GSS – Fall 2025

Abstract: In this talk we would like to introduce the core ideas driving the geometric Langlands program, and try to paint a picture as to where they came from. The main focus will be on motivating the transition from classical problems to geometry. We begin with two classic stories: class field theory and modular forms. We then introduce how geometrizing the problems lends us new tools which reinterpret the results. A great deal of this will come in the form of introducing the dictionary between the two worlds.

Abstract: In their 1984 work, Ash, Grayson, and Green compute the action of Hecke operators on a certain subspace of the cohomology of low-level congruence subgroups of $\textrm{SL}(3, \textbf{Z})$. This subspace contains the cuspidal cohomology, which is of primary interest. In this talk, we introduce a method that extends their work, allowing for computing the action of Hecke operators directly on the cuspidal cohomology.

Abstract: The set of orthogeodesics between convex subsets of a proper $\mathbb{R}$-tree equipped with a cocompact group action equidistribute with respect to a natural measure on the subsets. This was shown by Broise-Alamichel, Parkkonen, and Paulin, and depends heavily on a creation result of orthogeodesics in $\mathbb{R}$-trees. In this talk, we will prove the creation result and we aim to understand the statement of the equidistribution result. If time permits, we will explore the challenges that arise in settings other than $\mathbb{R}$-trees (namely CAT(0) cube complexes).

Abstract: Martin-Löf Type Theory (MLTT) has influenced the design of several programming languages and interactive theorem provers such as Rocq (formerly Coq), Lean, and Agda. In 1994, Martin Hofmann and Thomas Streiche demonstrated a surprising interpretation of MLTT in which identity types could be regarded as groupoids. The unexpected result of this and later scholarship is that MLTT is powerful enough to act as a synthetic theory of infinity-groupoids, leading to a somewhat shocking triple equivalence in the apparently unrelated fields of computer science, type theory, and algebraic topology. This talk will introduce MLTT, focusing on understanding the syntax and constructing basic types, including identity types. We will briefly see how to think of identity types as groupoids, hopefully (time permitting) concluding with a discussion of infinity groupoids and the relationship between MLTT and homotopy type theory (HoTT).

Abstract: An elliptic curve $E$ over a number field $K$ is said to be heavenly at the prime $\ell$ if the $\ell$-torsion points meet some arithmetic conditions. Consequently, we see that the Galois representation arising from the action on the $\ell$-torsion is upper-triangular. In previous work, Rasmussen and McLeman conjecture finiteness results for heavenly elliptic curves over quadratic fields, and determine a relationship between trace behavior and whether a curve has complex multiplication. In this talk, we will cover the background needed to discuss heavenly curves, and begin to look at some results for curves over number fields of odd degree.

Abstract: Fermat’s Last Theorem, also known as Fermat’s Conjecture, was a focal point for centuries of development in number theory; posed circa 1637, the theorem states that given $n$ a natural number, the equation $x^n + y^n = z^n$ has no integer solutions for $n>2$. This was proven by Andrew Wiles in a series of papers from 1993-1995, wherein he established the Modularity Theorem for semi-stable elliptic curves. Wiles’ result requires sophisticated tools from arithmetic geometry, and is highly technical. However, a special case of the theorem may be proven via machinery contemporary to the late 19th and early 20th centuries, namely by examining unique factorization in a number ring $\mathbb{Z}[\omega]$. We will follow an outline given in Daniel Marcus’ “Number Fields” to demonstrate that if $p$ is a regular prime and $p$ does not divide $x$, $y$, or $z$, then there are no integer solutions to $x^p + y^p = z^p$.

Abstract: The aim of this talk will be to give partial answers to the question what spaces arise as boundaries of hyperbolic groups. To do so, we will first give an overview of introductory ideas and tools in geometric group theory. We will develop intuition for the relationship between geometry and groups, and follow that intuition to define two different notions of the structure of a group at infinity. The first, called the ends of a group, will be done in generality for any finitely generated group. The second, called the boundary of a group, will require us to restrict our viewpoint to a class of groups which behave nicely under quasi-isometries, called hyperbolic groups. The theory on ends of a group will provide a nice classification theorem, and the boundary of a hyperbolic group will lead us to our motivating question, which is an interesting area of open research.

Abstract: In a finitely presented group $G$, how hard is it to solve the word problem? One way to answer this is with $G$’s Dehn function, which involves giving a notion of “area” to words that are trivial in the $G$. If we are given an automorphism $\Phi$ of $G$, we can construct a new group called the mapping torus of G with respect to $\Phi$, that captures information about both $G$ and $\Phi$. Studying how Dehn functions interact with mapping tori has been an active avenue of research in geometric group theory in recent years. One prolific class of groups in the world of geometric group theory (and elsewhere) are hyperbolic groups; these turn out to be exactly the groups that have the best possible Dehn function, which is linear. This year, Qianwen Sun showed that any mapping torus of a one-ended torsion-free hyperbolic group has, at worst, quadratic Dehn function. In this talk, we will introduce the concepts mentioned, and give an outline for how Sun’s proof of this powerful result works.

Abstract: In Gödel’s proof of the incompleteness theorem, a formula $\textrm{Bew}(x)$ is defined in the language of Peano Arithmetic such that $\textrm{Bew}(n)$ is true iff the sentence having Gödel number $n$ is provable in Peano Arithmetic. Provability Logic is built on top of propositional logic by closing all formulas under a box operator $\square$, which we will then use to interpret $\square A$ being true to mean that $A^*$ is provable in Peano Arithmetic where $A^*$ is some translation of $A$ into the language of arithmetic. The talk will go over results in the paper Strong Completeness of Provability Logic For Ordinal Spaces by Juan P. Aguilera and David Fernandez-Duque, in particular how to interpret these formulas over topological spaces, as well as several completeness results that have been proven with respect to different classes of spaces.

Abstract: The ideal class group of an integral domain, when it exists, is an abelian group which measures a ring’s failure to be a PID. In this talk, we will define ideal class groups in the context of number fields, and discuss what group structures are possible. If time permits, we will also showcase an intersection of ideal class groups and elliptic curves in class field theory.

Abstract: Ash, Grayson, and Green computed the action of Hecke operators on the cuspidal cohomology of congruence subgroups $\Gamma_0(3, p) \subseteq \mathrm{SL}(3, \mathbb{Z})$ for small $p$. Recently, we extended their work, gathering additional data for larger $p$ using a new technique which allows for computations directly on the space of interest. A natural question to ask is for what other congruence subgroups of $\mathrm{SL}(3, \mathbb{Z})$ can one perform analogous computations. In this talk, we detail techniques for certain congruence subgroups that are Iwahori at $p$, providing a framework for understanding the action of Hecke operators on the corresponding cohomology.

Abstract: This talk is a live reenactment, minus all the times I was not perfect, of me revising 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 content of this talk has already been formalized in my recent paper “Every Interesting Cubic Graph Has at Least Fourteen Vertices: A Revised Proof”. The idea for my recent paper was inspired by a final presentation given by Jay Lee during which he pointed out that many details of the proof for (2.1) were omitted that are thought-provoking. My aim is to leave novice graph theorists with an idea for the type of arguments often left to the reader in well-written graph theory papers along with tips on how to work through them themselves.