuss a "non-contraction" result for the holonomy
group of a solution to Ricci flow, namely, that if the reduced
holonomy of a complete solution of uniformly bounded curvature is
restricted to a subgroup of SO(n) at some non-initial time, it must be
restricted to the same subgroup at all previous times; it follows then
from existing results that the holonomy group is exactly preserved by
the equation. In particular, a solution may be Kahler or locally
reducible (as a product) on some time slice only if it is identically
so on its entire interval of existence. In contrast to the question of
"non-expansion" of holonomy, the problem of non-contraction cannot be
reduced completely to an application of the classification and
splitting theorems of Berger and De Rham and a series of appeals to a
relevant uniqueness theorem (here, backwards-uniqueness). However,
with an infinitesimal reformulation, we show that the problem can
nevertheless be reduced to one of unique continuation, and
specifically to one for a coupled system of partial- and
ordinary-differential inequalities of a form amenable to an approach
by Carleman inequalities. This reformulation also leads to an
alternative and essentially self-contained proof of the non-expansion
of holonomy via the analysis of a similar (albeit simpler and strictly
parabolic) system by means of the maximum principle.

\indent We will consider two systems of interacting diffusion processes, which go by the names rank-based and volatility-stabilized models in the mathematical finance literature. We will show that, if one lets the number of diffusion processes tend to infinity, the limiting dynamics of the system is described by the porous medium equation with convection in the rank-based case and by a degenerate linear parabolic equation in the volatility-stabilized case. In the first case we also provide the corresponding large deviations principle. The results can be applied in stochastic portfolio theory and for the numerical solution of partial differential equations. A part of the talk is joint work with Amir Dembo and Ofer Zeitouni.

Gradient Ricci solitons are those Riemannian manifolds whose
Ricci tensor is equal to a constant multiple of the metric plus the
hessian of a function. I will discuss some aspects of the literature
on complete gradient Ricci solitons assuming that the hessian of the function is not identically zero.

\indent The central curve of a linear program is an algebraic curve specified by linear and quadratic constraints arising from complementary slackness. It is the union of the various central paths for minimizing or maximizing the cost function over any region in the associated hyperplane arrangement. In this lecture we present joint work with Daniel Plaumann and Cynthia Vinzant on the geometry of central curves. We determine the degree, arithmetic genus and defining prime ideal of the central curve, thereby answering a question of Bayer and Lagarias. These invariants, along with the degree of the Gauss image of the curve, are expressed in terms of the matroid of the input matrix. Extending work of Dedieu, Malajovich and Shub, this yields an instance-specific bound on the total curvature of the central path, a quantity relevant for interior point methods.

\indent We study acts and modules of maximal growth over finitely generated free monoids and free associative algebras as well as free groups and free group algebras. The maximality of the growth implies some other specific properties of these acts and modules that makes them close to the free ones; at the same time, we show that being a strong infiniteness condition, the maximality of the growth can still be combined with various finiteness conditions, which would normally make finitely generated acts finite and finitely generated modules finite-dimensional.

The numerical solution of nonlinear elliptic equations is important in many applications; however, it is a challenging task to develop efficient software to solve general problems. This talk will describe the basic finite element method and develop an adaptive framework based on the SOLVE-ESTIMATE-MARK-REFINE iteration. A theory of convergence for this iteration, which allows the solver to be inexact, will be given as well as a new error estimator. Numerical results will be shown for a model problem arising in computational biochemistry.

\indent This is part two of an expository talk on the recent paper
of Tian and Zhu on the Kahler-Ricci flow on Fano manifolds.

Management policies for influenza outbreaks balance the expected
morbidity and mortality costs versus the cost of intervention policies. We
present a methodology for dynamic determination of optimal policies in a
stochastic compartmental model with parameter uncertainty. Our formulation
is based on Bayesian conjugate updating in conjunction with stochastic
control methods for optimal stopping. Facing a high-dimensional control
problem, we construct a new Monte-Carlo computational approach that
searches the full set of sequential control strategies. As a running
example, we study a stochastic SIR-model with isolation and vaccination as
two possible interventions. We also investigate the value of information
and effect of various cost structures. Numerical simulations demonstrate
the realized cost savings of choosing interventions based on the computed
dynamic policy over simpler decision rules.

\indent Discrete problems have a habit of being beautiful but difficult. In this talk we'll survey a menagerie of discrete analogues of classical operators arising in harmonic analysis, including singular integral operators, maximal functions, fractional integral operators, and Radon transforms. In general, discrete Radon transforms are the most difficult to treat, and we'll outline some of the methods that are breaking new ground in this direction. Key aspects of the methods presented come from number theory, and we will highlight the roles played by theta functions, exponential sums, Waring's problem, and the circle method of Hardy and Littlewood.

\indent We will review permutation sequences and allowable
permutation sequences and the theorem that every allowable permutation sequence can be realized by an arrangement of pseudolines.

\indent We introduce double permutation sequences which provide a combinatorial encoding of arrangements of convex sets in the plane. We shall also review the notion of a topological affine plane and several of its properties. In particular, we show that for every allowable double permutation sequence, there is a corresponding universal topological affine plane P, i.e. any finite arrangement of pseudolines is isomorphic to some arrangement of finitely many lines of P, and that every allowable double permutation sequence can be realized by an arrangement
of simply connected sets and pseudoline double tangenets to every pair of these sets. We conclude with some recent results using these methods.

\indent All of this is joint work with Jacob E. Goodman and some involves numerous other people including Raghavan Dhandapani, Adreas Holsmen, Shakhar Smorodinsky, Raphael Wenger, and Tudor Zamfirescu.

\indent It is an expository talk of the recent work of Wu and Zheng. Their work develop a systematic way to construct complete $U(n)$ invariant Kahler metrics of positive curvature on $C^{n}$. Studying the geometry of those metrics should be interesting. I will mention a simple application if time permits.

\indent Gradient Ricci solitons are those Riemannian manifolds whose Ricci tensor is equal to a constant multiple of the metric plus the hessian of a function. I will discuss some aspects of the literature on complete gradient Ricci solitons assuming that the hessian of the function is not identically zero.

