Everytopic Seminar

Seminar Information

Description: The Everytopic topic seminar features colloquium style talks by mathematicians, typically from the Boston area, in an informal environment. We encourage speakers to prepare 50 minute talks about their area of research, accessible to graduate students. Refreshments are provided before and during the talk. 

Time and Place: Our spring 2024 Seminars will take place biweekly, on Mondays from 12:15 PM to 1:15 PM in Goldsmith 300

Current Organizer: Tariq Osman

 Schedule (Spring 2024)

January 29th: Cancelled (Department Meeting)

Title: --

Abstract: --

February 12th: Guilherme Silva

Title: Intrasensitive Dice

Abstract: Intransitivity is an inherent facet of nature, being part of evolutionary equilibrium, of election prediction, sports leagues, interactions between medications, among many others. In mathematics, it produces rather interesting phenomena already from basic objects, for instance when playing dice. It is rather simple to produce three dice A, B and C with four faces each which are intransitive, in the sense that A is a better die to play with than B, B is better than C, and C is better than A. What about producing intransitive dice with larger numbers of faces? And what about producing four or more intransitive dice? In our talk, we will explain how combinatorics of words and a certain Central Limit Theorem for correlated random variables helped us answer these questions recently.

The talk is based on joint work with Luis G. Coelho, Tertuliano Franco, Lael V. Lima, João P. C. de Paula, João V. A. Pimenta and Daniel Ungaretti, which was carried out as part of an REU-like program in Brazil.

March 11th: Lior Alon (MIT)

Title: Fourier Quasicrystals via Lee-Yang Polynomials

Abstract: The concept of "quasi-periodic" sets, functions, and measures is prevalent in diverse mathematical fields such as Mathematical Physics, Fourier Analysis, and Number Theory. The Poisson summation formula provides a “Fourier characterization” for periodicity of discrete sets, and a Fourier Quasicrystals (FQ) generalizes this notion of periodicity: a counting measure of a discrete set is called a Fourier quasicrystal (FQ) if its Fourier transform is also a discrete atomic measure, together with some growth condition. 

Recently Kurasov and Sarnak provided a method for constructing one-dimensional FQs as the intersections of an irrational line in the torus with the zero set of a multivariate Lee-Yang polynomial. In this talk, I will show that the Kurasov-Sarnak construction generates all one-dimensional FQs.

I will also discuss the distribution of gaps between atoms in such FQs, showing that the countably many gaps equidistribute on an interval, with a distribution given explicitly in terms of ergodic dynamical systems on tori. 

In the last part, I will present a generalization of the Kurasov-Sarnak construction to any dimension, by introducing Lee-Yang varieties.

March 25th: Keaton Quinn (Boston College)

Title: Isometric immersions and the Gauss-Codazzi equations at infinity. 

Abstract: A pair of tensors (g,B) form the induced metric and shape operator of an immersion into hyperbolic space if and only if they solve the Gauss-Codazzi equations. Therefore, these equations can be seen as a kind of integrability condition for isometric immersions. Under certain curvature restrictions, the pair (g,B) induces a pair (g',B')---related to the ideal boundary at infinity of hyperbolic space---that solve a dual set of equations tied to the conformal geometry of g'. Moreover, any solution (g',B') produces a solution (g,B) to the Gauss-Codazzi equations, and hence produces an immersion into hyperbolic space. We review this construction (of (g',B') to immersion), survey some of its recent uses, and describe our work generalizing these ideas to higher dimensions.

April 15th: Lorenzo Ruffoni (Tufts University)

Title: Graphs, groups, and a recognition problem

Abstract: The right-angled Artin group (RAAG) associated to a graph is the group generated by the vertices and in which two generators commute when they are connected by an edge. There is a beautiful interplay between the combinatorial properties of the graph and the algebraic properties of the associated RAAG. This has made RAAGs very popular over the years, so it is not surprising that many other groups try really hard to look like RAAGs. We will discuss the problem of recognizing RAAGs among groups that look like RAAGs. While this problem is in general undecidable, we will describe a strategy that works Bestvina-Brady groups. The method is based on a combinatorial analysis of certain spaces of characters. This is based on joint work with Y.-C. Chang.

May 6th: Bingyu Zhang (University of Southern Denmark)

Title: Microlocal theory of sheaves: a new approach to symplectic geometry.

Abstract: The celebrated pseudo-homomorphic curve theory stands as a cornerstone in symplectic geometry. Yet, its intricate nature often demands a thorough exploration of the moduli space of solutions to certain nonlinear Cauchy-Riemann equations, a task fraught with difficulty. Conversely, a relatively softer approach, the microlocal theory of sheaves pioneered by Kashiwara-Schapira in the 1990s, has emerged as an alternative technical avenue within symplectic geometry, offering insights across various facets of the field.

In this talk, I will elucidate the fundamental concepts of the microlocal theory of sheaves and illustrate its utility in unraveling the mysteries of symplectic geometry. To enhance accessibility, my focus will primarily rest on concrete examples rather than delving into deep theorems.

 Schedule (Fall 2023)

October 23rd: Promit Ghosal (Brandeis University)

Title: A probabilistic dive into Liouville Quantum Blocks.

Abstract: Liouville Conformal field theory (LCFT) is an integral component of noncritical Bosonic string theory. Conformal blocks of 2D LCFT are fundamental objects in showing the conformal bootstrap program and are closely related to four dimensional supersymmetric gauge theory. Recently, there have been major breakthroughs in constructing LCFT using probabilistic ideas and proving conformal bootstrap programs.  In this talk, we discuss how to construct conformal blocks using probabilistic ideas and mention many of its properties and interesting connections. This talk will be based on two separate works with Guillaume Remy, Xin Sun and Yi Sun. 

November 6th: Jerson Caro Reyes (Boston University)

Title: Lower bounds for the Mordell-Weil rank

Abstract: In 1922, Mordell proved that the set of rational points of an elliptic curve defined over the rational numbers is a finitely generated abelian group. This implies that it has finite rank, known as the Mordell-Weil rank.

Obtaining lower bounds for the Mordell-Weil rank of an elliptic curve defined over Q is a relevant problem in number theory. For example, the question of whether the ranks of elliptic curves over Q are uniformly bounded or not remains open due to the lack of sharp bounds. 

In this talk, I will present two distinct methods for finding lower bounds. The first method is based on a result by Silverman concerning families of elliptic curves. The second method is based on results of Gao, Ge & Kühne about subvarieties of abelian varieties. These are joint works with N. Garcia-Fritz and H. Pasten.

November 20th: Yangyang Wang (Brandeis University)

Title: Mathematical analysis and models for understanding neural and other dynamics

Abstract: Central pattern generators (CPGs) are neural networks that are intrinsically capable of producing rhythmic patterns of neural activity and are adaptable to sensory feedback to produce robust motor behaviors such as breathing and swallowing. In this talk, I will discuss mathematical tools we developed for understanding complex bursting dynamics in CPG neurons as well as robust responses of motor systems to external perturbations, across multiple scales. Applications to biological problems and novel mathematical concepts inspired by these applications will be highlighted.