Spring 2023 

Date: Monday, February 6, 2023

Speaker: Ara Basmajian

Title: Curves on Riemann Surfaces, Moduli Spaces, and the Mumford Compactness Theorem

Abstract: A closed topological (orientable) surface of genus bigger than one is called a Riemann surface if it is equipped with a complex structure; and the space of all Riemann surfaces of a fixed genus is called the moduli space of that genus. When the genus is bigger than one the Riemann surface is also equipped with a unique hyperbolic structure. A natural question is how do these structures interact. After discussing the basics, the main focus of this talk will be to state and discuss the ideas involving the proof of the Mumford compactness theorem which serves as an excellent illustration of the interplay between complex analysis and hyperbolic geometry.

Date: Monday, February 27, 2023

Speaker: Martin Bendersky 

Title:  An introduction to the Vietoris-Rips complex of Hypercube graph

Abstract:  The Vietoris-Rips complex is a tool used to understand the shape of a metric space. It has found applications to a number of topics in applied topology. For example, one of the most computable forms of persistent homology of high-dimensional datasets is obtained by building a Vietoris-Rips complex on top of that dataset. It has also found application to coding theory.

I will discuss the Vietoris-Rips complex primarily as an excuse to introduce ideas from toric topology. Specifically the Moment Angle Complex of a simplicial complex, the Stanley-Reisner ring of a simplicial complex and the relation between the cohomology of the Moment Angle Complex and a Tor algebra constructed from the Stanley-Reisner ring (Hochster’s Moment Angle Complex Theorem). If there is time I might actually mention some applications to the Vietoris-Rips complex of Hypercube graphs which appear in coding theory.

Date: Monday, March 6, 2023

Speaker: Vladimir Shpilrain

Title:  Complexity of algorithms

Abstract: Complexity theory is at the core of theoretical computer science. In this talk, I will give examples of natural algorithmic  problems that admit very efficient solution (sublinear-time), as well as examples of NP-complete problems. The latter are relevant to the famous "P vs NP" problem. Then I will discuss three kinds of complexity: worst-case, average-case, and generic-case complexity, and their relevance in some real-life scenarios.

Date: Monday, March 13, 2023

Speaker: Abhijit Champanerkar

Title:  Unraveling knots - four perspectives

Abstract: A knot is an embedding of the circle in the 3-space and is usually represented by a knot diagram. In the late 19th century, motivated by physics, the celebrated Scottish physicist Peter Tait created the first knot tables. Knot theory has come a long way from its interesting origins. In this talk we will present four major perspectives on knots - diagrammatic, topological, geometric and quantum. We will see interesting theorems, and explore interactions of these different fields in the context of knots. Finally we will showcase some recent developments & generalizations and visit open problems in knot theory.

Date: Monday, March 20, 2023

Speaker: John Terilla

Title: Finding structure

Abstract: When dealing with a mathematical object X that has little or no structure, or just some unknown structure, it’s often helpful to replace X by something like “functions on X” which will have more structure.  X usually embeds into "the functions on X" so working with functions on X involves working with X and more.  For a literal example, think of replacing a set X by k^X, the functions from X to a field k.  Here, k^X is a vector space, mapping an element x in X to the indicator function of x defines an inclusion X \to k^X whose image is an independent spanning set.  I’ll discuss some similar examples and a few applications that I think are interesting.

Date: Monday, March 27, 2023

Speaker: Olga Kharlampovich

Title: Equations in random groups.

Abstract: We discuss different models of random groups. A random group in Gromov's density model with d<1/2 is known to be hyperbolic.  We prove that a random group, in the density model with d<1/16 (such a random group is a small cancellation group) satisfies with overwhelming probability a universal-existential first-order sentence $\sigma$ (in the language of groups) if and only if $\sigma$ is true in a nonabelian free group. This is based on our result that solutions of a system of equations in such a random group with overwhelming probability are images of solutions in a free group. We also discuss problems with higher densities that come from differences between hyperbolic and small cancellation groups. These are joint results with R. Sklinos.

Date: Monday, April 3, 2023

Speaker: Scott Wilson

Title: A survey of research on manifolds involving differential forms (along with a pizza problem).

Abstract: I'll describe several directions of research over the past 20 years or so, including progress, hangups, and open problems. This work is at the interface of geometry and topology on manifolds, which are intimately related via differential forms, equipped with various geometric structures.

