**Colloquium**

**Andrei Martinez-Finkelshtein**

University of Almeria

**Random Matrix Models, Non-intersecting random paths, and the Riemann-Hilbert Analysis** (pdf)

Random matrix theory (RMT) is a very active area of research and a great source of exciting and challenging problems for specialists in many branches of analysis, spectral theory, probability and mathematical physics. The analysis of the eigenvalue distribution of many random matrix ensembles leads naturally to the concepts of determinantal point processes and to their particular case, biorthogonal ensembles, when the main object to study, the correlation kernel, can be written explicitly in terms of two sequences of mutually orthogonal functions.

Another source of determinantal point processes is a class of stochastic models of particles following non-intersecting paths. In fact, the connection of these models with the RMT is very tight: the eigenvalues of the so-called Gaussian Unitary Ensemble (GUE) and the distribution of random particles performing a Brownian motion, departing and ending at the origin under condition that their paths never collide are, roughly speaking, statistically identical.

A great challenge is the description of the detailed asymptotics of these processes when the size of the matrices (or the number of particles) grows infinitely large. This is needed, for instance, for verification of different forms of “universality” in the behavior of these models. One of the rapidly developing tools, based on the matrix Riemann-Hilbert characterization of the correlation kernel, is the associated non-commutative steepest descent analysis of Deift and Zhou.

Without going into technical details, some ideas behind this technique will be illustrated in the case of a model of squared Bessel nonintersecting paths.

**Samuel Lisi**

Université de Nantes

**Symplectic Topology and Hamiltonian Dynamics**

In the 80s, the foundational work of Conley, Zehnder, Gromov and Floer opened up the field of symplectic topology. In subsequent developments, many tools have been developed to tackle questions about qualitative behavior of Hamiltonian dynamical systems. I will discuss a few of these applications, focusing on a recent discovery of a new obstruction to symplectic embedding.

**Maksym Derevyagin**

Katholieke Universiteit Leuven

**An Operator Approach to Pade Approximation**

The goal of the talk is twofold. First, a survey of results on convergence of Pade approximants for rational perturbations of the Cauchy transform of a measure will be given. The main feature of the results is that they can be obtained by using spectral properties of the underlying band matrices. It should be stressed here that these band matrices turn out to be perturbations of the Jacobi matrices corresponding to the Cauchy transform of a positive measure. Second, an operator approach for multipoint Pade approximation to the Cauchy transform of a positive measure will be presented.

**Timothy Ferguson**

Vanderbilt University

**Extremal Problems in Spaces of Analytic Functions**

Extremal problems have long played an important role in complex analysis. For example, the proof of the Riemann mapping theorem involves an extremal problem, and the famous Bieberbach conjecture (proved by de Branges) is about an extremal problem. I will discuss some of my recent results about extremal problems in spaces of analytic functions, particularly Bergman spaces and Fock spaces. The results will include regularity properties and some explicit solutions. I will also talk about a surprising connection between an extremal problem involving Toeplitz operators, the first eigenvalue of the Laplacian, and related quantity called the torsional rigidity.

**Carolyn Chun**

Brunel University London

**Inductive tools for handling internally 4-connected binary matroids**

A binary matroid is internally 4-connected if it does not break up as a 1-, 2-, or 3-sum. The class of such matroids includes the cycle matroids of internally 4-connected graphs, those 3-connected simple graphs that are 4-connected except for the possible presence of degree-3 vertices. Given internally 4-connected binary matroids M and N where N is a proper minor of M, we are interested in removing a small number of elements from M to obtain another internally 4-connected matroid that maintains an N-minor. As Johnson and Thomas noted in 2002, even for graphic matroids, we cannot always succeed in doing this. We are, however, able to show that we can obtain the desired theorem if we replace the notion of “removing a small number of elements” with “performing a small number of simple moves.” This talk will formalize these notions and give an update on our work toward this end.

