AMS Intro to Research Seminar has returned to Vincent 570!  Find us Wednesdays at 12:20.  🍕

Fall 2024 events:

10/23/2024- Speaker: Tyler Lawson

Title: "Algebra and equivalence"

Abstract: When we typically study algebraic structures, we do so via their elements and the operations we can apply to them. In this talk I'll discuss some example algebraic structures that originate in geometry, having both this "visible" aspect and a "subtext" that gives them extra structure.

10/08/2024- Speaker: Sasha Voronov

Title: "Algebra, geometry, and topology with a distant unifier: string theory"

Abstract: I will discuss a spectrum of problems in geometry and topology whose only relation to each other is that the ideas behind them came from string theory. One is a conjectural correspondence between del Pezzo surfaces and certain loop spaces, inspired by Mysterious Duality in string theory. Another one is constructing algebraic models of G-cyclifications. A totally different topic is supergeometry. The last one, time permitting, will be quantum deformation theory, a topic in homotopical algebra.

10/01/2024 - AMS Game Day

09/25/2024- Speaker:  Paul Garrett

Title: "Self-adjoint operators and the Riemann Hypothesis"

Abstract: In the early 20th century, both Hilbert and Polya (independently) speculated on the possibility of using the properties of self-adjoint operators on Hilbert spaces to prove the Riemann Hypothesis... which had come to be understood as not a mere question in complex analysis itself. In the early 1980s, Y. Colin de Verdiere resurrected this idea, with some proposals about self-adjoint operators on spaces of automorphic forms. It turns out (!?) that the simplest incarnations of those ideas definitely do not succeed (as E. Bombieri and I showed a few years ago). Nevertheless, the failure is subtle, and educational, suggesting further avenues of research.

01/31/2024- AMS Game Day

01/31/2024- AMS Game Day

01/24/2024- Speaker:  🎉Sasha Voronov🎉

Title: Drawing ideas and motivation from mathematical physics

Abstract: Mathematical physics has been an inspiration and motivation for mathematicians as long as mathematics existed. Newton, Leibniz, Euler, and Gauss developed remarkable theories in pure mathematics trying to understand the physical world around them. Modern physics, in particular elementary particle theory, brought even more to mathematics in the way of fruitful tools and beautiful ideas, as well as topics of research. Among the biggies are supergeometry, quantum groups, Donaldson theory, Floer homology, Seiberg-Witten invariants, mirror symmetry, and string topology. Throughout my mathematical career, I have been moving through various fields of pure mathematics, ranging from algebraic geometry to Lie theory to algebraic topology to algebra, following the influx of ideas from supersymmetry, quantum field theory, and string theory. I will talk about my current research projects in rational homotopy theory, algebraic supergeometry, and homotopical algebra and their math physics roots.

12/13/2023- AMS Game Day

12/6/2023- Speaker: Sarbartha Bhattacharya 

Title :  Representations - finite groups to padic. 

Abstract: What are padic groups and why do people care about their structure and representations ? I will start by giving a brief answer to these questions and try to give a glimpse into problems which serve as a motivation for their study. We will see how finite groups of Lie type served as a motivating idea behind some relatively recent geometric methods used in the study of representations of these groups. I will start from the basic concepts and use simple examples throughout the talk in an attempt to make it accessible. It would be a light and informal talk, and my hope is that everyone leaves the talk with at least a rough idea of the bigger picture. 

11/29/2023- AMS Game Day

11/22/2023- AMS Game Day

11/15/2023- Speaker: Alexander Watson 

Title: Mathematics of novel materials: from Scotch tape to superconductivity

Abstract: In 2018, twisted bilayer graphene, two layers of graphene stacked with a relative twist, was found to have very subtle electronic properties. This created huge interest in developing methods for mathematically modeling the quantum mechanics of this system. I will discuss recent progress on this, explaining how mathematical ideas such as multiple-scales analysis, homogenization, ergodicity, and the calculus of variations can be applied to this system.