Date: Monday, April 17, 2022

Speaker: Melvyn Nathanson

Title: Odd problems and results in number theory

Abstract: An introduction to some old and new solved and unsolved problems and results in different parts of number theory.

Date: Monday, April 24, 2022

Speaker: Krzysztof Klosin

Title: Introduction to modularity theorems

Abstract: Representations of the absolute Galois group of a number field arise naturally in the study of abelian varieties. There are long-standing conjectures asserting that these Galois representations are isomorphic to ones that are attached to certain analytic objects, called automorphic forms. One of the first breakthroughs in this field was Wiles's proof of the Taniyama-Shimura Conjecture making such a connection between representations attached to elliptic curves and classical modular forms. Analogous results for varieties of dimension 2 are still far from satisfactory and for dimensions three and higher essentially non-existent. This will be an introduction to modularity theorems and some techniques used to prove them. 

Fall 2022 

Date: Monday, August 29, 2022

Speaker: Christian Wolf

Title: Typical orbits, natural invariant measures and their computability properties.

Abstract: In this talk I give an introduction of various classes of typical orbits for a given dynamical system. Typical orbits are associated with certain natural invariant measures which are a central object in Ergodic Theory. In this context the term "natural" refers to measures which either maximize a relevant dynamical invariant such as entropy or dimension or to measures whose typical points have full Lebesgue measure. There are many open questions about natural invariant measures including their existence, uniqueness, properties, etc. If time allows we also discuss computability questions for natural measures.

Date: Monday, September 12, 2022

Speaker: Pablo Soberon 

Title:  Lifting methods in mass partitions

Abstract: In a mass partition problem, we seek to split evenly a family of measures in R^d given some geometric constraints on the partition.  In this talk we will discuss some recent results of this kind, related to partitions using parallel hyperplanes and their connections to algebraic topology.  In particular, we will show how results from equivariant topology using Stiefel manifolds can be used to obtain new mass partition results.

Date: Monday, September 19, 2022

Speaker: Jun Hu

Title:  Totally ramified rational functions: Speiser's graph, existence, and dynamics

Abstract: A rational function f (viewed as a map from the Riemann sphere to itself) is said to be totally ramified if for every critical value q, every preimage of q under f is a critical point. We introduce Speiser's graphs to study the existence of such rational functions. We will also mention the absence of certain dynamics for such rational maps.

Date: Monday, October 3, 2022

Speaker: Emma Bailey

Title:  Connections between random matrix theory and analytic number theory: an overview

Abstract: Random matrix theory provides a very profitable model for various number theoretic functions, notably including the Riemann zeta function.  I will present an overview of these theoretical (and time permitting numerical) connections and the ‘dictionary’ to translate between the two subjects.  

Date: Monday, October 17, 2022

Speaker: David Aulicino

Title: The Wonderful World of Translation Surfaces

Abstract: This talk will be a survey of some recent results on the topic of translation surfaces by me and my collaborators. We will start with basic examples, which include the torus. We will show how translation surfaces were used to solve a problem concerning the Platonic solids, and focus on the dodecahedron in particular. Next we will present a generalization of the remarkable fact that the proportion of primitive lattice points in a disc in the plane is approximately 6/pi^2. All necessary background will be given, and directions for future investigations will be presented. The collaborators include Jayadev Athreya, Pat Hooper, Martin Schmoll, Aaron Calderon, Nick Salter, Carlos Matheus.

Date: Monday, October 24, 2022

Speaker: Sandra Kingan

Title: Matroid Theory Research 

Abstract: Matroids are a modern type of synthetic geometry where the behavior of points, lines, planes, and higher dimensional surfaces are governed by combinatorial axioms. Hassler Whitney coined the term matroid in his 1935 paper "On the abstract properties of linear dependence". In defining a matroid Whitney captured the fundamental properties of independence that are common to graphs and matrices. In this talk I will focus on excluded minor results. For a given class of matroids with a specified structure, a minimal excluded minor is a matroid that is not in the class, but every single-element deletion and contraction is in the class. Such matroids are minimal obstructions for membership in the class. For example, the complete graph on five vertices and the complete bipartite graph with three vertices in each class are the minimal obstructions to planarity in graphs. I will give an overview of excluded minor results and describe some of my work in this area. I will also present ways to begin a research program in matroid theory. 

