GSS – Spring 2025

Abstract: The last hundred years of smooth topology have been incredibly successful, with geometric and gauge-theoretic methods leading to leaps in understanding about surfaces and 3-dimensional manifolds, and with surgery theory and homotopy theory answering many of the biggest questions one wants to know about higher-dimensional ($d\ge 5$) manifolds. This has left an interesting lacuna: what’s going on in dimension 4? The theory of 4-manifolds is incredibly rich, with them behaving in some ways like low-dimensional manifolds, in others like high-dimensional manifolds, and in yet others behaving in a way utterly their own.

This talk aims to give a broad introduction to the field of smooth 4-manifold theory, recapping important theoretical developments and giving an idea of a few of the modern research programs. We aim to be accessible to graduate students from all fields.

Abstract: In 2022, Kim, Lee, and Oh introduced the notion of a recoverable lattice: a positive definite integral lattice $L$ is said to be recoverable if every lattice $M$ that represents all proper sublattices of $L$, also represents $L$ itself. They provided various examples and established that indecomposable lattices of rank at most three are not recoverable. This result was generalized in 2023 by Chan and Oh, who showed that every indecomposable lattice is not recoverable. However, little is known about the recoverability of decomposable lattices. In this thesis, we study the recoverability of certain decomposable lattices. Specifically, we consider lattices of the form $aX$, where $X$ is an indecomposable lattice and $a\ge 1$, as well as lattices of the form $I_k\perp aR$, where $I_k$ is the sum of $k$ squares, and $R$ is an indecomposable root lattice. We prove that lattices of the form $aX$ are never recoverable. Furthermore, we completely characterize the recoverability of lattices of the form $I_k\perp aR$ in terms of the critical number of $R$, providing necessary and sufficient conditions under which these lattices are recoverable.

Abstract: This talk will present the history of studying the covolume of lattices within noncompact semisimple Lie groups. The area has a long history, beginning with Zassenhaus in the 1930s and remaining an open area to this day, with my advisor’s papers giving the current best known bounds on minimal volumes. We will demonstrate how the theory in this field was developed overtime, giving emphasis to how ideas inspire one another and fit into one long story of sorts. With this background established, we will proceed to discuss two original attempts at improving the current bounds. The first is via studying sectional curvature and volume calculations computationally using a program built to model such groups and which has led to fruitful results already, and the second exploits ergodic theory to give another way of attacking the problem.

Abstract: We characterize the virtually free groups that act geometrically on the trivalent tree. This characterization is motivated by the study of minimal volume entropy, a geometric and group invariant. We will define and motivate minimal volume entropy, explain the characterization via an example, and then present an application to Coxeter groups.

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.