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Title: Discrete Painlevé Equations and Orthogonal Polynomials Speaker: Aaron Dzhamay, University of Northern Colorado Time: 12:00pm–1:00pm Place: NES 102 or Zoom

Over the last decade it became clear that discrete Painlevé equations appear in a wide range of interesting applications. Thus, the question of recognizing a certain non-autonomous recurrence as a discrete Painlevé equation and understanding its position in Sakai’s classification scheme, recognizing whether it is equivalent to some known (model) example, and especially finding an explicit change of coordinates transforming it to such example, becomes one of the central ones. Fortunately, Sakai’s geometric theory provides an almost algorithmic procedure of answering this question. In this work we illustrate this procedure by considering two examples. The first example comes from the theory of discrete orthogonal polynomials. There are many connections between orthogonal polynomials and Painlevé equations, both differential and discrete. In particular, often the coefficients of three-term recurrence relations for orthogonal polynomials can be expressed in terms of solutions of some discrete Painlevé equation. In this example we study orthogonal polynomials with general hypergeometric weight and show that their recurrence coefficients satisfy, after some change of variables, the standard discrete Painlevé-V equation. We also provide an explicit change of variables transforming this equation to the standard form. This is joint work with Galina Filipuk (University of Warsaw, Poland) and Alexander Stokes (University College, London, UK). The second example is also related to the theory of orthogonal polynomials but the motivation comes from the Probability Theory. We consider the problem of tiling a hexagon by lozenges with some generalized weight and are interested in computing important statistical properties of this model called the gap probabilities. This model can be related to a q-Racah discrete orthogonal polynomial ensemble, and this computation is again done with the help of discrete Painlevé equation. This is a joint work with Alisa Knizel (Columbia University).

Title: Around a Theorem of Pólya, Part II Speaker: Louis Arenas Time: 12:00pm–1:00pm Place: NES 102 or Zoom

Title: Around a Theorem of Pólya Speaker: Louis Arenas Time: 12:00pm–1:00pm Place: NES 102 or Zoom

Random walks have cemented themselves as relevant objects of study in modeling diverse phenomenon ranging from Brownian motion to the prediction of European stock options through Black-Scholes. In a paper published in 1921 by George Pólya, he investigates walks on \(Z^d\) and asks when the walks are recurrent or transient. Meaning if there's probability 1 of picking a random walk which returns to the origin after finitely many steps. In this talk we present a recent (published in 2014) paper by Jonothan Novak which proves Pólya's theorem using tools that a 19th-century Mathematician would recognize.

Although Novak's proof is beautiful in its own right, our motivation will be to extend his proof techniques to a non-commutative setting. Specifically, we pose the problem of recurrence of random walks on the Cayley graph of the free group generated by \(d\) letters, \(F_d\). Random walks on non-commutative groups were of interest to people like Harry Kesten who under guidance of his advisor Mark Kac wanted criteria of amenability of finitely generated groups. Kesten would go on to formulate a criterion of amenability in his 1958 Ph.D. thesis.

Our focus will be to motivate the tools of free probability theory to answer the question of recurrence on \(F_d\).

Title: An Introduction to Sub-Riemannian Geometry Speaker: Thomas Bieske Time: 12:00pm–1:00pm Place: NES 102 or Zoom

Motivated by real-world activities, we will explore spaces where motion in one or more directions is restricted. Such spaces, called sub-Riemannian spaces, possess a geometry that behaves quite differently from traditional Euclidean \(R^n\). Our goal is to understand the interplay between this geometry and properties of solutions to partial differential equations. This talk will be accessible to advanced undergraduate students. Note: We will also use this talk to deconstruct the concept of a colloquium talk.

Title: An Introduction to Noncommutative Geometry Speaker: Nathan Hayford Time: 12:00pm–1:00pm Place: NES 102 or Zoom

Non-commutative geometry is a subject that grew out of the need to extend results about commutative Banach algebras to the non-commutative setting. It was pioneered by Alain Connes and his collaborators, and has since found applications in many areas of mathematics and physics.

In this talk, we provide a gentle introduction to non-commutative geometry by studying one the simplest instances of it — the non-commutative torus. We compare the results of our calculations to what happens in the commutative case. Further, we demonstrate how the problem connects to the quantum mechanics of 2D electrons in an external magnetic field, and how this connection allows us to derive (formally) an old and famous result due to P. Levy on the area distribution of random walks in two dimensions.

Title: Algebraic Curves over Finite Fields: Theory and Applications Speaker: Vincenzo Pallozzi Lavorante Time: 12:00pm–1:00pm Place: NES 102 or Zoom

The study of algebraic curves defined over a finite field has attracted much interest in recent times. The purpose of this talk is to introduce the basic notions related to algebraic curves over finite fields, together with some examples of how it finds a concrete and common application in different research areas. In particular we will explore the connection with coding theory, permutation polynomials and see an interesting application in finite geometry.