Below is a list of talks given at GSS during the Fall 2023 semester. Click here to learn more about GSS. Use the arrows on the left to read the abstract for a given talk.
12/7: An Introduction to Measure Theoretic Entropy – Lydia Ahlstrom
Abstract: A dynamical system is a non-empty set, $\mathsf{X}$, and a map $\mathsf{f} : \mathsf{X} \to \mathsf{X}$. The goal of dynamical systems is to study the long term behavior of the system $\mathsf{(X, f)}$. One might wonder if there is a way to quantify how “chaotic” a system is. The answer is entropy. In this talk, $\mathsf{X}$ will be a measure space with a measure-preserving map, $\mathsf{f}$. In this setting, we will construct the definition of measure-theoretic entropy and form an intuitive understanding of the concept using information theory. We will conclude by calculating entropy for examples involving the circle.
11/16: Computational Complexity of Pattern Exhibitability – Michele Bailetti
Abstract: Patterns (of consistency and inconsistency) are essentially descriptions of abstract configurations of sets. In model theory, patterns can give a general framework to talk about dividing lines in the universe of first-order theories. In this talk we will view the basic definitions around patterns, show that not every pattern is exhibitable and analyze the computational complexity of the decision problem of pattern exhibitability.
11/9: How Computers are Shaping Math – Zack Porat
Abstract: Given any starting position of a Rubik’s Cube, how many moves are required to solve it? The solution to this problem was found using computational methods. In this talk, we will briefly explain this problem and its solution as motivation for a discussion on computers in mathematics. In particular, we will talk about methods mathematicians use to communicate with computers and how computations benefit us, primarily in the realm of computational algebra and number theory. If time permits, we will also discuss proof assistants and their role in determining math accuracy.
11/2: Pseudo-Finite Fields – Roberto Torres
Abstract: This talk will be an introduction to the theory of pseudo-finite fields. I will give an axiomatic description of the theory of finite fields and define pseudo-finite fields as infinite models of that theory.
10/26: A Survey of the Marked Length Spectral Rigidity Question – Elvin Shrestha
Abstract: This talk will be a survey of the Marked Length Spectral Rigidity question, which asks whether the ‘marked’ lengths of closed geodesics determine the geometry of a Riemannian manifold. Several affirmative answers that have been found in dimension two. We will discuss the key ideas and tools––most notably, geodesic currents––that go into tackling this problem. We present a proof outline of the simplest case, and hopefully along the way, highlight the interplay between geometry and smooth dynamics.
10/19: Group Actions on Trees – Arianna Zikos
Abstract: A big goal of geometric group theory is to study finitely generated groups as geometric objects. To this end we associate metric spaces to our group and ask if geometric properties of the space can tell us algebraic information about the group. In this talk, we will introduce Bass-Serre theory, which is the study of group actions on trees. We will present some results which relate splittings of groups with actions on trees with certain properties. Finally, we discuss how Bass-Serre theory can be used as a tool to understand virtually free groups.
10/12: Recoverable Lattices: Rank 2 Case and Beyond – Felipe Valdes (Winni)
Abstract: Let $\mathsf{f}$ be a positive definite quadratic form over the ring of integers, we say $\mathsf{f}$ is recoverable if for any form $\mathsf{g}$ that represents all proper sub-forms of $\mathsf{f}$, then $\mathsf{g}$ must also represent $\mathsf{f}$. It was proved recently that for any indefinite indecomposable form $\mathsf{h}$, $\mathsf{h}$ is not recoverable, however little is known about the positive definite case. We will show some examples of binary lattices which are recoverable and some that are not recoverable and also a few examples on higher ranks.
10/5: Automorphisms of Free Groups – Eleanor Rhoads
Abstract: Free groups are the building blocks of infinite groups, and so they have a rich history throughout mathematics. In this talk, we’ll examine the group of automorphisms for a free group of finite rank, $\mathsf{Aut(F_n)}$. We’ll discuss a few tools that are used in modern study of $\mathsf{Aut(F_n)}$, and then zoom in on one in particular–Stallings foldings. With foldings, our goal will be to give a generating set for $\mathsf{Aut(F_n)}$. This talk will largely follow material from “Topology of Finite Graphs” by John Stallings and expository material by Karen Vogtmann.
9/28: The Riemann-Roch Theorem – Suzanne O’Hara
Abstract: In this talk we will explore some of the background of the Riemann-Roch Theorem. Riemann-Roch is a tool that helps translate between topological and algebraic information, and tells us about the existence of functions with given poles and zeros on our curve. We will discuss equivalence of varieties, vector spaces of divisors, and the genus of an algebraic curve.
9/21: Flows of Graphs – Ian Gossett
Abstract: The study of graph flows yields many deep and surprising connections between flows, colorings, and connectivity. We give an introductory survey of graph flows and present open problems and conjectures.
9/14: Quadratic Forms, Lattices and Representations – Felipe Valdes (Winni)
Abstract: Let $\mathsf{V}$ be a finite dimensional vector space over a field $\mathbb{F}$ with $\textsf{char}(\mathbb{F}) \neq \mathsf{2}$, let $\mathsf{B}$ be a symmetric bilinear form $\mathsf{V}$ to $\mathbb{F}$, then $\mathsf{B}$ defines a unique quadratic map $\mathsf{Q}$ from $\mathsf{V}$ to $\mathbb{F}$ and for any vector $\mathsf{v}$ in $\mathsf{V}$ we call $\mathsf{Q}(\mathsf{v})$ the norm of $\mathsf{v}$. If $\mathbb{F}$ is the field of rational number and $\mathsf{V}$ has a basis $\{\mathsf{v_1,v_2, \ldots, v_n}\}$, we can define the lattice associated to $\mathsf{V}$ by the construction $\mathsf{L}=\mathbb{Z}\mathsf{v_1} + \mathbb{Z}\mathsf{v_2} + \cdots + \mathbb{Z}\mathsf{v_n}$. The goal is to know how the norms of elements in $\mathsf{L}$ behave.