GSS – Spring 2025

Abstract: After extensive exploration, we find that C*-algebras are an incredibly unique structure within the world of functional analysis and operator algebras. A C*-algebra is a Banach algebra together with an involution called “*” [which can be thought of as an “adjoint” operation] such that ||a*a|| = ||a||^2 holds for any element in the C*-algebra. Some results about Banach algebras extend naturally. In particular, the functional calculus and therefore Spectral Mapping Theorem for Banach algebras has a natural extension for Abelian C*-algebras. We primarily focus on building up to the Spectral Mapping theorem and comparing its statement in both environments, ultimately proving that one is an extension of the other. The second part of the thesis leads up to proving the GNS-Construction and eventually the GNS-Theorem. This is the piece that allows us to connect C*-algebras to sub-algebras of bounded linear operators over Hilbert spaces, otherwise known as the algebra of observables in the world of quantum physics. The GNS-Theorem is also an essential piece in proving that C*-algebras are axiomatizable in the language of model theory. The axiomatization is exhibited at the conclusion of this thesis.

Abstract: Given a standard Sudoku puzzle, what is the minimal number of clues needed to create a valid puzzle? 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.

Abstract: In 2022, Professors Kim, Lee and Oh gave the following definition: A lattice $L$ is not recoverable if there is a lattice $M$ that represents all proper sublattices of $L$ but not $L$ itself. In the same paper they worked on binary $\mathbb{Z}$-lattices (dimension 2) and showed the existence of recoverable and not recoverable lattices, moreover, they showed that any indecomposable $\mathbb{Z}$-lattice $L$ whose rank is at most 3 is not recoverable. In 2023, Professor Chan and Professor Oh generalized this last statement to the following: Any indecomposable positive definite lattice $L$ is not recoverable. We want to show that there is a slightly more general statement, that is: If $L$ is an indecomposable $\mathbb{Z}$-lattice, then the lattice defined as the finite orthogonal sum of $L$ with itself is not recoverable.

Abstract: In this talk, we will discuss two main topics relating to elliptic curves. I will give a brief introduction to curves with complex multiplication (CM) and curves defined over the complex numbers. During the process, we will discuss the structure of the endomorphism ring of an elliptic curve and visualize why elliptic curves have genus 1. If we have time at the end, we will discuss extra structure we get when studying curves with CM by a maximal order that are defined over $\mathbb{C}$.

Abstract: In 2000, the New York Mets owed player Bobby Bonilla \$5.9M from his contract. They decided to defer the payments. Instead of getting paid \$5.9M in 2000, Bobby Bonilla instead, starting in 2011, received \$1,193,248.20 every July 1’st, continuing until 2035. This effectively turned \$5.9M into almost \$30M. In this mini-talk, we will explore the math behind this deferral. 

Abstract: Language models are clearly pretty important in the modern era… but how do they work? In this talk, we’ll approach this question by looking at a method called word2vec. This is a technique that produces word embeddings, vector representations for words, which serve as the foundation for NLP (natural language processing).

Abstract: In the world of homology, we use algebraic tools and constructions to understand spaces. We don’t often see the same tools applied to homotopy groups, which may seem to suggest that homotopy groups do not play nicely with these algebraic techniques. While this is true for most cases, there are dimensional constraints that we can impose in order to get meaningful information from the usual algebraic constructions. We will define relative homotopy groups, long exact sequences, and homotopy excision (where it works). From there we will state the Freudenthal Suspension Theorem, and give an introduction into stable homotopy groups.

Abstract: A combing of a group is a choice of a “normal form” for each group element, viewed either as a word representing that element, or a path leading to that element (in the Cayley graph). Combings are useful in geometric group theory in that they naturally connect the algebraic and geometric viewpoints, and combings can be studied instead of the group as a whole in certain contexts. In this talk, we’ll set up the basics of combings, and examine some of the ways they can be used.

Abstract: We will discuss results from the 2008 paper A Finiteness Conjecture on Abelian Varieties with Constrained Prime Power Torsion by Chris Rasmussen and Akio Tamagawa. A variety that is heavenly at a prime $p$ is defined by a field inclusion of the $p$-power torsion in a field called heaven. Using an action of Galois groups and an identification with modular curves, we will show that there exists only finitely many of examples of $\mathbb{Q}$-isomorphism classes of heavenly elliptic curves defined over the rational numbers.