**Daniel Fiorilli**

University of Michigan

**Nuclear Physics and Number Theory**

While the two fields named in the title seem unrelated, there is a strong link between them. Indeed, random matrix theory makes predictions in both fields, and some of these predictions can be verified rigorously on the number theory side. This amazing connection came to life during a meeting between Freeman Dyson and Hugh Montgomery at the Institute for Advanced Study. Random matrices are now known to predict many statistics about zeta functions, such as moments, low-lying zeros and correlations between zeros. The goal of this talk is to discuss this connection, focusing on number theory. We will cover both basic facts about the zeta functions and recent developments in this active area of research.

**Stan Dziobiak**

University of Mississippi

**On excluded and unavoidable minors in graphs**

A classical result of Kuratowski and Wagner states that a graph G is planar if and only if K5 and K3;3 are not minors of G. (A minor of a graph G is a graph obtained from G by any sequence of the following operations: contracting edges, deleting edges, and deleting vertices). It is widely regarded as the starting point of graph minor theory. In the mid 80’s, Robertson and Seymour proved one of the deepest theorems in combinatorics, known as the Graph Minor Theorem (GMT): in any infinite set of graphs, at least one graph is a minor of another. Equivalently, a class of graphs is closed under minors if and only if it has a finite number of excluded minors (known as the obstruction set for the class). Knowing the obstruction set for a given minor-closed class of graphs is important as it allows for a polynomial-time algorithm to test membership in the class. Unfortunately the proof of the GMT is non-constructive, and hence gives no method of

nding such obstruction sets. In this talk, we will present certain classes of graphs for which the obstruction sets are known, including our characterization of apex-outerplanar graphs, and progress on the characterization of a bigger class, namely that of apex-series-parallel graphs. We will also discuss some results on unavoidable minors in graphs, including our result on the sizes of the unavoidable minors in 3-connected graphs.

**Sevak Mkrtchyan**

Carnegie Mellon University

**The dimer model on the hexagonal lattice**

The dimer model is the study of random perfect matchings on graphs, and has a long history in statistical mechanics. On the hexagonal lattice it is equivalent to tilings of the plane by lozenges and to 3D stepped surfaces called skew plane partitions – 3 dimensional analogues of Young diagrams with a partition removed from the corner. This particular instance of the model has been intensely studied in the past 15 years by Kenyon, Okounkov, Reshetikhin and many co-authors. I will discuss the scaling limit of the model under a certain family of measures called “volume”-measures, the phase-transition phenomenon in this model, the effects of varying the boundary conditions on the limit shape, the nature of local fluctuations in various regions of the limit shape and connections with random matrix theory.

**Bernard Lidicky**

University of Illinois at Urbana-Champaign

**Applications of Flag Algebras in Hypercubes and Permutations**

Flag algebras is a method, recently developed by Razborov, designed for attacking problems in extremal graph theory. There are recent applications of the method also in discrete geometry or permutation patterns. The aim of talk is to give a gentle introduction to the method and show some of its applications to hypercubes and permutations. The talk is based on a joint works with J. Balogh, P. Hu, H. Liu, O. Pikhurko, B. Udvari, and J. Volec.

**Sean O’Rourke**

Yale University

**Singular values and vectors under random perturbation**

Computing the singular values and singular vectors of a large matrix is a basic task in high dimensional data analysis with many applications in computer science and statistics. In practice, however, data is often perturbed by noise. A natural question is the following. How much does a small perturbation to the matrix change the singular values and vectors?

Classical (deterministic) theorems, such as those by Davis-Kahan, Wedin, and Weyl, give tight estimates for the worst-case scenario. In this talk, I will consider the case when the perturbation is random. In this setting, better estimates can be achieved when our matrix has low rank. As an application, I will discuss several matrix reconstruction problems including a Netflix-type problem. This talk is based on joint work with Van Vu and Ke Wang.

**Kevin McGoff**

Duke University

**Stochastic Properties of Dynamical Systems**

In this talk I discuss recent work on both statistical and probabilistic aspects of dynamical systems. In many scientific settings, one observes data that is believed to be generated by a dynamical system from within a class of model systems, and the statistical problem is to infer the generating system from the data. In recent work with S. Mukherjee, A. Nobel, and N. Pillai, we show that maximum likelihood estimation provides a consistent inference procedure for some classes of systems.