\indent The Allen-Cahn equation is a popular PDE model for phase transition and phase separation. The level sets of solutions represent interfaces between two materials. A fine analysis of interfaces relies on solutions of Allen-Cahn equation in the entire spaces. In this talk, I will give a brief survey on results for local minimizers of the related Allen-Cahn energy functional, in particular in connection with the De Giorgi conjecture which relates the level set of monotone solutions to minimal surfaces. I will further discuss a classification scheme for finite morse index solutions of Allen-Cahn equation in the whole plane. The level sets of such solutions may represent intersecting interfaces, and the energy of such solutions displays a quantization effect. In particular, I will show that saddle solutions must have even symmetry.

We discuss a natural form of Ricci-Flow conjugation between
two distinct general relativistic data sets given on a compact
$n$-dimensional manifold. The Ricci flow generates a form of $L^2$
parabolic averaging, of one data set with respect to the other, with a
number of desiderable properties: (i) Preservation of the dominant
energy condition; (ii) Localization by a heat kernel, (associated with
the linearized Ricci flow), whose support sets the scale of averaging;
(iii) Entropic stability.

The partial fraction algorithm of Guoce Xin
has recently led to a breakthrough in the theory
of Tesler matrices. In particular we now have a beautiful
formula for the polynomials enumerating the families
of Tesler matrices with positive hook weights. The Xin
algorithm also yields a very illuminating new proof of
the original Tesler matrix formula for the
Hilbert series of Diagonal Harmonics due to Jim Haglund. We will try
to give a glimpse of these developments in a
self contained manner.

This is an expository talk. I will give an introduction to
the weak Kahler-Ricci flow.

\indent The group of diffeomorphisms of the circle arises naturally
in many places in mathematical physics. In this talk, I will first
discuss various topologies of the group and its Lie algebra. Then I
will talk about the construction of a Brownian motion in this group
using a very strong metric.

\small
Recent experimental studies reveal that significant additional improvement in the power conversion efficiency of organic photovoltaic devices is possible through better morphology control of the organic thin film layer during the manufacturing process. A set of computational tools that can (a) predict the evolving three dimensional morphology within the active layer during the fabrication process; and (b) relate the structure with device properties would significantly augment current experimental efforts and strengthen the pursuit of this vision of high power conversion efficiency devices.

In this talk, I will discuss recently developed multiscale computational strategies that link fabrication process, nanostructure and property of thin films. The topics covered in this talk are:

1) A computational framework that effectively acts like a virtual "stereological microscope" to visualize morphology evolution from early stages of phase separation until the formation of the stable morphology. This multiscale framework is based on a continuum description of evaporation-induced phase-separation in ternary systems and is able to resolve nano-morphological features while being able to simulate device scale domains.

2) A suite of morphology descriptors that encode the various physical processes that affect the total power conversion efficiency of a photovoltaic cell.

3) A virtual performance characterization framework that efficiently "interrogates" the morphology to investigate relationships between the morphology at the nano-scale with the device performance.

Baskar Ganapathysubramanian is the William March Assistant Professor of Mechanical Engineering and Electrical and Computer Engineering at Iowa State University. His research interests are in stochastic analysis, multiscale modeling, and design of materials and processes using computational techniques. Ganapathysubramanian completed his PhD and MS from Cornell University and holds a BS degree from the Indian Institute of Technology-Madras.

\indent The goal of this work is to describe principles for optimal encoding of multidimensional stimuli using neural populations. I will describe an analytic framework for finding maximally informative boundaries that separate stimuli according to combinations of neural responses. We find that for Gaussian signals optimal decision boundaries are planar, regardless of neural noise level. For non-Gaussian (or sparse) signals that are typical of our sensory environment, optimal decision boundaries are curved, and their shape depends both on the number of neurons in the network and on noise in individual neurons. Finally, I will describe geometric properties of these decision boundaries that can be used as indicators of whether the network will be well described using pair-wise Ising model, and if so, what are the values of the pairwise interactions.

High-dimensional numerical simulation problems arise in diverse contexts, including in population balance models, an application that motivates our numerical algorithm design. To enable discretization in tens of dimensions, we focus on particle (meshless) methods with Markov jump process dynamics. While numerical methods for such systems have been known since the mid-1970s, efficient techniques that enable large-scale simulation are much more recent.

In this talk we present advances in three key aspects. First, we show how multiscale rate functions can be efficiently simulated by applying importance-sampling ideas to tau-leaping time-discretizations. This is particularly relevant for systems with a continuum of scales from slow to fast, without clear scale separations that would enable homogenization.

The second development we discuss is variable resolution in the sample-space, where non-uniform particle samplings can be used to achieve variance reduction for particular observables of the process. We show how such non-uniform or variable sampling can be driven by local error estimators, allowing adaptive resolution for the system state discretization.

Finally, we present a new parallelization technique for Markov jump processes, based on particle diffusion across well-chosen network paths between processor nodes. While the Markov process naively involves dense communication, we show how sparse communication can give accurate approximations with near-linear scaling.

\indent The space of positive definite symmetric matrices has been studied extensively as a means of understanding dependence in multivariate data along with the accompanying problems in statistical inference. Many books and papers have been written on this subject, and more recently there has been considerable interest in high- dimensional random matrices with particular emphasis on the distribution of certain eigenvalues. Our present paper is motivated by modern data acquisition technology, particularly, by the availability of diffusion tensor-magnetic resonance data. With the availability of such data acquisition capabilities, smoothing or nonparametric techniques are required that go beyond those applicable only to data arising in Euclidean spaces. Accordingly, we present a Fourier method of minimax Wishart mixture density estimation on the space of positive definite symmetric matrices.

\indent In the last 25 years, Elliptic Curve Cryptography has become a mainstream primitive for cryptographic protocols and applications. This talk will give a survey of elliptic curve cryptography and its applications, including applications of pairing-based cryptography which are built with elliptic curves. One of the information-theoretic applications I will cover is a solution to prevent pollution attacks in content distribution networks such as BitTorrent which use network coding to achieve optimal throughput. Another application I will cover is to privacy of electronic medical records and billings systems.

\indent A generalized complex structure (as introduced by Hitchin) consists of a complex structure on the direct sum of the tangent and cotangent bundles of a manifold, satisfying a certain integrability condition. A complex manifold can be regarded as a generalized complex manifold in a canonical way. This leads to an enlarged space of deformations: in addition to deformations as a complex manifold, there are also non-commutative and gerby deformations.

\indent Symplectic manifolds can also be regarded as generalized complex manifolds. For K3 surfaces, a complex structure can be deformed via generalized complex structures to a symplectic structure. It appears that these deformations connect pairs of Fourier-Mukai equivalent K3s to pairs of mirror K3s.

