## September

21 – Leon Liu (MUST)

Title: The Moduli Space of Triangles

Abstract: Moduli spaces are ubiquitous in geometry. This talk is intended to give an introduction to the rich theory of moduli spaces by examining the case of triangles. We are going to discuss the notion of a moduli space and why it matters, as well as how the notion of stack appears even in the case of classifying triangles. We will give different models of (coarse) moduli spaces (for different notions of triangles) and describe their compactifications.

28 – Zachary Gardner (MUST)

Title: A Topological Proof of Bézout’s Lemma

Abstract: Bézout’s Lemma is a result of elementary number theory that states the following: given relatively prime integers p and q, there exist integers a and b such that a*p+b*q=1. Bézout’s Lemma is an extremely useful result that, among other things, allows us to prove that Z is a PID (principal ideal domain). The standard proof involves reverse-engineering the Euclidean algorithm. We will present an entirely topological proof of Bézout’s Lemma that uses the structure of the 2-torus T^2 and a gadget called the oriented intersection number. Our aim is to provide a small glimpse into the deep connections between number theory and topology.

## October

9 – Tom Gannon

Title: Arrow’s Impossibility Theorem

Abstract: In honor of the last day to register to vote in Texas, we will discuss all of the mathematics of voting. We will discuss the mathematics of all voting systems used around the world and discuss a key result known as Arrow’s Impossibility Theorem. Arrow’s Impossibility Theorem roughly says that there are no “reasonable” voting systems. We will discuss what a “voting system” is, what it could mean for a voting system to be reasonable, and then prove the theorem. This talk requires no prerequisite knowledge, and anyone is welcome to stop by!

12 – David Green (MUST)

Title: The Property F Conjecture

Abstract: Modular tensor categories are an important mathematical tool used in the study of topological quantum computing. We will discuss a little bit about what these tools are, with the goal of stating an open conjecture known as the Property F conjecture.

17 – Zachary Gardner (MUST)

Title: The Extension Problem

Abstract: The notion of building more complex structures from a collection of simpler objects is an idea that shows up across all of math. One very useful instance of this notion is the idea of extending a group by another group. We will investigate the structure of such extensions and motivate the concept of extension through example applications, in both the abelian (module) and non-abelian worlds. If time allows, we will sketch some general solution methods for extension problems.

## November

9 – Demetrius Rowland (MUST)

Title: Computational Neuroscience

Abstract: This talk will cover my work with Professor Thibaud Taillefumier in mathematical neuroscience. I have divided the talk into four sections.

- How we model the evolution of voltages in a neural network, which happens much faster than the evolution of synaptic weights
- The evolution of the weights after the set of voltages has settled
- A derivation of a formula for the weight changes in a two node network and a treatment of the formula’s implications
- A look at some of the code used to create the neural network

13 – Dr. Eileen Martin

Title: Extracting Green’s Functions from Random Data

Abstract: Cross-correlation of time-series data arises in many time-series regression analysis problems. In particular, in systems governed by partial differential equations (like the heat or wave equation), cross-correlations of data due to random sources tell us about the physics of the system. Sensor pairs’ cross-correlations yield an estimate of data that would be recorded on one sensor due to an impulse source at the other sensor, roughly a Green’s function of the PDE. Thus the cost and labor of traditional controlled-source experiments typically needed to infer material properties (thermal conductivity or wave speed) is no longer necessary.

The trade-off for using random sources is more expensive computational data processing: quadratic with the number of sensors, and difficult to parallelize. Compounding this issue, new technologies have made dense arrays of many vibration sensors much cheaper for experimentalists. Motivated by recent experiments in permafrost thaw monitoring and urban earthquake hazard analysis, I will show a new easily parallelizable linear algorithm for inferring wave speeds from random vibration data. A further challenge is that cross-correlations of vibration data only yield Green’s function estimates under certain assumptions on the random sources, and checking these assumptions is a mostly-manual process. I will show how we are using a mix of signal processing and machine learning to speed up and partially automate the interpretation of the wide variety of frequently changing noise sources found in urban areas and around infrastructure.

30 – Zachary Gardner (MUST)

Title: Completion: metric and algebraic

Abstract: Completeness is an important notion across much of mathematics and is useful because it allows for such things as sequential approximation. We will first discuss completeness of metric spaces. We will then characterize the completion of a metric space and prove existence and uniqueness. Next, we will discuss Krull topology and algebraic notions of completeness and completion. We will conclude by comparing both approaches in the context of the p-adic integers.

## December

7 – Hunter Stufflebeam (MUST)

Title: A primer in geometric measure theory

Abstract: None given