In a separate project joint with R. Pavlov, we consider dynamical systems from a probabilistic perspective. In particular, we study random shifts of finite type, which are dynamical systems whose rules of evolution are chosen at random. In this setting, we describe some likely properties of a system chosen at random.

**Hehui Wu**

Simon Fraser University

**Dicromatic number and fractional chromatic number** (pdf)

Given an undirected graph $G$, the chromatic number $\chi(G)$ is the minimum order of partitions of $V(G)$ into independent sets. Given a directed graph $D$, a vertex set is acyclic if it does not contain a directed cycle. The chromatic number $\chi(D)$ is the minimum order of partitions of $V(G)$ into acyclic sets. The dichromatic number of an undirected graph $G$, denoted by $→\chi(G)$, is the maximum chromatic number over all its orientations. Erdös and Neumann-Lara proved that $C_1n/\log n≤→\chi(K_n)≤ C_2n/\log n$ for some constants $C_1,C_2$. They conjectured that if the dichromatic number of a graph is bounded, so is its chromatic number.

Let $I(G)$ be the set of all independent sets of $G$, and let $I(G,x)$ be the set

of all those independent sets which include vertex $x$. For each independent set I, define a nonnegative real variable $x_I$. The fractional chromatic number $\chi_f(G)$ is the minimum value of $\sum_{I\in I(G)}x_I$, subject to $\sum_{I\in I(G,x)}x_I\geq 1$ for each vertex $x$. We prove that there is a constant $C$ such that for any graph $G$, we have $→\chi(K_n) ≥ CN/\log n$. This is joint work with Professor Bojan Mohar in Simon Fraser University.

**Javid Validashti**

University of Illinois at Urbana-Champaign

**Algebraic Methods in Geometric Problems** (pdf)

In this talk, I will describe some algebraic methods to study problems in geometry, where there is a beautiful interplay between ideas from commutative algebra, algebraic geometry and combinatorics. The main goal is to understand how geometric ideas manifest in various algebraic structures and invariants. It turns out there are often elegant characterizations of geometric properties in terms of algebraic conditions, that can be captured numerically. For instance, certain equisingularity conditions are translated into algebraic conditions known as integral dependence of modules, that are governed by numerical invariants referred to as multiplicities.

**Nikolai Chernov**

University of Alabama at Birmingham

**Dynamical Models for Superdiffusion and Superconductivity** (pdf)

Electrical current is modeled by a periodic Lorentz gas under an external field. One can also picture this as a pinball machine or a Glaton board. Depending on the size and positions of the pins (heavy molecules) the resulting current may be normal (satisfying Ohm’s law and the Einstein relation) or ballistic (exhibiting superconductivity). Most of the facts were first observed by physicists in computer experiments and recently proven rigorously by mathematics. A few facts, though, were first discovered mathematically and then verified experimentally.

**Guillermo Reyes**

Polytechnic University of Madrid

**The Cauchy Problem for The Porous Medium Equation with Variable Density. Asymptotic Behavior of Solutions** (pdf)

**Joseph Brennan**

University of Central Florida

**Commutative Algebras Associated to Graphs** (pdf)

**Richard M. Aron**

Kent State University

**Smooth Surjections from Non-Separable Banach Spaces** (pdf)

**Susan Cooper**

University of Nebraska

**Powerful Invariants of Points** (pdf)

**David Farmer**

American Institute of Mathematics

**Differentiating Polynomials** (pdf)

**Guillermo Curbera**

University of Sevilla, Spain

**Mathematicians of the World: Unite!** (pdf)

**Bernardo Cascales**

Universidad de Murcia, Spain

**Scalar, vector and multi-valued integration** (pdf)

**Sean Sather-Wagstaff**

North Dakota State University

**Structure of homomorphism sets, and generalizations** (pdf)

**Michal Karonski**

Adam Mickiewicz University, Poland

** On the 1-2-3-conjecture** (pdf)

**Florian Enescu**

Georgia State University

**Multiplicity in algebra and geometry** (pdf)

**Vlad Timofte**

University of Mississippi

**The solution of a long standing open problem: Finding a good differentiation theory on locally convex spaces** (pdf)

**Maxim Zinchenko**

California Institute of Technology

**Spectral Theory for Jacobi Matrices** (pdf)