GSS – Spring 2024

Abstract: This talk will present results by Manning regarding the relationship between the topological entropy of the geodesic flow on a compact Riemannian manifold and the volume growth entropy of the universal cover of the manifold.

Abstract: Ash, Grayson, and Green computed the action of Hecke operators on a certain subspace of the cohomology of low-level congruence subgroups of $\mathsf{SL(3, \mathbb{Z})}$.  This subspace contains the cuspidal cohomology, which is of primary interest.  For prime level less than 100, they found four levels at which nonzero cuspidal classes arose and determined local factors for the $\mathsf{L}$-functions.  In this talk, we extend their work, introducing a method that allows for computing the action of Hecke operators directly on the cuspidal cohomology.  Using this method, we obtain data for prime level less than 1500, finding four additional levels at which nonzero cuspidal classes appear and calculating local factors for two of these levels.

Abstract: The boundary of a metric space describes the space of directions to infinity. We will make this definition precise and provide lots of examples. Then we will describe the ‘fellow travel’ topology on the boundary. We will discuss how to associate a boundary to a group, and why this boundary is well-defined up to homeomorphism for delta hyperbolic groups. A natural question then is, what can the boundary tell us about a metric space or a group? We will present some results which answer this question.

Abstract: We give necessary and sufficient conditions for a group to act on the trivalent tree. These conditions are given in terms of reduced graphs of groups and the existence of subgroup series with some index requirements. We prove that every virtually free right-angled Coxeter group satisfies these conditions, and therefore acts geometrically on the trivalent tree. As an application, we show that minimal volume entropy is multiplicative for such groups and we give an explicit calculation.

Abstract: Graph acyclicity has been extended to several notions of acyclicity in hypergraphs: $\alpha$-acyclicity, $\beta$-acyclicity, and $\gamma$-acyclicity. It has been shown that many fundamental database problems that are NP-hard are tractable when the underlying hypergraph is acyclic. Despite the usefulness of all degrees of acyclicity, $\beta$-acyclicity and $\gamma$-acyclicity have been given less attention, particularly when it comes to generalizing these tractable results beyond acyclicity. We will explore a generalization of $\beta$-acyclicity introduced by Lanzinger called nest-set width and expand upon this discussion by providing observations gleaned from studying a large data set of hypergraphs.

Abstract: The chaos game is a process of plotting points in Euclidean space, which after some time, produces the image of a fractal. We are motivated by recent papers on the chaos game to study other random processes on fractals and deterministic processes on circles. The main question is this: how long should one expect to wait before the random points become $\delta$-dense in the relevant space? In this talk, we state and prove a theorem computing the expected wait time for the $\delta$-density of random points in a self-similar fractal. 

Abstract: The classical Lie groups $\textsf{SO}(n), \textsf{SU}(n)$, and $\textsf{Spin}(n)$ can be viewed as isometry groups of projective spaces over the skew-fields $\mathbb R, \mathbb C$, and $\mathbb H$. The exceptional Lie groups could not be realized as isometry groups of any known mathematical object when they were first discovered. Nowadays, it is well known that we can understand $\mathsf{G_2}$ as the automorphism group of the octonions. In this talk, we construct the octonion division algebra $\mathbb O$ and give an outline of the proof of the fact that the set of derivations on the octonions is isomorphic to an exceptional Lie algebra of type $\mathsf{G_2}$.

Abstract: In the early 20th century David Hilbert posed a list of problems that were unsolved at the time, and most of them became very influential in 20th century mathematics. In this talk, we will focus on the proof of the 17th problem on that list that states the following: Given a multivariate polynomial that takes only non-negative values over the reals, can it be represented as a sum of squares of rational functions? A positive result was proved by Artin in 1927 and generalized for polynomials over real closed fields. I’ll present an alternative proof using some properties of the theory of Real Closed Fields. 

Abstract: Let $\mathbb{Z}^\mathsf{n}$ be the lattice span by the set $\mathsf{{\epsilon_i}, 1\leq i \leq n}$ with bilinear form $\mathsf{B} \colon \mathbb{Z}^\mathsf{n} \times \mathbb{Z}^\mathsf{n} \to \mathbb{Z}$ given by the identity matrix $\mathsf{I_n}$. The lattice $\mathsf{A_{n-1}}$ is defined as the set of vectors in $\mathbb{Z}^\mathsf{n}$ whose coordinates add up to zero, while $\mathsf{D_n}$ is the set of vectors whose coordinates add up to an even number. We will present some diagrammatic representations for the elements of the reflection group of $\mathsf{A_n}$ and diagrams for the proper full rank root sublattices of $\mathsf{D_n}$.

Abstract: For a finitely presented group with a solvable word problem, a natural question to ask is how hard it is to actually solve the word problem? Our primary tool to address this is the group’s Dehn function, which describes how much “area” a trivial word can capture relative to its word length. In this talk, we’ll interpret what Dehn functions are from both a computational and geometric standpoint, and examine some of their uses in geometric group theory.

Abstract: DP-colorings of graphs (originally called correspondence colorings) are a generalization of list colorings that were introduced by Dvorak and Postle in 2015 and have since been a popular topic of study. The Alon-Tarsi Theorem and Bondy-Boppana-Siegel Lemma are two well-known results that relate orientations of graphs to list colorings, and though it is known that these theorems do not carry over “as is” to DP-colorings, we characterize some classes of correspondence assignments that admit DP-coloring analogs and generalizations of these theorems.

Abstract: Last week, we constructed $\mathbb{Z}_\mathsf{p}$ using inverse limits. For this talk, we’ll define a $\mathsf{p}$-adic valuation and absolute value. These tools will characterize all norms on $\mathbb{Q}$, and let us give an alternative definition of $\mathbb{Z}_\mathsf{p}$ as the ring of integers for a complete field $\mathbb{Q}_\mathsf{p}$.

Abstract: Ultraproducts play a fascinating role in mathematics, connecting algebraic structures to model theory. These well-behaved structures preserve essential first-order properties, making them models with “big” cardinalities for specific theories.

Abstract: The Tate module is an object useful for understanding isogenies on elliptic curves. This talk will review some algebraic background necessary to construct the Tate module. We will discuss multiplication maps on elliptic curves and their torsion points. We will also describe the process of inverse limits with respect to p-adic integers and the Tate module.