11/8/2023- Speaker: Gregory Handy

Title: A day in the life of a mathematical biologist

Abstract:In this talk, I will demonstrate the interdisciplinary nature of mathematical biology research by diving into two of my previous research projects. While both can be viewed as an investigation into stochastic fluctuations that drive biological processes, we see that they use drastically different mathematical frameworks to derive insight into acutely different applications. In the first part, we will consider a novel stochastic process motivated by astrocytes, glial cells that ensheath neuronal synapses and can rapidly remove diffusing signaling molecules from the synaptic cleft. In the second part, we will explore how the brain leverages interneuron diversity and noisy recurrent connections to assist with cortical computations. Throughout the talk, I will also highlight current problems being tackled in my lab and briefly highlight the structure and content of Math 8540: Topics in Mathematical Biology that I will be teaching next semester. Special note: I currently have funding available to support new graduate students in my lab.

10/25/2023- Speaker: Maru Sarazola

Title: What does research in category theory look like?

Abstract: Category theory has become an essential language in modern mathematics, touching upon a variety of other fields such as algebra, topology, and number theory, to name a few. However, this sometimes makes it look like category theory is all about repackaging things from other subjects, and we are left to wonder: what does a category theorist really do in terms of research? In this talk, I will attempt to give a glimpse into the world of a pure category theorist, and the type of problems that we like to think about. No previous knowledge of category theory required!

10/18/2023- Pre-talk for GeMM Colloquium on 11/19/2023

Speaker:  Laura Escobar  (Washington University St. Louis)  

Title: Volumes of polytopes and Erhart theory

Abstract: A (convex) polytope is the convex hull of a finite set.  There are many ways to measure a polytope: volume, number of vertices, number of lattice points inside the polytope, etc.  In this talk we will focus on the volume of a polytope and its discrete version, the Ehrhart polynomial.  We will also discuss how algebraic geometry provides tools to compute volumes of polytopes.  The talk will have many examples and pictures.

10/11/2023- Speaker: Max Engelstein 

Title: Random walks, cold corridors and lungs

Abstract: Given the temperature on the outside of the house, can you figure out how cold it will be at a specific point inside the house? Aside from being a question of great practical importance in Minnesota, this question has been the driving force behind the development of a subfield of math known as Harmonic Analysis for over 250 years. We will discuss Fourier's original solution to the problem (which works for circular houses) and move on to more modern development (which works even for houses with fractal walls). Along the way we will draw connections with Brownian motion, lungs and geometry. No prerequisites assumed; I'll use the word measure a few times, but I won't assume you know anything about measures. 

09/27/2023 - Speaker: Michelle Chu

Title: "Arithmetic hyperbolic manifolds and their finite covers"

Abstract: Hyperbolic manifolds are Riemannian manifolds of constant sectional curvature -1. The geometry of hyperbolic manifolds is completely determined by their topology, and also completely determined by their fundamental group. This means that we can study these manifolds from the perspective of geometry, topology, or group theory. In this talk I will introduce these concepts and focus on a particular class of hyperbolic manifolds which are constructed from integer points in algebraic groups. I will also discuss interesting research questions one can ask about their finite covers, or equivalently, the finite index subgroups of their fundamental groups.

09/20/2023 - Speaker: Paul Garrett

Title: "Self-adjoint operators and the Riemann Hypothesis"

Abstract: Ever since Hilbert and Polya more than 100 years ago, people have speculated on possible applications of spectral theory of self-adjoint operators to proving the Riemann Hypothesis (one of the most significant unresolved issues in mathematics). By this year, there are some definite connections known, between RH and eigenvalues of variants of the Laplacian on spaces generalizing the (non-Euclidean) modular curve (parametrizing isomorphism classes of elliptic curves). But, as E. Bombieri and I proved in 2013, after making the heuristic set-up rigorous (using global automorphic Sobolev spaces), the simplest approaches definitely cannot succeed (unless RH is violently false!). Ok, but, then, what about less-simple approaches? :)  I will also explain what RH is, and its mathematical significance (apart from its notoriety).