GSS – Fall 2021

Abstract: The objects of my research are points on modular curves. We will go through the basic ideas and construction of modular curves and their connection to elliptic curves.

Abstract: In this talk, I want to introduce Hilbert Metric and its properties. Further, I will mention some examples of $\mathsf{3}$-manifolds which arise as Hilbert geometries which have a non-Riemannian and non-uniformly hyperbolic geometric structure.

Abstract: Constructive mathematics is mathematics done without the law of excluded middle, the law that for any proposition $\mathsf{P}$, either $\mathsf{P}$ or not $\mathsf{P}$. In this talk I will attempt to explain why anyone would want to do this.

Abstract: Geometric group theorists study groups by studying (geometric/topological) spaces on which they act. Therefore, we might ask which properties of groups are seen in the spaces on which they act and vice versa. This talk will introduce a particular space on which a group acts, which is taken to be as “nice” as possible. This space is called the Cayley graph. Then we will discuss growth of groups, and in doing so will answer some of the questions posed above.

Abstract: A $\mathsf{k}$-list assignment of a graph $\mathsf{G}$ pairs each vertex $\mathsf{v \in V(G)}$ with a list $\mathsf{L(v)}$ of $\mathsf{k}$ colors. We say $\mathsf{G}$ is $\mathsf{k}$-list colorable when it is possible to assign each vertex a color from its list such that no two vertices connected by an edge receive the same color. $\mathsf{G}$ is equitably $\mathsf{k}$-list choosable if for every possible $\mathsf{k}$-list assignment, $\mathsf{G}$ has a $\mathsf{k}$-list coloring in which no color is used more than $\mathsf{\lceil \frac{\mid V(G) \mid}{k} \rceil}$ times. We will discuss the equitable choosability of prism graphs $\mathsf{\Pi_n}$ and supporting lemmas.

Abstract: Robertson and Seymour’s Graph Minor Theorem is one of the deepest theorems in mathematics. In this talk, we present the statement of the theorem, discuss some of its implications, and introduce the notion of treewidth – a powerful tool used in the proof of the theorem.

Abstract: Magnitude is an isometric, real-valued invariant of metric spaces. It is of particular interest since it has been shown to encode other features such as dimension, volume, and curvature, as well as bearing a strong connection to persistence. This talk will seek to introduce magnitude, providing foundational information such as well-definedness, key features, and examples, then move on to an introduction of magnitude dimension, magnitude homology, and recent results.

Abstract: Elliptic curves are an interesting area of study in mathematics, laying at the intersection of algebra, geometry, and number theory. They are a powerful tool, having applications in everything from Andrew Wiles’ proof of Fermat’s Last Theorem to cybersecurity. In this talk, we first provide an introduction to elliptic curves by discussing their geometry and associated group structure. We then narrow our focus, further investigating the j-invariant of an elliptic curve.

Abstract: Let $\mathsf{(W,S)}$ be a Coxeter system. For each pair $\mathsf{(u,w)}$ where $\mathsf{u}$ is an element of $\mathsf{W}$ and $\mathsf{w}$ is a word in the alphabet $\mathsf{S}$ we associate a family of polynomials $\mathsf{R_{u, w}(t)}$ . These polynomials are obtained as a subset of Libedinsky’s light leaves for the pair $\mathsf{(u,w)}$ and coincides with the Kazhdan-Lusztig polynomials for the same. The idea is to find a formula using a diagrammatic approach.

Abstract: A recurring pattern in homotopy theory is that we are stuck between two extremes. Good objects (e.g. CW complexes up to homotopy equivalence) form categories with bad properties. But good categories (e.g. all topological spaces and continuous maps) contain objects with bad properties. The theory of model categories is a way to have the best of both, letting us work in the good category and correct objects into good ones when necessary. The goal of the talk is to give a sense of what model categories are, and to use them to try and justify the definition of ‘Tor’.

Abstract: Expander graphs are highly connected and sparse graphs that have a lot of applications in networks and computers. This property over a graph is equal to another property over a matrix related to the graph. In this talk, we explain two different ways to construct an infinite family of expander graphs. One uses Kazhdan property on a family of groups and gives us Cayley graphs which are expander. Another way family of expanders can be constructed is by induction and graph products. We extend the definition of an expander for hypergraphs.

Abstract: In some of his recent work Hedden made the following conjecture that has since been attributed to him: The only endomorphisms of the knot concordance group which are induced by satellite operators are the zero homomorphism, the identity homomorphism, and the involution that takes each class to its inverse. In this talk I will introduce Hedden’s conjecture (and all the relevant terminology) from a predominantly diagrammatic point of view, and build towards a statement of a generalization of the conjecture for when the domain is the 2-component string link concordance group.