\indent This talk will be an introduction to generalized complex geometry, leading to a description of the above phenomena.

\indent Models and estimators for covariance matrices are very well studied. For non-Gaussian distributions, simply studying covariance gives an incomplete picture. Extending the Edgeworth series gives the pxpxp skewness tensor, the pxpxpxp kurtosis tensor, and so on. We describe a strategy for building multilinear factor models of cumulant tensors using subspace varieties. This leads to a difficult optimization problem and a fully implicit, gradient-based numerical optimization method using parallel transport on the Grassmannian to perform estimation. We also discuss some of the associated statistical challenges and applications.

\indent Pfaffian circuits are a new, geometrically motivated, and simplified construction of Valiant's holographic algorithms. These algorithms exploit dual Spinor varieties to simulate certain quantum computations (fermionic linear optics) classically, and provide a means to probe the conjectured classical-quantum boundary. Combinatorial problems addressed include planar NAE-SAT, lattice path problems and evaluation of certain Tutte polynomials. Basis change is one route to superposition-like effects, and some of the geometric considerations in analyzing Pfaffian circuits under arbitrary basis change will be discussed. Connections are made to the sum-product algorithm, SLOCC
equivalent entangled states, and monoidal categories.

Consider a population of fixed size that
evolves over time. At each time, the genealogical
structure of the population can be described by a
coalescent tree whose branches are traced back to
the most recent common ancestor of the population.
This gives rise to a tree-valued stochastic process.
We will study this process in the case of populations
whose genealogy is given by the Bolthausen-Sznitman
coalescent. We will focus on the evolution of the
time back to the most recent common ancestor and the
total length of branches in the tree.

Let f be a holomorphic selfmap of a complex manifold X. A point z in X
is said to lie in the Fatou set if the family of iterates is a normal
family in a neighborhood of z. A connected component of the Fatou set
is called a Fatou component. As all the orbits of a Fatou component
behave similarly, understanding Fatou components is an important step
in understanding complex dynamical systems. For rational functions in
the Riemann sphere Fatou components are quite well understood: every
Fatou component is preperiodic and periodic Fatou components have been
completely classified. Neither is true in higher dimensions, although
there has been some progress towards the description of periodic Fatou
components.

I will review what is known in the literature and present some recent
results. The talk will be geared towards complex analysts, not
dynamicists. This is joint work with Mikhael Lyubich.

\indent Variational integrators form a general class of structure preserving numerical algorithms for simulating dynamics. In this talk, we will present a new variational integrator, which combines techniques from spectral methods with the galerkin variational integrator framework. It will be shown that, under certain conditions, variational integrators constructed in this way inherit both the excellent convergence properties of classical spectral methods as well as structure preservation.

\indent In this talk, I will present two deblurring methods, one exploits the spatial interactions in images, i.e. the self-similarity; and the other explicitly takes into account the sparse characteristics of natural images and does not entail solving a numerically ill-conditioned backward-diffusion.

\indent In particular, the self-similarity is defined by a weight function, which induces two types of regularization in a nonlocal fashion. Furthermore, we get superior results using preprocessed data as input for the weighted functionals.

\indent The second part of the talk is based on the observation that the sparse coefficients that encode a given image with respect to an over-complete basis are the same that encode a blurred version of the image with respect to a modified basis. Following an ``analysis-by-synthesis'' approach, an explicit generative model is used to compute a sparse representation of the blurred image, and the coefficients of which are used to combine elements of the original basis to yield a restored image.

Data obtained describing terrorist events are particularly difficult to analyze, due to the many problems
associated with the both the data collection process, the inherent variability in the data itself, and the
usually poor level of measurement coming from observing political actors that seek not to provide
reliable data on their activities. Thus, there is a need for sophisticated modeling to obtain reasonable
inferences from these data. Here we develop a logistic random effects specification using a Dirichlet process to
model the random effects. We first look at how such a model can best be implemented, and then we use the
model to analyze terrorism data. We see that the richer Dirichlet process random effects model, as compared
to a normal random effects model, is able to remove more of the underlying variability from the data,
uncovering latent information that would not otherwise have been revealed.

\indent In this talk I will be discussing an extension of the analytic tools so that they work for problems on subdomains that might not even be Lipschitz and characterize the rates of convergence of domain decomposition methods.

\indent The lattice of non-crossing partitions of $[n]$, $NC(n)$, is a highly-studied, highly-symmetric playground for both algebraic and enumerative combinatorics. First seriously studied by Kreweras (1972), it has more recently come to light in two very different contexts: as the combinatorial underpinnings of the new, fruitful field of Free Probability, and having new unexpected connections to Coxeter groups and hyperplane arrangements (both discoveries made in the last 15 years).

\indent In this lecture, we will discuss non-crossing pair partitions of $[2n]$, $NC_2(2n)$. This set is in natural bijection with $NC(n)$. Consider the subset of those pair partitions that only pair $1$s to $0$s in some random bit-string of length $2n$. The enumerative properties of such classes of pairings are extremely important to some hard problems in free probability theory and random matrix theory. We will discuss what is known about this enumeration (including a "tight" theorem on symmetry and maximization), as well as some surprising and suggestive algebraic properties of posets associated to these pairings.

\indent This is joint work with Mahrlburg, Rattan, and Smyth, as well as Chou, Fricano, Poh, Shore, Whieldon, Wong, and Zhang.

\indent Gradient Ricci solitons are those Riemannian manifolds whose Ricci tensor is equal to a constant multiple of the metric plus the hessian of a function. I will discuss some aspects of the literature on complete gradient Ricci solitons assuming that the hessian of the function is not identically zero.

\indent We discuss a portion of the paper by Song and Yuan titled
"Metric Flips with Calabi Ansatz." In particular, they give an example of where the Kahler-Ricci flow performs a flip as an analytic analogue to Mori's minimal model program.

\indent One of the greatest accomplishments of the theory of general
relativity in the past century is the proof
of the positive mass/energy theorem for asymptotically flat spacetime.
This provides the theoretical
foundation for the stability of an isolated gravitating system.
However, the concept of mass/energy remains a challenging
problem because of the lack of a quasilocal description. Most
observable physical models are finitely extended spatial
regions and measurement of mass/energy on such a region is essential
in many fundamental issues. In fact, among Penrose's
list of major unsolved problems in classical general relativity, the
first one is ``Find a suitable quasi-local
definition of energy-momentum in general relativity". In this talk,
I shall describe a
new proposal of quasi-local mass/energy by Shing-Tung Yau of Harvard
University and myself.

