GSS Home

Spring 2024 GSS will be held weekly on Thursdays 2:20 - 3:20 pm ET in Goldsmith 300

The Graduate Student Seminar (GSS) is a weekly seminar in the Brandeis Mathematics Department run by the GSS organizers: Joshua Perlmutter and Sarah Dennis, on the behalf of the Graduate Student Representatives.

GSS is a great way to learn how to give research talks

It's difficult to overstate the importance of communicating your research effectively.  You can speak about whatever you're reading now, your research, or something you think all math grad students should know. So, join us and learn the skill many mathematicians never learn.

The seminar is by grad students and for grad students only! Faculty are not allowed to come.

You can view some previous semesters' schedules by using the navigation at the top of the page.

Spring 2024

Title: Job Talk

Abstract: In this talk we will highlight some of the research I have done throughout my Ph.D. and contextualize it within mathematics. We will go through some of the background and origins of topics in which I am interested, go through some results in broad detail, and finish with some current open questions on which I am working. 

Title: Schmidt game and nontrivial Schmidt diagrams

Abstract: The (\alpha, \beta)-game was introduced by Wolfgang Schmidt as a convenient tool for working with badly approximable numbers and some similar sets and, with several modern modifications, proved itself to be a useful instrument in Diophantine approximations and dynamics. We will not focus on this useful applications, but instead will discuss some questions about the properties of the original game and one recent result about it.

Title: Stochastic Collapse: An Implicit Bias of SGD for Deep Neural Networks

Abstract: Deep neural networks are known for their surprising generalization capabilities, even when they appear to be overparameterized. A potential explanation for this phenomenon is that the training process (stochastic gradient descent, or SGD) has an implicit bias for "simpler" models, which effectively have a lower parameter count. I'll be presenting the results from a 2023 paper by Chen, Kunin, Yamamura, and Ganguli which provides a mathematical framework for a new form of implicit bias termed "stochastic collapse," resulting from parameter-dependent gradient noise.

Title: Math History is Black History: A brief analysis of the fundamental mathematical achievements of African-Americans and their obstacles

Abstract: History is recounted, not only as a collection of significant events, but contributions that have expanded our knowledge and inspired further action, particularly in the arts and sciences. However, power and prejudice have invariably dictated the influence of certain groups and in turn, have left potentially groundbreaking works to be neglected and obscured. Mathematics is no stranger to this unfortunate consequence of history, as shown by the overwhelming majority of Eurocentrism. In observance of Black History Month, I will not only discuss the fundamental discoveries of a select few of many under-celebrated African-American mathematicians, but I also encourage an introspective analysis of the considerable paucity of African-Americans in mathematical society. 

Title: From Eisenstein Series to Eisenstein Series

Abstract:  Very rough introduction regarding Eisenstein series, from its original version and its applications to the modern (generalized) ones, which are central objects in many research areas nowadays and is still a mystery in many (most) senses (especially for the speaker)

Title: Representation Dimension of Artin Algebras

Abstract: Many properties of rings and algebras are described in terms of their modules; examples include semisimple algebras, Noetherian, Artinian rings and more. One of the most convincing examples of this may be Morita equivalence of rings. Two rings are said to be Morita equivalent if their (left) module categories are equivalent. Another such example that we will discuss is called representation finiteness, which is when the algebra has finitely many indecomposable modules up to isomorphism. Building off of this concept, Auslander introduced the "representation dimension" of an algebra which is intended to measure how far an algebra is from being representation finite. In this talk, we will motivate and define the representation dimension and discuss some of its properties.

Title: Smallest non-cyclic quotients of braid groups

Abstract: The braid group on n strands has a quotient isomorphic to the symmetric group on n elements, induced by a natural projection map. Margalit conjectured that this symmetric group is the smallest non-cyclic quotient of a braid group, and his conjecture was proved in recent work of Kolay. In this talk, I will give an overview of Kolay’s proof and discuss some applications of the result.

Title: Pi day special: Everything a mathematician should know about the moon

Abstract: Chances are you have seen the moon before. But how does it work? In this talk I will cover the basics of the moon: how phases work, cycles, what the earth looks like from the moon, change of phases, observed tilt angle, and the moon tilt illusion. Whether you like calendars, astrology, stargazing, optical illusions, or vectors, there is something for everyone! This will be a very accessible talk. The goal is not to introduce listeners to a different part of math, but rather to provide a mathematical way of thinking about something that we all observe but many don't know much about

Title: The Ax-Grothendieck Theorem: Studying Hypersurfaces with Gödel's Completeness Theorem

Abstract: The field of mathematical logic has a reputation for being secluded from the rest of mathematics: there aren't many well-known examples of the results from the field being applied to solve problems in other areas of math. In this talk, we'll look at an exception: Grothendieck's proof of the Ax-Grothrndieck theorem, a fun application of logic to solve a problem in algebraic geometry.

Title: Walking to Infinity Along Some Number Theory Sequences

Abstract: An interesting open conjecture asks whether it is possible to walk to infinity along primes, where each term in the sequence has one digit more than the previous. We present different greedy models for prime walks to predict the long-time behavior of the trajectories of orbits, one of which has similar behavior to the actual backtracking one. Furthermore, we study the same conjecture for square-free numbers, which is motivated by the fact that they have a strictly positive density, as opposed to primes. We introduce stochastic models and analyze the walks' expected length and frequency of digits added. Lastly, we prove that it is impossible to walk to infinity in other important number-theoretical sequences or on primes in different bases.

Title: Numerical simulation of moffatt vortices in thin fluid films

Abstract: The Navier-Stokes equations are a system of nonlinear PDE's modeling the physical properties of a fluid (volume, velocity, pressure, density etc). Under the assumptions of lubrication theory, where fluid is thin and flow is slow, the full Navier Stokes equations can be simplified to Stokes equations, and further simplified to Reynolds equations. Despite the simplification, there is still no exact solution to either Stokes or Reynolds equations for an arbitrary setting. I will summarize some of the numerical tools we have developed for solving the Stokes and Reynolds equations. I will then discuss the differences in fluid behavior observed by the two models. In particular, domains with sharp corners pose an issue. The Stokes equations predict the formation of an infinite sequence of vortices when the fluid is driven through the corner, whereas Reynolds equation does not capture this behavior. These vortices, also known as Moffatt eddies, are a concern for lubrication applications as not all fluid is being flushed through the domain. I will present some preliminary results we have for the error between Reynolds equation and Stokes equation for a triangular cavity-driven flow. These early results indicate that despite the Moffatt vortices being small in size and magnitude, the effect on the global fluid behavior is significant. 

Title: An Introduction to Length Spaces

Abstract: This talk will provide viewers with an introduction to the theory of length spaces. A length space structure imbues a metric space with a well-defined notion of continuous paths. I will walk through the definition, as well as examples and non-examples, and the talk will explore key properties of length spaces. If there is time, I will also discuss the notion of Gromov-Hausdorff convergence and the result that the Gromov-Hausdorff limit of a length space is a length space.