Date: Monday, October 31, 2022

Speaker: Russell Miller

Title: Hilbert’s Tenth Problem for Subrings of the Rational Numbers 

Abstract: Hilbert's original Tenth Problem demanded an algorithm that decides, for an arbitrary diophantine equation in arbitrarily many variables over Z, whether there exists an integer solution to that equation.  In 1970, in a triumph of computability theory, Matiyasevich completed work by Davis, Putnam, and Robinson, showing that there is no such algorithm.  That being the case, attention shifted to the question of deciding whether such an equation has a rational solution.  It remains open whether an algorithm exists for deciding this question.

In this talk, after introducing Hilbert's Tenth Problem and its history, we will discuss it not just for the rational numbers, but for arbitrary subrings of Q, producing another subring (much larger than Z) where the problem remains undecidable.  These subrings form a topological space, with a form of Lebesgue measure and also with the property of Baire, and one can therefore ask whether various properties of rings (usually involving HTP) hold on a small or large class of the subrings of Q.  Potentially the answers about the sizes of these classes might lead to an answer regarding Q itself.

Date: Monday, November 7, 2022

Speaker: Ilya Kapovich

Title: Groups as geometric objects 

Abstract: For most of the 20th century group theory has been considered a part of algebra and groups have been studied primarily as algebraic objects. The advent of the theory of word-hyperbolic groups in early 1990s changed that point of view and reconnected group theory to some of its geometric roots going back to the work of Cayley, Klein and Lie. Geometric group theory seeks to relate algebraic properties of groups to geometric properties of spaces on which these groups admit “nice” actions by symmetries. In this talk we will discuss some of the themes and ideas of geometric group theory and some of the tools used there.

Date: Monday, November 14, 2022

Speaker: Vincent Martinez

Title: The Mystery of Turbulence: A mathematician’s perspective

Abstract: Turbulence occurs everywhere, from the mundane to the spectacular, in smoke emanating from a pipe or the wake of a fastball, to the gas ejected from black holes that helps to form galaxies around us. In spite of its ubiquity and the fact that the equations that describe fluid motion were written down more than 200 years ago, Turbulence remains one of the great unsolved problems of physics. This talk will present a short account of the mathematical study of turbulence and the fundamental issues we are confronted with from one mathematician’s point of view.

Date: Monday, November 21, 2022

Speaker: Renato Ghini Bettiol

Title: Positive curvature 

Abstract: In this talk, I will explain a few different ways to think about smooth manifolds with positive (sectional) curvature, such as round spheres, and why they keep puzzling geometers. I will give an overview of what is currently known and not known about them, including the main conjectures in the field, and how some seemingly unrelated methods from optimization and real algebraic geometry might lead to new insights here. 

Date: Monday, November 28, 2022

Speaker: Louis-Pierre Arguin

Title: Large values of the Riemann zeta function on the critical line

Abstract: I will give an account of the recent progress in probability and in number theory to understand the large values of the zeta function on the critical line. This problem has interesting connections with extreme value theory in probability, as well as random matrix theory. These connections will be emphasized.

Date: Monday, December 5, 2022

Speaker: Nicholas Vlamis

Title:  An invitation to big mapping class groups 

Abstract: The mapping class group of a 2-manifold M is the group of isotopy classes of self-homeomorphisms of M.  These groups have been heavily studied for over a century, but mostly in the context of finite-type surfaces (i.e. the interiors of compact surfaces).   Recently, there has been a growing body of work studying the mapping class groups of infinite-type surfaces, which are often referred to as big mapping class groups.  In this talk, we will give a quick summary of why mapping class group are of interest historically, introduce the classification of surfaces in terms of topological ends, and give a brief introduction to big mapping class groups. 

Date: Monday, December 12, 2022

Speaker: Stephen Preston

Title: Riemannian Geometry on Lie Groups

Abstract: The Euler equations for a rigid body, and the Euler equations for ideal fluids, have in common that they describe geodesics on a Lie group with a left- or right-invariant metric. Arnold showed how to describe the geometry of these groups in 1966, and since then there has been a lot of work in both finding new geometric interpretations of differential equations, and on their properties such as curvature, stability, completeness, and conjugate points. I will give a survey of some results and open problems, particularly about the difficulties one can face in infinite-dimensional geometry.