GSS – Fall 2025

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.