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 2025 Seminars will take place biweekly, on Mondays from 12:15 PM to 1:15 PM in Goldsmith 117

Current Organizer: Tariq Osman

 Schedule (Spring 2025)

Feb 10th: Thomas Ng (Brandeis)

Title: Residual finiteness in low-dimensional topology

Abstract: Residually finite groups can be approximated arbitrarily well by their finite quotients.  Classical work of Mal’cev in 1958 established residual finiteness as a criterion for linearity. We will discuss both the importance of residual finiteness in topology for promoting immersions to virtual embeddings with connection to the resolution of the virtual Haken conjecture for hyperbolic 3-manifolds, and also the challenges involved in constructing residually finite groups.

Feb 24th: Robin Zhang (MIT)

Title: Conway-Coxeter friezes as positive integral points

Abstract: In the 1970s, Coxeter and Conway introduced infinite arrays of positive integers called frieze patterns. In the 2000s, it was discovered that these friezes could be extended to general Dynkin types with deep connections to representation theory. Most types of positive integral friezes have been completely enumerated using methods from discrete geometry, algebraic combinatorics, and cluster algebras. We describe how to view these enumeration problems as a question of bounding the number of positive integral points on certain n-dimensional affine varieties associated to Cartan matrices.

Mar 10th: Kiyoshi Igusa (Brandeis University)

Title: From algebraic K-theory to ghost modules

Abstract: In my paper ``Generalized Grassmann invariant-redrawn'' (arXiv:2502.19147), I decided to redraw the pictures from my old paper so that they match the ``stability diagrams'' for representations of quivers which are a popular concept in representation theory. But, the diagrams don't quite match. One piece of the picture (one root of the associated Lie algebra) is missing. Using a modified version of Bridgeland stability, I will bring back the missing root as a ``ghost module''. But two ghosts appear! I will keep the exposition simple, drawing lots of pictures. I will show you how to look for ``ghosts'' with possible applications to algebraic K-theory.

Mar 24th: Claire Frechette (Boston College)

Title: Lattice Models: How Chemistry Reveals Information about Groups

Abstract: The study of lattice models, a visualization tool for molecular chemistry, began with the desire to understand how local energy states of one molecule could coalesce to create global phenomena like temperature or potential energy of a material. Shockingly, these same constructions allow us to study mathematical functions showing up in the representation theory of groups like the general linear group, giving us new tools that provide elegant proofs of symmetry-type identities and reveal additional information about these mathematical objects. In this talk, we will start with one popular case (Schur polynomials), which show up all over combinatorics, group theory, and representation theory, before diving into the flexibility of these chemistry-inspired models and the surprising algebraic information they encode.

April 7th: Malavika Mukundan (Boston University)

Title: Marked cycle curves for quadratic polynomials and rational maps

Abstract: Given any holomorphic dynamical system with a marked periodic cycle, we may track the changes in the marked cycle under perturbations of the system in some ambient parameter family. This gives rise to a marked cycle curve, which is a branched covering over the parameter family. In this talk, we discuss marked cycle curves of fixed period over two families: quadratic polynomials, and quadratic rational maps with a critical 2-cycle. The topology of these curves is highly related to the topology of the Mandelbrot set. In particular, we shall give cell-decompositions for these curves based on the combinatorics of special parameters in the Mandelbrot set. This is joint work with Caroline Davis, Daniel Stoll and Giulio Tiozzo. 

April 21st: TBA

Title: TBA

Abstract: TBA

Schedule (Fall 2024)

October 7th: Josh Wayne Southerland (Indiana University)

Title: Diophantine properties of affine diffeomorphisms of a lattice surface

Abstract: A translation surface is a polygon in the plane with an even number of sides, where we identify sides by translation (side come in pairs where each side in the pair has the same length). Some translation surfaces have large groups of affine diffeomorphisms which are diffeomorphisms mapping the surface to itself. On the polygonal representation of the translation surface, these affine diffeomorphisms look like linear maps applied to the polygon (where we cut and re-glue the transformed polygon back to the original). Lattice surfaces, whose collection of derivatives of these affine maps comprise a lattice in SL2(\R), are surfaces with large groups of affine diffeomorphisms. In this talk, we will discuss ongoing work that shows that the affine diffeomorphisms of lattice surface have Diophantine-like properties. This is ongoing work with Chris Judge. 

October 21st: Tal Malinovitch (Rice University)

Title: Twisted Bilayer Graphene in Commensurate Angles

Abstract: Graphene is an exciting new two-dimensional material. Though it was considered theoretical for a long time, it was isolated about 20 years ago. Since then, it has drawn much attention due to its numerous exciting properties. More recently, it was discovered that when twisting two layers of graphene with respect to each other, at certain angles called ``magic angles", exotic transport properties emerge. The primary tool for studying this thus far is the famous Bistritzer-MacDonald model, which relies on several approximations.

This work aims to build the first steps in studying magic angles without using this model. Thus, we study a model for TBG without the approximations mentioned above in the continuum setting, using two copies of potential with the symmetries of graphene, sharing a common origin and twisted with respect to each other (so-called TBG in AA stacking). We describe the angles for which the two twisted lattices are commensurate and prove the existence of Dirac cones for such angles. Furthermore, we show that for small potentials, the slope of the Dirac cones is small for commensurate angles that are close to incommensurate angles. This work is the first in a series of works to build a more fundamental understanding of the phenomenon of magic angles.  

In this talk, I will introduce the main phenomena of twisted bilayer graphene and state our main results.

October28th: Dani Alvarez-Gavela (Brandeis University)

Title: A Geometric Introduction to K_1

Abstract:  I will give a gentle introduction to the first algebraic K-theory group K_1(R) of a ring R from a geometric perspective and explain its relation to Smale’s proof of the topological Poincaré conjecture in high dimension.

November 11th: Canceled

Title: --

Abstract: --

November 25th: Nima Hoda (Tufts University)

Title: Nonpositively Curved Groups

Abstract: Geometry has been a tool in the study of groups for as far back as the work of Max Dehn on fundamental groups of surfaces in the early 20th century.  In this introductory talk, I will discuss what is meant when we speak of "nonpositively curved groups" in the broad sense.  I will start by introducing group presentations and van Kampen's Lemma, which provides us with a combinatorial geometric way of understanding groups.  I will then give some historical background, discussing Dehn's work exploiting hyperbolic geometry to solve algorithmic questions for surface groups.  Finally, I will discuss more recent developments in this line of thinking, which has emerged as a major subfield of geometric group theory.

December 9th: Yun Shi (Brandeis University)

Title: Bridgeland stability conditions and stable objects on some elliptic surfaces

Abstract: Bridgeland stability conditions, defined by Bridgeland, is a stability condition defined on an arbitrary triangulated category. In this talk, I will give a brief introduction to Bridgeland stability conditions on the derived category of coherent sheaves on smooth varieties, focusing on the cases of curve and surface. I will also discuss results on Bridgeland stable objects on some surfaces. This is a joint work with Tristan Collins, Jason Lo, and Shing-Tung Yau.

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.