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Random Matrix Models, Non-intersecting random paths, and the Riemann-Hilbert Analysis

Andrei Martínez-Finkelshtein

Universidad de Almería

Almería, SPAIN

3:00pm-4:00pm

NES 103

E. A. Rakhmanov

**Abstract**

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.

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Random Matrices, Integrable Wave Equations, Determinantal Point Processes: a Swiss-Army Knife Approach

Marco Bertola

Concordia University

3:00pm-4:00pm

CMC 130

Seung-Yeop Lee

**Abstract**

Random Matrix models, nonlinear integrable waves, Painlevé transcendents, determinantal random point processes seem very unrelated topics. They have, however, a common point in that they can be formulated or related to a Riemann-Hilbert problem, which then enters prominently as a very versatile tool. Its importance is not only in providing a common framework, but also in that it opens the way to rigorous asymptotic analysis using the nonlinear steepest descent method. I will briefly sketch and review some results in the above-mentioned areas.

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Virtual Endomorphisms of Groups

Said Sidki

Universidade de Brasilia

3:00pm-4:00pm

CMC 130

Dmytro Savchuk

**Abstract**

A virtual endomorphism of a group \(G\) is a homomorphism \(f:H\rightarrow G\) where \(H\) is a subgroup of \(G\) of finite index \(m\). A recursive construction using \(f\) produces a so-called *state-closed* (or, *self-similar* in dynamical terms) representation of \(G\) on a \(1\)-rooted regular \(m\)-ary tree. The kernel of this representation is the \(f\)-\(\mathrm{core}(H)\); i.e., the maximal subgroup \(K\) of \(H\) which is both normal in \(G\) and is \(f\)-invariant, in the sense that \(K^{f}\leq K\).

Examples of state-closed groups are the Grigorchuk \(2\)-group and the Gupta-Sidki \(p\)-groups in their natural representations on rooted trees. The affine group \(\mathbb{Z}^{n}\mathrm{GL}(n,\mathbb{Z)}\) as well as the free group \(F_{3}\) in three generators admit faithful state-closed representations. Yet another example is the free nilpotent group \(G=F(c,d)\) of class \(c\), freely generated by \(x_{i}\) \((1\leq i\leq d)\): let \(H=\left\langle x_{i}^{n}(1\leq i\leq d)\right\rangle\) where \(n\) is a fixed integer greater than \(1\) and \(f\) be the extension of the map \(x_{i}^{n}\rightarrow x_{i}\) \((1\leq i\leq d)\).

We will discuss state-closed representations of general abelian groups and of finitely generated torsion-free nilpotent groups.

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The Interactions of Solitons in the Novikov-Veselov Equation

Jen-Hsu Chang

UC Riverside and National Defense Univ., Taiwan

3:00pm-4:00pm

CMC 130

Wen-Xiu Ma

**Abstract**

Using the reality condition of the solutions, one constructs thereal Pfaffian N-solitons solutions of the Novikov-Veselov (NV) equation using the tan function and the Schur identity. By the minor-summation formula of the Pfaffian, we can study the interactions of solitons in the Novikov-Veselov equation from the Kadomtsev-Petviashvili (KP) equation's point-of-view, that is, the totally non-negative Grassmannian. Especially, the Y-type resonance, O-type and the P-type interactions of X-shape are investigated; furthermore, the Mach-type solton is studied to describe the resonance of incident wave and reflected wave. Also, the maximum amplitude of the intersection of these line solitons and the critical angle are computed and one makes a comparison with the KP-(II) equation.

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Geometric curve flows and integrable systems

Stephen Anco

Department of Mathematics and Statistics

Brock University

Ontario, CANADA

3:00pm-4:00pm

CMC 130

Wen-Xiu Ma

**Abstract**

The modern theory of integrable soliton equations displays many deep links to differential geometry, particularly in the study of geometric curve flows by moving-frame methods.

I will first review an elegant geometrical derivation of the integrability structure for two important examples of soliton equations: the nonlinear Schrödinger (NLS) equation; and the modified Korteweg-de Vries (mKdV) equation. This derivation is based on a moving-frame formulation of geometric curve flows which are mathematical models of vortex filaments and vortex-patch boundaries arising in ideal fluid flow in two and three dimensions. Key mathematical tools are the Cartan structure equations of Frenet frames and the Hasimoto transformation relating invariants of a curve to soliton variables, as well as the theory of Poisson brackets for Hamiltonian PDEs.

I will then describe a broad generalization of these results to geometric curve flows in semi-simple Klein geometries \(M=G/H\), giving a geometrical derivation of group-invariant (multi-component) versions of mKdV and NLS soliton equations along with their full integrability structure.

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Ordering free groups and free products

Zoran Šunić

Texas A&M University

3:00pm-4:00pm

CMC 130

Milé Krajčevski

**Abstract**

We utilize a criterion for the existence of a free subgroup acting freely on at least one of its orbits to construct such actions of the free group on the circle and on the line, leading to orders on free groups that are particularly easy to state and work with.

We then switch to a restatement of the orders in terms of certain quasi-characters of free groups, from which properties of the defined orders may be deduced (some have positive cones that are context-free, some have word reversible cones, some of the orders extend the usual lexicographic order, and so on).

Finally, we construct total orders on the vertex set of an oriented tree. The orders are based only on up-down counts at the interior vertices and the edges along the unique geodesic from a given vertex to another. As an application, we provide a short proof of Vinogradov´s result that the free product of left-orderable groups is left-orderable.

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Recent developments in Quantum invariants of knots

Mustafa Hajij

Louisiana State University

3:00pm-4:00pm

CMC 130

Mohamed Elhamdadi

**Abstract**

Quantum knot invariants deeply connect many domains such as lie algebras, quantum groups, number theory and knot theory. I will talk about a particular quantum invariant called the colored Jones polynomial and some of the recent work that has been done to understand it. This invariant takes the form a sequence of Laurent polynomials. I will explain how the coefficients of this sequence stabilize for certain class of knots called alternating knots. Furthermore, I will show that this leads naturally to interesting connections with number theory.

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The valence of polynomial harmonic mappings

Erik Lundberg

Florida Atlantic University

3:00pm-4:00pm

CMC 130

Dmitry Khavinson

**Abstract**

While working to extend the Fundamental Theorem of Algebra, A. S. Wilmshurst used Bezout’s theorem to give an upper bound for the number of zeros of a (complex valued) harmonic polynomial. Although the bound is sharp in general, Wilmshurst conjectured that Bezout’s bound can be refined dramatically. Using holomorphic dynamics, the conjecture was confirmed by D. Khavinson and G. Swiatek in the special case when the anti-analytic part is linear. We will discuss recent counterexamples to other cases as well as an alternative probabilistic approach to the problem.