Meets on (some) Mondays at 4:30 p.m. in Stevenson 1432

Organized by: Denali, denali dot d dot relles at vanderbilt dot edu

Monday, April 20th

Kristina Dahle

A friendly tour of continuous functional calculus

The continuous functional calculus allows us to apply a continuous function to a normal element in a C* algebra. In this talk, we'll introduce some basic ideas of C* algebras, what continuous functional calculus is, and what uses it has.

Monday, March 30th

Darrion Thornburgh

Title: Determining those Boolean functions whose restrictions to affine spaces are plateaued

Abstract: It is well-known that bent Boolean functions (i.e. maximally nonlinear Boolean functions over \(\mathbb{F}_2^n\) with \(n\) even) have the property that their restrictions to affine hyperplanes are plateaued (i.e. have a Walsh transform taking at most one nonzero absolute value). Quadratic Boolean functions (i.e. Boolean functions of algebraic degree at most two), and as we prove in this talk, partially-bent functions (i.e. affine extensions of bent functions) also share this property with bent functions. We characterize partially-bent (resp., quadratic) Boolean functions as those functions that are plateaued on any affine hyperplane (resp., any affine subspace of dimension \(k\), where \(3 \leq k \leq n-2\), while these are all Boolean functions for \(0\leq k\leq 2\)). Our characterization of partially-bent (resp., quadratic) functions extends to strongly plateaued vectorial functions. We state an open question on vectorial functions that happens to be related to an important one on crooked functions. This work is in collaboration with Claude Carlet.

Monday, March 23th

Will McDermott

The Characteristic Initial Value Problem

Monday, March 2nd

Jayashree Kalita

Title: Asymptotics for Almost Alternating Sign Patterns via the Circle Method

Abstract: Computer experiments led Andrews, in 1986, to conjecture striking sign patterns and growth phenomena for the coefficients of five partition-theoretic q-series from the Ramanujan's Lost Notebook. The first of these functions, the now-famous series \[ \sigma(q):=\sum_{n\geq0}\frac{q^{n(n+1)/2}}{(-q;q)_n} \] exhibits remarkable growth and vanishing behavior, which was proven by Andrews, Dyson, and Hickerson, by tying this series to the arithmetic of the quadratic field \(\mathbb{Q}(\sqrt{6})\). Cohen further uncovered that the numerical phenomenon was due to the q-series being what we would now call, thanks to work of Lewis-Zagier, a period integral of a Maass waveform. This example also foreshadowed the modern theories of mock Maass theta functions initiated by Zwegers, and quantum modular forms introduced by Zagier. \\ However, the other four q-series remained largely unexplored until recent work of Folsom, Males, Rolen, and Storzer, who proved some of the Andrews’ conjectures for the series \[ v_1(q):=\sum_{n\geq0}\frac{q^{n(n+1)/2}}{(-q^2;q^2)_n}. \] Jointly with Kundu, Storzer and Wang, we established almost alternating sign patterns for coefficients of the remaining three q-series along with proving a 40-year-old conjecture of Andrews. Using analytic techniques like the circle method, we derived asymptotics for the coefficients, whose alternating and oscillatory behavior explains the observed patterns. We also introduced a new family of q-series exhibiting similar phenomena. In this talk, I will give a non-technical overview of the main ideas.

Monday, February 16th

Daniel Lee

Title: An Introduction to Optimal Transport

Abstract: Optimal transport is a rapidly growing area of mathematics with connections to analysis, probability, statistics, and computer science. In this talk, we will introduce the basic ideas of optimal transport theory, discuss why it has attracted so much attention in recent years, and survey some of the types of problems and applications that motivate current research.

Monday, February 2nd

Denali Relles

Title: Model Categories in Homotopy Theory

Abstract: Homotopy theory distinguishes itself by classifying objects up to homotopy, rather than the stricter notion of homeomorphism. In this talk, we will explore the problems this creates, and how these problems are resolved using the concept of a model category.

Thursday, November 20th

Darrion Thornburgh

Title: An introduction to Boolean functions, vectorial Boolean functions, and Fourier analysis over \(\mathbb{F}_2^n\)

November 10th

Brian Morton

Title: The Mathematics of Relativistic Fluid Dynamics

Abstract: In this talk, I will outline the basic propositions of classical fluids. Then, I introduce relativistic inviscid fluids: the relativistic Euler equations. There is special attention paid to the free-boundary situation with the physical vacuum condition. Finally, I consider the linearized physical vacuum boundary problem and show the energy estimate and existence of solutions.

October 27th

Mohao Yi

Title: Kakeya Problem over finite fields

Greg Borissov

October 13th

Title: The Doomsday Algorithm

Alberto Magana

September 29th

Title: Introduction to Tropical Geometry