\indent Recent advances in mixed-integer nonlinear optimization and the solution of optimization problems with differential equation constraints have heightened interest in methods that may be warm started from a good estimate of a solution. In this context, we present a regularized sequential quadratic programming (SQP) method based on a primal-dual augmented Lagrangian function. Trial steps are computed from carefully chosen subproblems that utilize relationships between traditional SQP, stabilized SQP, and the augmented Lagrangian. Each subproblem is well defined regardless of the rank of the Jacobian, and (to some extent) we challenge the belief that large penalty parameters should be avoided.

This is an expository talk on the behavior of the
Kahler-Ricci flow on a smooth minimal model of general type. I will
show that the flow converges to a singular Kahler-Einstein metric.

Recent years have seen the initial success of the variational implicit-solvent modeling of biomolecules. The dielectric boundary or the solute-solvent interface is the key quantity in such modeling. This work concerns the effective dielectric boundary force that contributes significantly to the conformation and dynamics of an underlying biomolecular system. Such a force is the "normal velocity" in the level-set numerical computation of equilibrium biomolecular structures. Precisely, the dielectric boundary force is defined as the shape derivative of the electrostatic free energy. Both the Poisson-Boltzmann free energy, and its Coulomb-field and Yukawa-field approximations are considered. Analytical formulas of the corresponding dielectric boundary force are derived. This is joint work with Hsiao-Bing Cheng, Li-Tien Cheng, and Xiaoliang Cheng, and Zhengfang Zhang.

Talk begins at 11:15 AM.

For many reasons, there exists an ongoing, healthy tension and competition (usually friendly!) between research communities that develop partial-differential equation (PDE) and boundary-integral equation (BIE) models and implementations. In this talk, I will describe two of my recent research efforts employing BIE to model the electrostatic component of biomolecule solvation, emphasizing the connections between PDE and BIE approaches in the hopes of encouraging deeper and more substantive communication and collaboration towards our shared goals: understanding experiments and the underlying mathematics. The first project addresses an electrostatic model I call BIBEE (boundary-integral based electrostatics estimation), which relies on a rigorous operator approximation of the boundary-integral formulation for the common mixed-dielectric Poisson PDE. BIBEE represents the BIE formulation of an earlier approximation by the Borgis group, and here BIE offers novel insights such as the fact that the approximation gives a provable upper bound to the true answer. In the second project, we explore the implications of nonlocal dielectric response by the solvent. This more sophisticated solvation model and its BIE formulation are relatively recent, and a combination of PDE and BIE approaches will undoubtedly offer a much more efficient route to identify and understand the limitations and strengths of this class of solvation models, as well as its connections to other theories of solvation.

It is very well known that every graph on $n$ vertices
and $m$ edges admits a bipartition of size at least $m/2$.
This bound can be improved to
$m/2 + (n-1)/4$ for connected graphs, and $m/2 + n/6$
for graphs without isolated vertices,
as proved by Edwards, and Erd\H{o}s, Gy\'arf\'as, and
Kohayakawa, respectively. A bisection
of a graph is a bipartition in which the size of the two parts
differ by at most 1. We prove
that graphs with maximum degree $o(n)$ in fact contain
a bisection which asymptotically achieves the above bounds.
These results follow from a more general theorem,
which can also be used to answer several questions
and conjectures of Bollob\'as and Scott on
judicious bisections of graphs.

Joint work with Po-Shen Loh and Benny Sudakov

Internet-scale quantum repeater networks will be heterogeneous
in physical technology, node functionality, and management. The
classical control necessary to use the network will therefore face
similar issues as Internet data transmission. In this talk, I will
describe the basic ideas behind quantum repeater networks, some of
their uses, the open problems that must be addressed, and our current
research on the Quantum Recursive Network Architecture, or QRNA.
As an example of the collision of quantum and classical engineering, I
will present our analysis of multiplexing schemes for repeater
networks, and show that Internet-style statistical multiplexing
improves aggregate throughput of the network, is robust and fair, and
can be easily implemented.

Bio: Rodney VAN METER received a B.S. in engineering and applied
science from the California Institute of Technology in 1986, an M.S. in
computer engineering from the University of Southern California in
1991, and a Ph.D. in computer science from Keio University in 2006.
His main research is architectures for distributed quantum
computation. Additional interests include storage systems,
networking, and post-Moore's Law computer architecture. He has held
positions in both industry and academia in the U.S. and Japan. He is
now an Associate Professor of Environment and Information Studies at
Keio University's Shonan Fujisawa Campus. Dr. Van Meter is a member
of AAAS, ACM and IEEE.

When Segre proved his celebrated characterization of conics (``every set of $q+1$ points in $PG(2,q)$, $q$ odd, no three
of which are collinear, is a conic''), he did more than proving a beautiful and interesting theorem; he in fact provided the starting point of a new direction in combinatorial geometry. In this branch of combinatorics the idea is to provide purely combinatorial characterizations of objects classically defined in an algebraic way.\\

In this talk we consider the following question:

\begin{quote}
Is it possible to characterize finite classical polar spaces by their intersection numbers with
respect to hyperplanes and subspaces of codimension $2$?
\end{quote}

{\bf Remark: I do not assume knowledge of finite geometry.}

\emph{Cops and Robbers} is a vertex pursuit game played on graphs which has
gained considerable recent attention. An intruder (or \emph{robber})\ is
loose on a network and some number of agents (or \emph{cops}) are trying to
capture him. The players are restricted to vertices, and move to
neighboring vertices along edges at alternate ticks of the clock. The
minimum number of cops needed to capture the robber is called the \emph{cop
number}. The most famous open problem on the cop number is \emph{Meyniel's
conjecture}, which claims that for a connected graph $G$ of order $n,$ $
c(G)=O(\sqrt{n}).$ We consider this and other conjectures and problems related to graphs with large cop number, algorithms for computing the
cop number, and the cop number of planar graphs.

We use a branching Markov process on the space of finite configurations
to solve a Dirichlet problem associated with the operator $\Delta u + u^2$.
We follow the pioneering works of M. Nagasawa, N. Ikeda, S. Watanabe,
M.L. Silverstein, and the approach of E.B. Dynkin.

In this talk, we will discuss a family of graphs related to
the high girth graphs D(k,q). The graphs D(k,q) were originally
constructed by utilizing a bilinear product based on the root system of an affine Lie algebra.? We give a modification of this
construction which applies to Generalized Kac-Moody algebras of rank 2. Using these constructions we obtain a new lower bound on the maximum number of edges in graphs without 14-cycles.? This is joint work with Art Terlep.

