GSS – Spring 2023

Abstract: Correspondence colorings, a generalization of list colorings, were introduced by Dvorak and Postle in 2018 as a tool to prove that planar graphs without cycles of lengths 4 to 8 are 3-choosable. They have since been the object of study of many researchers, and have proved to be interesting in their own right. It is known that the famous orientation theorems for list colorings–the Bondy-Boppana-Siegel lemma and the Alon-Tarsi Theorem–do not apply to correspondence colorings “as is.” However, if we restrict to certain classes of correspondence colorings, it is indeed possible to obtain generalizations. In this talk, we characterize three classes of correspondence assignments that admit generalizations of the orientation theorems.

Abstract: Minimal volume entropy is a Riemannian manifold invariant defined by Gromov. Bregman and Clay defined minimal volume entropy for groups of finite type. Right-angled Coxeter groups (RACGs) are not groups of finite type because they contain torsion elements. We will discuss how to correctly define minimal volume entropy for RACGs and give an outline of how to compute it for the simplest class of RACGs.

Abstract: Since Fox and Milnor first defined the knot concordance group, $\mathsf{C}$, in the 1960’s, one of the central motivations of the field has been understanding the group structure. Towards that end, satellite constructions have been an indispensable tool. Intuitively, a satellite construction can be thought of as grabbing a handful of a knot $\mathsf{R}$ and tying in another knot $\mathsf{K}$. If we fix $\mathsf{R}$ and the location where we are grabbing, this induces a well-defined operator from $\mathsf{C}$ to itself. The resourcefulness of satellite constructions in the study of concordance has generated interest in the properties of satellite operators themselves. In particular, many have noted that satellite operators are seldom homomorphisms, and recently this has culminated in the conjecture that the only homomorphisms induced by satellite operators are the zero map, the identity map, and the involution that takes each class to its inverse.

In this talk we are concerned with a generalization of this conjecture to the setting of the string link concordance groups, $\mathsf{C^m}$, and operators induced by string link infection. Intuitively, string link infection can be thought of as grabbing a knot $\mathsf{R}$ in $\mathsf{m}$ handfuls and tying in the pattern of an $\mathsf{m}$-component string link $\mathsf L$. As above, if we fix $\mathsf R$ and the $\mathsf m$ locations we are grabbing, this induces a well-defined operator from $\mathsf{C^m}$ to $\mathsf C$. Infection operators are a generalization of satellite operators in the sense that $\mathsf{C^1}$ is isomorphic to $\mathsf C$, thus infecting by a 1-component string link can be thought of as a satellite construction. Note that like satellite operators, infection operators are seldom homomorphisms. We will propose a new conjecture for homomorphisms induced by string link infection, and provide evidence towards the conjecture by considering certain classes of interesting patterns. Our main tool will be invariants of the algebraic concordance group.

Abstract: Measures of the diversity of metric spaces are well studied, and have long been speculated to bear connection to persistent homology. One such connection, persistent magnitude, is inspired by the connection between magnitude, an isometric invariant of metric spaces, and persistent homology, through blurred magnitude homology. In this talk, I will introduce the tools from persistence required to understand persistent magnitude, as well as alpha magnitude, the persistent magnitude of the alpha complex. Examples of note will be shown, as well as a conjecture linking alpha magnitude to the dimension of compact positive definite metric spaces. If time allows, I will show some early results of a promising clustering algorithm which utilizes alpha magnitude.

Abstract: In this talk, we explore the fascinating interplay between orientations and list colorability of graphs.

Abstract: Algebraic geometry is a field which studies the solutions to polynomial equations over fields. One of the first major theorems you may come across is Hilbert’s Nullstellensatz. The theorem is often quoted as it establishes a one-to-one correspondence between radical ideals and algebraic sets. In this talk we will introduce the definitions of algebraic sets, their ideals, and work through a complete a proof of the Nullstellensatz.

Abstract: Évariste Galois is known for his contributions to algebra, and his early demise. In this talk, we first will learn about his (short) life. We will be introduced to a famous cast of characters along the way, all of whom played pivotal roles in Galois’s story. Next, we will connect his seminal work with the subject we now know as Galois Theory. Finally, we will end with a brief overview of Galois Theory and its applications.

Abstract: Given two Hermitian forms with entries in the ring of integers of a number field, can we determine whether or not they are equivalent? In other words, can we, in a reasonable amount of time, find an invertible, integral matrix taking one to the other, or determine that no such matrix exists? In this talk, I will present the proof for the number field $\mathsf{\Q(\sqrt{2})}$, and emphasize the interplay between number theory, ergodic theory, representation theory, and homogeneous dynamics.

Abstract: In this talk, we will begin by defining elliptic curves and their arithmetic. We will then give a survey of relevant topics in the study of elliptic curves, including $\mathsf{N}$-division fields, Galois representations, and the theory of complex multiplication. We will conclude with a brief overview of ongoing research into Galois representations attached to elliptic curves with complex multiplication.

Abstract: Indiscernible sequences (or more generally, indiscernibles) are a central notion in model theory. Roughly speaking, an indiscernible sequence is an infinite sequence of tuples from a model of a first-order theory that can not be distinguished by certain formulas. In this talk, we introduce the basic definitions and techniques regarding Indiscernible sequences, and we will use Ramsey’s Theorem to prove their existence.

Abstract: Probabilities are often determined based on conditions and whether said conditions have been met. In many cases it is interesting to look at certain occurrences based on states, such as the probability that someone will get sick in 10 days given that they are currently healthy. To do this, mathematicians, statisticians and actuarial scientists often use state processes, in particular stochastic to determine such probabilities. In this talk we will explore topics regarding conditional probability and state processes, including Markov chains.