GSS – Spring 2026

Abstract: In their 1984 work, Ash, Grayson, and Green compute the action of Hecke operators on the cohomology of congruence subgroups $\Gamma_0 (3, p) \leq \textrm{SL} (3, \textbf{Z})$ for small p. A natural extension is to work with other congruence subgroups of $\textrm{SL} (3, \textbf{Z})$. In the first part of the talk, we detail techniques for working with congruence subgroups that are Iwahori at p, computing the action of Hecke operators on the corresponding cohomology. In the second part of the talk, we return to the $\Gamma_0 (3, p)$ setting, introducing a new technique which allows for computations directly on the cuspidal cohomology.

Abstract: In the classification of complete first-order theories, dividing lines are defined to separate theories with different complexities and behaviors. The tame sides of these dividing lines are often defined by the absence of local combinatorial properties, given by specific configurations of definable sets. Using the concept of pattern of consistency and inconsistency, we describe a general framework for studying these combinatorial properties. Then, reversing the usual approach of trying to extend structural theorems from a tame class of theories to a wider one, we start from the strongest combinatorial property that a theory can have. We then weaken this property and define new ones giving more structure to the wild side of previously defined dividing lines and matching back to the usual picture. Finally, we present some partial results toward a new dichotomy, thematically related to a known dichotomy, which states that every NSOP theory is either stable or it has IP.

Abstract: In this talk we will aim to introduce the basics of Grothendieck style modern algebraic geometry focusing around schemes and perfectoid (really adic, but we will point out the difference later) spaces. The hope is to show how we aim to coherently glue together data from familiar algebraic objects to give insights into questions we care about. We will start with some comments on smooth manifolds as a motivating picture, but then dive into schemes proper. Algebraic constructions like ideals, localizations, and valuations will be used to reinterpret familiar geometric objects through the language of scheme theory. Then we will study the failures of our definitions thus far to properly handle p-adic spaces, leading us to building Huber’s adic spaces to account for that, with some hand-waving at how these are properly upgraded to perfectoid spaces. Throughout the talk hints at the categorical insights of Grothendieck coming from Yoneda will be provided to point to where objects like algebraic stacks or Scholze’s diamonds might arise from.

Abstract: We’ll discuss recent work which uses, among other tools, real number analogues of Erdős’ integer residue systems to construct approximating functions with a prescriptive approximating behavior when considered with two sets of inhomogenous parameters. These approximating functions are inhomogenous analogues of Duffin and Schaeffer’s original counter example to removing the monotonicity condition in Khintichine’s theorem, which we’ll introduce as background to the subject.

Abstract: This talk will present results by Jacek Świątkowski from “Trees of manifolds as boundaries of spaces and groups.” Prior to this paper, few explicit topological spaces were known to be homeomorphic to Gromov boundaries of hyperbolic groups. The list included spheres of arbitrary dimension and a number of small dimensional compact spaces. The goal of this paper was to extend the list to trees of manifolds of arbitrary dimension. To do so, Świątkowski proves that the visual boundaries of certain pseudomanifolds are trees of manifolds. These pseudomanifolds can then be hyperbolized, allowing us to construct hyperbolic groups with tree of manifolds boundaries.

Abstract: My paper “Every Interesting Cubic Graph Has at Least Fourteen Vertices: A Revised Proof” revises the proof for (2.1) in the paper “Girth Six Cubic Graphs Have Petersen Minors” by Neil Robertson, Paul D. Seymour, and Robin Thomas which states that every interesting cubic graph has at least fourteen vertices. The experience of writing this paper left me with two questions. The first: Can I pass off the tedious work in my proofs to a computer? The second: How can I efficiently come up with examples of interesting graphs? This led me on a journey of expanding my background by taking the course “COMP 360A: Introduction to Automated Reasoning” taught by Robert Rose at Wesleyan University during the 2025 fall semester. I began battling with both questions in my final project for the course. This final project ended up not being so final as it has evolved into an ongoing collaboration with Robert Rose. After introducing a new concept called interesting graph skeletons, we were able to pass off the entire proof for (2.1) to the automated modeling tool Alloy and we were able to get Alloy to spit out examples of interesting graph skeletons which can be easily modified to be examples of interesting graphs. In addition, our code has been a great starting point to begin passing off the proof of an additional fact mentioned in the original paper to Alloy. (Fact: There is exactly one interesting graph on fourteen vertices. This graph has girth six, and is usually called the Heawood graph.) My aim is to inspire audience members to consider adding Alloy to their research toolkit by demonstrating how useful it has been since being added to mine.