\indent The continuum self-consistent field theory is applied to the study of the adsorption of flexible polyelectrolyte (PE) onto the surfaces of two two-dimensional charged square objects with a constant electric field strength immersed in a weakly charged polyelectrolyte solution. The dependences of the different chain conformations, i.e., bridging, loop, tail and train, and in particular, the bridging chain conformation, on various system parameters (the charge fraction of the PE chains, the surface charge density, the object size, the salt concentration, etc.) are investigated. The efficient Multigrid method is adopted to numerically solve the modified diffusion equation and the Poisson equation. It is found that, the thickness L{\scriptsize B} of the boundary layer of the adsorbed PE chains is independent of the chain length, and scales with the surface charge density $\sigma$ and the fraction of charges on PE chains $\alpha_{p}$ as L{\scriptsize B}$\sim\sigma ^{0.36}$ and L{\scriptsize B}$\sim\alpha _{p}^{0.36}$, respectively. Simulation results reveal that, the total amount of bridging chain conformation in the system scales linearly with respect the size of the charge objects, and scales linearly with the chain length in the long polymer chain regime. Simulation results reveal that, the total amount of the bridging chain conformation in the system scales with the charge fraction of PE chains as a power law, and the scaling exponent is dependent on all the other system parameters. Simulation results show that, the total amount of charges on the adsorbed chains in the system can over-compensate the surface charges for relatively long chains with high charge fractions.

\indent I will talk about a proof of Lubotzky's conjecture on the quantitative version of Wang's theorem. Roughly the conjecture says that the asymptotic growth of the number of lattices in G a simple Lie group with covolume at most x, up to an automorphism of G, is the same as the subgroup growth of any lattice in G.

\indent I will discuss the behavior of the Kahler-Ricci flow on
projective bundles. We show that if the initial metric is in a
suitable Kahler class, the fibers collapse in finite time and the
metrics converge subsequentially in the Gromov-Hausdorff sense to a metric on the base. This is a joint work with J. Song and G.
Szekelyhidi.

\indent Multicomponent vesicle adhesion and fusion play important roles in many biological processes such as exocytosis, endocytosis. Many experimental and theoretical studies have focus on this subject. In this talk, we will first briefly review the biological background of the lipid bilayer vesicle membranes and the existing works on modeling the vesicle membranes, mainly the sharp interface model and the diffuse interface model. we will then consider the adhesion of multicomponent vesicle membranes. By using geometric description (sharp interface model) to represent the vesicle surface, and a phase field labeling function to distinguish the different components on the vesicle, the total energy, governing the equilibrium shapes of the vesicle, is set up. By solving the Euler-Lagrange equations, we present a number of typical adhered axisymmetric two-component vesicle profiles. A numerical experiment is conducted to show that adhesion may promote phase separation for a multicomponent vesicle. Thirdly, vesicle-vesicle adhesion and fusion process are discussed. By incorporating the adhesion effect, we mainly focus on the prefusion and postfusion states in the fusion process. Numerical experiments reveal that there can be many interesting equilibrium configurations of the prefusion and postfusion states. By carrying our simulations based on the gradient flow of the associated energy functional, we are also able to elucidate the dynamic transitions between the prefusion and postfusion states.

\indent More than 40 years ago, Erdos asked to determine the maximum possible number of edges in a $k$-uniform hypergraph on $n$ vertices with no matching of size $t$ (i.e., with no $t$ disjoint edges). Although this is one of the most basic problem on hypergraphs, progress on Erdos' question remained elusive. In addition to being important in its own right, this problem has several interesting applications. In this talk we present a solution of Erdos' question for $t < \dfrac{n}{(3k^2)}$. This improves upon the best previously known range $t = O \dfrac{n}{k^3}$, which dates back to the 1970's.

Joint work with P. Loh and B. Sudakov.

\indent I will explain how $p$-adic methods (the so-called $p$-adic local Langlands correspondence) can be used to prove the existence of an analytic continuation for complex $L$-functions.

\indent In $p$-adic Hodge Theory, comparison morphisms relate $p$-adic etale cohomology of varieties over local fields of mixed characteristic $(0,p)$ with their de Rham cohomology. I will present a construction of such a morphism that uses Chern classes from motivic cohomology into etale and de Rham cohomology.

\indent Let $k$ be of a field of characteristic $0$. The first Weyl algebra $A_1(k) = k/(yx-xy-1)$ is $Z$-graded with deg$(x) = 1$, deg$(y) = -1$, and is a simple ring of $GK$-dimension $2$. Sierra has studied its category of graded modules and shown how to find all $Z$-graded algebras with an equivalent graded module category. Smith has also shown how the geometry of this example is related to a certain stack. Our goal is to study more general classes of $Z$-graded simple rings to find more examples which may have interesting algebraic and geometric properties. Specifically, we study the structure of $Z$-graded simple rings $A$ with graded quotient ring $Q$ such that $Q_0$ is a field with trdeg $Q_0 = GK A - 1$. As a special case, we can classify all $Z$-graded simple rings of $GK$-dimension $2$. This is joint work with Jason Bell.

\indent Generalized barycentric coordinate functions allow for novel, flexible finite element methods accommodating polygonal element geometries. The Sobolev-norm error estimates associated to such methods, however, require varying levels of geometric criteria on the polygons, depending on the definition of the coordinate functions. In this talk, I will discuss these criteria for a variety of coordinate definitions and discuss the practical tradeoffs between enforcing geometric constraints and computing finite element basis functions over polygons.

\indent In a recent paper, Rains and Vazirani used Hecke algebra techniques to develop $(q,t)$-generalizations of a number of well-known vanishing identities for Schur functions. However, their approach does not work directly at $q=0$ (the Hall-Littlewood level). We discuss a technique that is more combinatorial in nature, and allows us to obtain generalizations of some of their results at $q=0$ as well as a finite-dimensional analog of a recent summation formula of Warnaar. We will also briefly explain how
these results are related to $p$-adic representation theory. Finally, we will explain how this method can be extended to give an explicit construction of Hall-Littlewood polynomials of type $BC$.

\indent I will discuss results in analysis on a compact Lie group that can be obtained using Brownian motion. These include a ``Hermite expansion" and an analog of the Segal-Bargmann transform. Both results can be understood by lifting Brownian motion in the group to Brownian motion in the Lie algebra. I will also briefly discuss an open problem concerning the infinite-dimensional limit of these results.

