Fall 2023

Fall 2023

Title: Direct Products Are Morse Limited

Abstract: Geometric group theory is the study of groups by understanding an associated metric space, known as a Cayley graph. In a metric space, Morse quasi-geodesics generalize the concept of quasi-convexity. I will walk through a proof that in the Cayley graphs of direct products of groups, the lengths of Morse quasi-geodesics is necessarily bounded (this is known as being Morse limited). This talk will be a good introduction to some of the concepts explored in geometric group theory.

Title: Secure Multiset Queries & Clustering

Abstract: I'll present two projects from 2022 in bioinformatics and mathematical biology. In the first project, my co-author and I demonstrated a methodology for increasing security guarantees on approximate count queries using Ring Learning with Errors (RLWE) cryptography. I'll briefly cover bounds of the approximation algorithm and security of (RLWE) which is analogous to the Shortest Vector Problem (SVP) on a lattice from number theory. Next, I'll present on a project clustering the medical data of Austism Spectrum Disorder (ASD) patients to identify subtypes of the disorder. We map patient records into a reduced subspace of words, and then apply the Latent Dirichlet Allocation (LDA) language model to cluster the reduced feature vectors. 

Title: The Modified Artin Complex

Abstract: Artin groups, a generalization of braid groups, are easy to define but hard to understand. There are many long-standing open questions, few of which have been answered for Artin groups in general. One of these questions involves the intersection of parabolic subgroups. This talk will highlight the geometric group theory technique of constructing and studying complexes with nice geometry and nice group action as a means to begin to answer this open question. We will discuss the class of “locally reducible” Artin groups, and define the “modified Artin complex” for these groups. Lastly, we will learn about a condition known as systolicity, a combinatorial analogue for CAT(0)-ness and show that the modified Artin complex for locally reducible Artin groups is systoli

Title: Extracting Eilenberg-MacLane Coordinates via Principal Bundles

Abstract: I am going to present an application of the theory of principal bundles to the problem of nonlinear dimensionality reduction in data analysis. More explicitly, given data with nontrivial underlying topology, we may derive coordinates for data points on an Eilenberg-MacLane space K(G,q) from persistent cohomology computation on dimension q with coefficients in G.

Title: Combinatorial Nullstellensatz

Abstract:  In this talk, I will present the Combinatorial Nullstellensatz method, developed by Noga Alon in the 1990s, and show its true power by proving theorems in Number Theory, Combinatorics, and (most importantly) give a criterion for the solvability of a Sudoku puzzle! 

Title: Thin film fluid dynamics for a square wave

Abstract: Under the assumptions of the thin fluid film approximation, the Navier Stokes equations for fluid dynamics can be simplified to a Poisson-like partial differential equation known as the Reynolds equation. The Reynolds equation describes the pressure of an incompressible Newtonian fluid in relation to space and time. In general, there is no exact solution for Reynolds equation. In the particular application to an N-step square wave height function, we can find an exact solution through a linear system of O(N) equations. By applying a Schur complement LU decomposition, this linear system can be solved efficiently. We are further interested in generalising the square wave solution to a discrete approximation of an arbitrary continuous height function. However, the thin film assumptions are delicate, and there is more to understand about how discontinuous height functions lead to error in the thin film approximation compared to a full Navier-Stokes solution.

Title: Combinatorics of exceptional collections in type $\tilde{\mathbb{A}}$ 

Abstract: We will define quivers of type $\tilde{\mathbb{A}}$, their representations, and exceptional collections of these representations. We will then introduce a combinatorial model of these representations, based on the one constructed by Garver, Igusa, Matherne, and Ostroff for type $\mathbb{A}$, by drawing strands on a copy of $\mathbb{Z}$. We will see that collections of strands called strand diagrams are in bijection with exceptional collections in type $\tilde{\mathbb{A}}$. After, we will introduce a different combinatorial model consisting of arcs on an annulus that is also in bijection with exceptional collections in type $\tilde{\mathbb{A}}$. Using the arc diagrams, we will place exceptional collections into finitely many infinite families, and then using a labeling of the strand diagrams, we will show that for a certain orientation of the quiver, the number of families is counted by a generalization of the Catalan numbers known as the Rothe numbers, or the Rothe-Hagan coefficients of the first type.

Title: Contravariantly finite subcategories

Abstract: The concept of contravariant finiteness was introduced by Auslander and Smalo in 1980. About a decade later, Auslander and Reiten were the first to relate it to the finitistic dimension (findim) conjecture. Specifically, they proved that if the subcategory of modules with finite projective dimension is contravariantly finite, then the findim conjecture holds. We will introduce representations of quivers/algebras, discuss their proof, and provide examples where this contravariant finiteness condition holds or fails.

Title: Hierarchical hyperbolicity and finite-type Artin groups

Abstract: Historically, a useful technique for studying groups has been to study a "nice" action of a particular group on a hyperbolic metric space. This technique is powerful (providing a solution to the word problem, among other things) but restrictive, since many interesting groups do not admit "nice" actions on hyperbolic metric spaces. One method of generalizing the technique is to look at actions of groups on metric spaces which are not hyperbolic but can be described by using many hyperbolic metric spaces to form a coordinate system. Such spaces are called hierarchically hyperbolic. In this talk, I will give a brief overview of hierarchical hyperbolicity, present some preliminary results towards finding a "nice" action of finite-type Artin groups on a hierarchically hyperbolic metric space, and explain what the word "nice" means in this context.

Title: Variations on a theme: geometry and representations

Abstract: The representation theory of semisimple Lie algebras is a powerful, elegant, and wide-reaching theory which is especially useful in geometric contexts. In this talk I'll introduce three closely-related theorems, spanning roughly thirty years, in which geometry gives back to representation theory by realizing representations inside of some geometric object. We'll look at some examples together and even see glimmers of the famed Riemann-Hilbert correspondence.

Title: On the relationship between t-structures and exceptional collections over hereditary algebras

Abstract: In this talk, we fix a finite dimensional hereditary algebra \Lambda. We will introduce the bounded derived category of finitely-generated modules over \Lambda, denoted \mathscr{D}^b. We will explore some of the properties of \mathscr{D}^b such as triangles, t-structures, simple-minded collections, exceptional sequences of objects in \mathscr{D}^b, and exceptional sequences of \Lambda modules. We will use Koenig and Yang's bijection between `algebraic' t-structures and simple-minded collections to construct a new bijection between equivalence classes of `algebraic' t-structures of \mathscr{D}^b and exceptional collections of \Lambda modules.