Abstract: When an elliptic curve $E/K$ is heavenly at $\ell$, there are arithmetic constraints that guarantees an upper-triangular representation of $G_K$ on the $\ell$-torsion of the curve. We hope that such simple representations are sparse on the absolute Galois group of a number field $K$. Previous work has shown that in many cases, for a fixed field $K$ there are only finitely many $K$-isomorphism classes of elliptic curves heavenly at all primes. Recent work of McLeman and Rasmussen explores a “bi-directional” finiteness result where both the prime of interest and the quadratic field of definition is allowed to vary. In this talk we will continue to explore my thesis research of extending this bi-directional finiteness result to cubic fields. In addition we will see how we lift finiteness results from isomorphism classes over the algebraic closure of $\mathbb{Q}$, to isomorphism classes over distinct prime degree fields.

Abstract: Developed by Jean Leray while a prisoner of war in 1940s Germany, a sheaf is a mathematical object used to keep track of additional structure on a topological space, namely by assigning to each open set a collection of data in a compatible manner. This talk will provide an overview of the very basics of sheaf theory, and in particular describe how morphisms between them are determined by local data at every point.

Abstract: In this talk, our goal is to determine the conformal dimension of a geometric structure called a hyperbolic building. This follows the Marc Bourdon paper, “Immeubles hyperboliques, dimension conforme et rigidité de Mostow,” which I will be expected to know for my candidacy exam. We will begin with a brief introduction into hyperbolic buildings, and how we can view them as a generalized reflections group. Then, we will consider geodesics through the space, so that we can determine the boundary of a hyperbolic building. Finally, we will introduce conformal dimension, which measures the dimension of the boundary using Hausdorff dimension. By placing a combinatorial version of a visual metric on the boundary, we can calculate the conformal dimension of a hyperbolic building.

Abstract: Let $\psi: \mathbb{N} \to \mathbb{R}$ be a decreasing function, and ${\gamma_q}_{q \in \mathbb{N}}$ be a sequence of real numbers and consider the number of integer solutions, $(q,p)$, to $|qx-p-\gamma_q| < \psi(q)$. If $W(\psi, \gamma)$ is the set of $x$ which have infinitely many integer solutions, then it has long been known through the Inhomogeneous Khintchine Theorem that the Lebesgue measure of said set is either 0 or 1 depending on whether $\sum_{q \in \mathbb{N}} \psi(q)$ converges or diverges if $\gamma_q$ is a constant sequence. However, it is unknown if this result is true for a nonconstant sequence. In this talk we will explore the progress toward a Khintchine type result when $\gamma_q$ is not a constant sequence.

Abstract: A variety of games can be played on a knot diagram that has no crossing information. Players take turns determining the crossing information to achieve their assigned goals (to either create the unknot or a nontrivial knot). We will discuss winning strategies in two such games: To Knot or Knot to Knot and Much Ado About Knotting. The talk will end with a list of additional games that have not yet been investigated.  We will conclude by pairing up and playing knot games!

Abstract: Artin groups are a family of groups that are prevalent in geometric group theory. However, they are defined abstractly in a manner which lacks motivation. In this talk, we’ll look for motivation for Artin groups by examining two concepts which seem unconnected at first glance: reflections and braids.