\indent We present the context and theorems for noncommutative maps, or dimension free maps evaluated on tuples of matrices. This turns out to be much more rigid than functions from the classical commutative case of several complex variables or several real variables. We present some theorems of Helton et al on free analytic maps in the context of change of variables.

\indent Ionic size effects are significant in many biological systems. Mean-field descriptions of such effects can be efficient but also challenging. When ionic sizes are different, explicit formulas in such descriptions are not available for the dependence of the ionic concentrations on the electrostatic potential, i.e., there are no explicit, Boltzmann type distributions. This work begins with variational formulations of the continuum electrostatics of an ionic solution with such non-uniform ionic sizes as well as multiple ionic valences. An augmented Lagrange multiplier method is then developed and implemented to numerically solve the underlying constrained optimization problem. Extensive numerical tests demonstrate that the mean-field model and numerical method capture qualitatively some significant ionic size effects, particularly those for multivalent ionic solutions, such as the stratification of multivalent counterions near a charged surface. The ionic valence-to-volume ratio is found to be the key physical parameter in the stratification of concentrations. All these are not well described by the classical Poisson--Boltzmann theory, or the generalized Poisson--Boltzmann theory that treats uniform ionic sizes. Finally, various issues such as the close packing, limitation of the continuum model, and generalization to molecular solvation are discussed. This is joint work with Zhongming Wang and Bo Li.

\indent A rigorous convergence theory for Galerkin methods for a model Helmholtz problem in $R^{d}, d=1,2,3,$ is presented.
General conditions on the approximation properties of the approximation space are stated that ensure quasi-optimality of the method. As an application of the general theory, a full error analysis of the classical hp-version of the finite element method (hp-FEM) is presented where the dependence on the mesh width $h$, the approximation order $p$, and the wave number $k$ is given explicitly. In particular, it is shown that quasi-optimality is obtained under the conditions that $kh/p$ is sufficiently small and the polynomial degree $p$ is at least $O(log k)$. This result improves existing stability conditions substantially.

\indent We will consider two recent open problems stating that
certain statistics on various sets of combinatorial objects are
equidistributed. The first, posed by Anders Claesson and Svante
Linusson, relates nestings in matchings of $2n$ points on a line to
occurrences of a certain pattern in permutations in $S_n$. The second, posed by Miles Jones and Jeffrey Remmel, relates occurrences of a large class of consecutive permutation patterns to occurrences of the same pattern in the cycles of permutations. We will prove an extension of the Garsia-Milne involution principle and use it to solve both problems.

\indent If a variety $X$ is a quotient of a smooth variety by a finite group, it has quotient singularities---that is, it is \emph{locally} a quotient by a finite group. In this talk, we will see that the converse is true if $X$ is quasi-projective and is known to be a quotient by a torus. In particular, all quasi-projective simplicial toric varieties are global quotients by finite groups! Though the proof is stack-theoretic, the construction of a smooth variety $U$ and finite group $G$ so that $X=U/G$ can usually be made explicit purely scheme-theoretically.

\indent To illustrate the construction, I'll produce a smooth variety $U$ with an action of $G=\mathbb{Z}/2$ so that $U/G$ is the blow-up of $\mathbb{P}(1,1,2)$ at a smooth point. This example is interesting because even though $U/G$ is toric, $U$ cannot be taken to be toric. This is joint work with Matthew Satriano.

\indent An $r$-graph is like a graph except that every edge contains $r$ vertices instead of two. We'll talk about how to find a large independent set in an $r$-graph, which means a set of vertices not containing any edge. Conversely, we'll also discuss how to generate a hypergraph where large independent sets do not exist. Some of research is joint with Jacques Verstraete.

\indent I will begin by giving a broad overview of the geometric Langlands progam, with emphasis on the local to global results of Beilinson, Drinfeld, Frenkel, and Gaitsgory. Their methods are particularly compelling because they do not have analogues in the original Langlands program. Using geometric Langlands as motivation, I will introduce the Hitchin fibration and describe recent results regarding the geometry of the global nilpotent cone (i.e., Hitchin fiber over zero). These results may be viewed in the
context of a global analogue of Springer theory, which suggests many future directions. If there is time, I will also explain relations to classical and higher rank Brill-Noether theory.

\indent It is widely known the classical Poisson-Boltzmann theory fails to describe electrostatic interactions in the cases of multivalent counterions, highly charged surfaces, and low temperatures, as the theory ignores many-body ion-ion correlations. Surprising phenomena due to correlation effects include the charge inversion and the attraction between two likely charged colloids, both of which are recent focuses of theoretical and experimental study. In this talk, we will present some theoretical and simulation results on these problems by studying the role of image charges on spherical colloids, and discuss the method to include the correlation effects in the mean-field PB theory.

\indent A``geometric'' construction of $SU(2) (2+1)-$dimensional TQFTs associates to a surface the vector space spanned my multi-curves modulo local relations. The relevant local relations are defined by the Jones-Wenzl projectors. This talk will outline an approach to categorification of this picture construction of TQFTs, in particular categorification of the Jones-Wenzl projectors, and it will explain how the evaluation at a root of unity may be viewed as a localization of a category. (Joint work with Ben Cooper).

\indent We will discuss the notion of arithmetic matroid, whose main example is provided by a list of elements of a finitely generated abelian group. Guided by the geometry of toric arrangements, we will present a combinatorial interpretation of the associated arithmetic Tutte polynomial, which generalizes Crapo's formula for the classical Tutte polynomial.

\indent The aim is to show a convergence theorem of the Kahler-Ricci flow: if the initial metric is sufficiently close to a shrinking
Kahler-Ricci soliton with respect to a holomorphic vector field, then
the modified Kahler-Ricci flow by this holomorphic vector field will
converge to a shrinking Kahler-Ricci soliton.

\indent In chemical systems, one studies the concentrations of chemical species over time. Of particular interest are steady-state concentrations, and more specifically, when systems admits multiple equilibria. Recent results of Helton partially characterize this possibility in terms of the sign pattern of the stoichiometric matrix, where the stoichiometric matrix describes the chemical species' interactions.

\indent Many biological and physiological processes involve
self-regulating mechanisms that prevent too much growth while ensuring against extinction: the rate of growth is somewhat random (``noisy"), but the distribution depends on the current state of the system. Cancer growth and neurological control mechanisms are just a few examples. In finance, as well, markets self-regulate since people want to "buy low" and "sell high".

\indent Some questions that we'd like to answer are: does the system have a well-defined average? In more technical terms, we want to know if the system is ergodic. How does this long-term average compare to the long-term behavior of the deterministic (not random) system? What can we say about the distribution of ``survival times", i.e. the distribution of times until the system reaches a particular value?

\indent In this talk we will look at (and listen to) a simple example of a noisy, discrete dynamical system with parametric noise and explore ways to answer these questions analytically. We prove ergodicity for a class of growth models, and show that the randomness is harmful to the population in the sense that the long-term average is decreased by the presence of noise. When systems obeying noisy growth laws are connected together as a coupled lattice, the long-term effects of the
noise can have damaging effects on the organism as a whole, even though local interactions might favor growth in a particular area. We will present simulations that highlight the effect of both the noise and the local coupling on the survival of the organism.

\indent We use toric symmetry and degeneration to study the GW theory of $X = {\bf P^1} \times {\bf P^1} \times {\bf P^1}$ . There exists a toric blowup $Y$ of $X$ whose polytope is the permutohedron. The permutohedron admits a symmetry which is manifest in the GW theory of $Y$. We use degeneration to show this symmetry descends to $X$ itself. This also shows a subtle relationship between the GW theories of ${\bf P^1} \times {\bf P^1} \times {\bf P^1}$ and ${\bf P^3}$. All this is joint work with Dhruv
Ranganathan.

\indent Starting with $n$ particles at the origin in $Z^d$, let each particle in turn perform simple random walk until reaching an unoccupied site. Lawler, Bramson and Griffeath proved that with high probability the resulting random set of n occupied sites is close to a ball. We show that its fluctuations from circularity are, with high probability, at most logarithmic in the radius of the ball, answering a question posed by Lawler in 1995 and confirming a prediction made by chemical physicists in the 1980's. Joint work with David Jerison and Scott Sheffield.

\indent I will first review the notion of Galois averages of Rankin-Selberg $L$-functions, in particular those of Rankin-Selberg $L$-functions of weight-two cusp forms times theta series associated to Hecke characters of imaginary quadratic fields. I will then present a conjecture about the behaviour of these averages with the conductor of the character, of which the nonvanishing theorems of Rohrlich, Vatsal and Cornut-Vatsal are special cases. Finally, I will explain a strategy of proof, at least in the setting where the class number is equal to one.

\indent This talk discusses the derivation and analysis of mathematical models motivated by the experimental induction of contour phosphenes in the retina. First, a spatially discrete chain of periodically forced coupled oscillators is considered via reduction to a chain of scalar phase equations. Each isolated oscillator locks in a 1-2 manner with the forcing, so there is intrinsic bistability, with activity peaking on either the odd or even cycles of the forcing. If half the chain is started on the odd cycle and half on the even cycle ("split state"), then with sufficiently strong coupling a wave can be produced which can travel in either direction due to symmetry. Numerical and analytic methods are employed to determine the size of coupling necessary for the split state solution to destabilize such that waves appear. Next we take a continuum limit, reducing the chain to a partial differential equation. We use a Melnikov function to compute, to leading order, the speed of the traveling wave solution to the partial differential equation as a function of the form of coupling and the forcing parameters and compare our result to numerically computed discrete and continuum wave speeds. This is joint work with Bard Ermentrout and Jonathan Rubin, published in Physica D volume 240, issue 7 as a paper under the same name.

\indent A tropical curve is a vertex-weighted metric graph. It is hyperelliptic if it admits an involution whose quotient is a tree. Assuming no prior knowledge of tropical geometry, I will develop the theory of tropical hyperelliptic curves and discuss the relationship with classical algebraic curves and their Berkovich skeletons. Along the way, we will see some nice combinatorics, including an analogue for graphs of holomorphic maps of Riemann surfaces.

\indent We discuss the challenge of understanding the WILD representation theory of Lie algebras over fields of positive characteristic. Even very explicit examples lead to difficult, if not impossible, problems. One can make some computations, but how does one give structure to these computations? Recent joint work with Julia Pevtsova introduces algebro-geometric invariants for such representations, an approach which leads to algebraic vector bundles on familiar (and not so familiar) algebraic varieties.

\footnotesize

The talk is devoted to the problem of characterization of integrals as linear functionals. The main idea goes back to Hadamard. The first well known results in this field are the F.Riesz theorem (1909) on integral presentation of bounded linear functionals by Riemann-Stiltjes integrals on the segment and the Radon theorem (1913) on integral presentation of bounded linear functionals by Lebesque integrals on a compact in Rn. After papers of I.Radon, M.Fréchet and F.Hausdorff the problem of characterization of integrals as linear functionals is used to be formulated as the problem of extension of Radon theorem from Rn on more general topological spaces with Radon measures. This problem turned out to be rather complicated. The history of its solution is long and rich. It is quite natural to call it the Riesz-Radon-Fréchet problem of characterization of integrals. The important stages of its solution are connected with names of S.Banach (1937-38), Sacks (1937-38), Kakutani (1941), P.Halmos (1950), Hewitt (1952), Edwards (1953), N.Bourbaki (1969), and others. Some essential technical tools were developed by A.D.Alexandrov (1940--43), M.Stone (1948--49), D.Fremlin (1974), and others.
In 1997 A.V.Mikhalev and V.K.Zakharov had found a solution of Riesz-Radon-Fréchet problem of characterization for integrals on an arbitrary Hausdorff topological space for nonbounded positive radom measures.
The next modern period of this problem for arbitrary Radon measures is connected mostly with results by A.V.Mikhalev, T.V.Rodionov, and V.K.Zakharov. A special attention is paid to algebraic aspects used in the proof.

\indent We consider a model of asexually reproducing individuals. The birth and death rates of the individuals are affected by a fitness parameter. The rate of mutations that cause the fitnesses to change is proportional to the population size, $N$. The mutations may be either beneficial or deleterious. In a paper by Yu, Etheridge and Cuthbertson (2009) it was shown that the average rate at which the mean fitness increases in this model is bounded below by $\text{log}^{1-\delta}N$ for any $\delta > 0$. We achieve an upper bound on the average rate at which the mean fitness increases of $O(\text{log}\hspace{1pt}N/\text{log log} \hspace{1pt}N)$.

\indent Ask a physicist. You will be told that it's a manifold with some anticommuting coordinates. I will explain this cryptic answer by addressing the following natural questions.

(1) What does this mean? That is, in what mathematical context is it meaningful?

(2) Why would anyone want to do this? That is, what applications does it have?

\small
A variety of linear algebras over a field is said to be Schreier if any subalgebra of a free algebra of this variety is free. The variety of all algebras, the variety of all commutative algebras, the variety of all anti-commutative algebras, the variety of all Lie algebras, the variety of all Lie superalgebras, varieties of all Lie p-algebras and Lie p-superalgebras are the main types of Schreier varieties of algebras.
Let A(X) be the free algebra of a Schreier variety of algebras with the set X of free generators. A system of elements $u_1,…, u_n$ of A(X) is primitive if there is a set Y of free generators of the free algebra A(X) such that $u_1,…, u_n$ belong to Y.
An element u of A(X) is said to be almost primitive if u is not a primitive element of A(X), but u is a primitive element of any subalgebra of A(X) which contains it.
Algorithms to recognize primitive systems of elements of free algebras of the main types of Schreier varieties of algebras are constructed. We obtain also algorithms to construct complements of primitive systems of elements with respect to free generating sets. Series of almost primitive elements are constructed.
This talk is based on joint works with C.Champagnier, A.A.Chepovskii, A.V.Klimakov, A.V.Mikhalev, I.P.Shestakov, U.U.Umirbaev, J.-T.Yu, and A.A.Zolotykh.

\indent If we have a smooth compact manifold, then its cotangent bundle has a natural symplectic form. A smooth affine variety also has a natural symplectic form. One can ask the following question: which cotangent bundles are symplectomorphic to smooth affine varieties? We construct many cotangent bundles that are not symplectomorphic to smooth affine varieties.

\indent The main tool used to distinguish these objects is called wrapped Floer cohomology.

\indent Given a Pfaffian system on a smooth manifold, we shall discuss the notion of reduced submanifold and how to find them. This was motivated from the problem of deciding the minimality of generic CR manifolds. As best known by the Noether's theorem conservation laws arise from the symmetry of differential equations. We approach the conservation laws from the viewpoint of the reduction of Pfaffian systems and discuss some possible applications.

\indent Counting curves on threefolds has been defined in several conjecturally equivalent instances, by integrating with respect to a virtual class on a moduli space (of stable maps for Gromov-Witten theory, ideal sheaves for Donaldson-Thomas theory, and stable pairs for Pandharipande-Thomas theory). The analogous picture for K3 surfaces is incomplete. The Gromov-Witten theory has been calculated by Maulik, Pandharipande, and Thomas, and was shown to give rise to modular forms. In joint work with Benjamin Bakker we define and compute an analog of DT/PT theory on K3 surfaces via stable pairs and show that it similarly gives rise to modular forms on $\Gamma(4)$.

\indent I will first describe a generalization of a result by Haagerup and Thorbjornsen on the asymptotic norm of non-commutative polynomials of random matrices, in the case of unitary matrices. Then I will show how such results help us refine our understanding of the outputs of random quantum channels. In particular one obtains optimal estimates for the minimum output entropy of a large class of typical quantum channels. The first part of this talk is based on joint work with Camille Male, and the second part is based on joint work with Serban Belinschi and Ion Nechita.

It is well-known from Weil that the isomorphism classes of rank n vector bundles on an algebraic curve can be written as the set of certain double cosets of GL(n,A), where A is the adeles of the curve. I will introduce such presentation in the realm of algebraic geometry and discuss two of its applications: the first is the Tamagawa number formula (proved by Gaitsgory-Lurie), and the second is the Verlinde formula in positive characteristic

\indent One-parameter convolution semigroups of probability measures on Euclidean space are related to limits of partial products of infinitesimal triangular systems of measures, in the sense that such limits are embeddable into one-parameter convolution semigroups. It is a long-standing problem related to the central limit theorem that on an arbitrary locally compact group such a result cannot be tackled unless the infinitesimal system is commutative and additional conditions on the underlying group and/or the limiting measure are satisfied. We shall develop the main steps towards the solution of the problem of embeddable limits and connect the problem with the embedding of infinitely divisible probability measures on the group. The problem, in full generality, is still open.

\indent We consider the question of global in time existence and uniqueness of solutions of the infinite depth full water wave problem. We show that the nature of the nonlinearity of the water wave equation is essentially of cubic and higher orders. For any initial data that is small in its kinetic energy and height, we show that the 2-D full water wave equation is uniquely solvable almost globally in time. And for any initial interface that is small in its steepness and velocity, we show that the 3-D full water wave equation is uniquely solvable globally in time.

We investigate the transversality of holomorphic mappings between CR submanifolds of complex spaces. In equidimension case, we show that a holomorphic mapping sending one generic submanifold into another of the same dimension is CR transversal to the target, provided that the source manifold is of finite type and the map is of generic full rank. In different dimensions, the situation is more delicate. We will show that under certain restrictions on the dimensions and the rank of Levi forms, the mappings for which the set points where the map has degenerate rank is of codimension at least 2, is transversal to the target. In addition, we show that under more restrictive conditions on the manifolds, finite holomorphic mappings are transversal.

We show that any $k$-uniform hypergraph with $n$ edges contains two
edge
disjoint subgraphs of size $\tilde{\Omega}(n^{2/(k+1)})$ for $k=4,5$
and
$6$. This result is best possible up to a logarithmic factor due to a
upper bound construction of Erd\H{o}s, Pach, and Pyber who show there
exist $k$-uniform hypergraphs with $n$ edges and with no two edge
disjoint isomorphic subgraphs with size larger than
$\tilde{O}(n^{2/(k+1)})$. Furthermore, this result extends results
Erd\H{o}s, Pach and Pyber who also established the lower bound for $k=2$
(eg. for graphs), and of Gould and R\"odl who established the result for

I will introduce a new geometric flow on complex, non-Kahler
manifolds. I will exhibit Perelman-type functionals for this flow, and some regularity results. Finally I will present an optimal regularity conjecture and discuss its relationship to the long open problem of the classification of Class VII surfaces. This is joint work with G.Tian.

Given any closed hyperbolic surface of a fixed genus, we
establish an identity involving dilogarithm of lengths of simple
closed geodesics in all embedded pairs of pants and one-holed tori in
the surface. One may consider this as a counterpart of McShane's
identity for closed hyperbolic surfaces. This is a joint work with Ser
Peow Tan.