In 2024 Fall, the Math Department Colloquium Series will be held on Wednesdays 3:30 pm at CU 218. Tea and refreshments available at 3:00 p.m. in the Assmus Conference Room (CU 212).
If you have any questions, please reach out to the organizors Taeho Kim (tak422 AT lehigh DOT edu) and Ao Sun (aos223 AT lehigh DOT edu).
Date: 09/04/2024
Speaker: Si Tang (Lehigh University)
Title: On the Thouless-Anderson-Palmer (TAP) equations of spin glass models
Abstract: Spin glasses are disordered systems that were originally invented by theoretical physicists to study the ferromagnetic phenomenon as a result of local magnetic interactions of atoms. They have been found close connections with various directions of mathematics (e.g., random matrix theory), statistics (e.g., Bayesian inferences), and biology (e.g., network modeling and evolution). In this talk, I will discuss classic results, predictions and applications of spin glass models, with a focus on my recent progress on the Thouless-Anderson-Palmer (TAP) equations, a system of fixed-point equations in terms of the spin magnetization. The iterative scheme based on the TAP equations is a special example of the popularly adopted approximate message passing (AMP) algorithm in statistical inference and machine learning.
Date: 09/18/2024
Speaker: Angela Hicks (Lehigh University)
Title: An expression for the Frobenius map in the fundamental basis
Abstract: The Schur functions are a well known basis for the symmetric polynomials; many well studied families of symmetric polynomials are positive when expressed in the Schur basis. A key tool in the representation theory of the symmetric group is the classical Frobenius map, which encodes symmetric group modules as symmetric polynomials and irreducible representations as Schur functions. When a polynomial is a positive sum of Schur functions, it thus is the image, under this map, of a representation of the symmetric group.
We discuss a new formula for the Frobenius map on permutation representations in terms of Gessel's fundamental basis, and how it can be used to show polynomials are Schur positive. We also discuss its role in connecting the classical Frobenius map to a similar map of Krob and Thibbon on the representations of the 0-Hecke algebra.
Date: 09/25/2024
Speaker: Grant Barkley (Harvard University)
Title: The combinatorial invariance conjecture
Abstract: Let u and v be two permutations of the numbers 1,...,n. Associated to u and v is a polynomial P_uv, called the Kazhdan-Lusztig polynomial, which encodes numerical invariants that are central in the field of geometric representation theory. The coefficients of P_uv simultaneously describe the singularities of Schubert varieties, the structure of Hecke algebras, and the representation theory of Lie algebras. Associated to u and v is another object, the Bruhat graph of (u,v), which is a directed graph describing the transpositions taking u to v.
The combinatorial invariance conjecture (CIC) of Dyer and Lusztig asserts that the Bruhat graph of (u,v) uniquely determines P_uv. Recently, Geordie Williamson and Google DeepMind applied machine learning techniques to this problem. Using those techniques, they conjectured an explicit recursion that would compute P_uv from the Bruhat graph and thereby prove the CIC. In joint work with Christian Gaetz, we prove the Williamson-DeepMind conjecture in the case where u is the identity permutation. The talk will not assume familiarity with any of these topics.
Date: 10/02/2024
Speaker: Scott Williams (SUNY Buffalo)
Title: Lehigh Plus Fifty
Abstract: In 1964 I entered grad school at Lehigh University with intention to earn a PhD in Algebra. In the next fifty years I
1. solved problems in Topology and Dynamics.
2. worked at SUNY Buffalo; Charles University in Czech Republic and Beijing Normal University in China.
What mathematics did I do?
Date: 10/09/2024
Speaker: Ken Monks (University of Scranton)
Title: Proof Verification with Lurch
Abstract: Would your students benefit from an easy-to-use, open-source, web-based word processor that could check their assigned mathematical proofs? In this talk we introduce Lurch, our software project designed specifically for this purpose. We will explain how you can use this software and accompanying course materials, and customize it for your own purposes. While existing proof verification tools like Lean, Isabelle, Coq, and Mizar are powerful and effective, they often have steep additional learning curves and can be difficult to customize. We will explain how the custom Lurch validation algorithm overcomes these challenges, and pose some questions for future work.
Date: 10/16/2024
Speaker: Jason Hattrick-Simpers (University of Toronto)
Title: TBD
Abstract: TBD
Date: 10/23/2024
Speaker: Tara Trauthwein (Oxford University)
Title: The Central Limit Theorem, Stein's Method and Spatial Random Graphs
Abstract: The classical Central Limit Theorem states that sums of independent random variables behave more and more like normally distributed random variable when we increase the number of summands. This simple result has astonishingly wide consequences, since it means that a huge number of real phenomena observed in large quantities can be safely estimated by using the Gaussian distribution. Stein's Method, introduced in 1972, is a hugely convenient tool for the estimation of the theoretical distance between any random variable, and a Gaussian. In this talk, we will give a gentle introduction to Stein's Method and show how it can be used, not only in a context of sums of independent random variables, but in the rich topic of (spatial) random graphs.
Date: 10/30/2024
Speaker: Nick Mayers (NC State University)
Title: The quantum k-Bruhat order
Abstract: Finding combinatorial interpretations for the structure constants of Schubert polynomials is a long-standing open problem in algebraic combinatorics. In the case where one of the Schubert polynomials is a Schur polynomial, the structure constants are encoded in a poset called the “k-Bruhat order”. In studying the k-Bruhat order, Bergeron and Sottile were led to introduce a monoid which encodes the chain structure of the k-Bruhat order. Using the monoid structure, the authors were able to establish properties and descriptions of certain structure constants. In this talk, after outlining the developments discussed above, we discuss ongoing work concerning an analogous story for quantum Schubert polynomials and an associated quantum k-Bruhat order. This is joint work with Laura Colmenarejo.
Date: 11/06/2024
Speaker: Jinxin Xue (Tsinghua University)
Title: Singularities and global dynamics of the N-body problem.
Abstract: The N-body problem is a fundamental model in classical mechanics. It continues to play an important role in modern physics and mathematics due to its simplicity and richness of dynamical behaviors such as the existence of chaos, noncollision singularities, etc. In this talk, we give an overview of the dynamics of the N-body problem and explain our work on the existence of noncollision singularities and superhyperbolic orbits.
Date: 11/20/2024
Speaker: Linjun Zhang (Rutgers University)
Title: Statistical Perspectives of Algorithmic Fairness
Abstract: Algorithmic fairness in machine learning has recently garnered significant attention. However, several pressing challenges remain: (1) The fairness guarantees of existing fair classification methods often rely on specific data distributional assumptions and large sample sizes, which can lead to fairness violations when the sample size is moderate—a common situation in practice. (2) Due to legal and societal considerations, using sensitive group attributes during decision-making (referred to as the group-blind setting) may not always be feasible. (3) Useful machine learning models nowadays have complicated structures, making it challenging to develop practical theories. In this talk, we will share partial solutions to these questions.
Date: 12/04/2024
Speaker: Steven Groen (Lehigh University)
Title: The Schottky problem in characteristic p.
Abstract: First, we discuss what Abelian varieties are and why number theorists and algebraic geometers are interested in them. We then see that Jacobians of curves are important examples of Abelian varieties, but not all Abelian varieties are isomorphic to the Jacobian of a curve. This leads to a natural question, known as the Schottky problem: can you characterize the Abelian varieties that are isomorphic to the Jacobian of a curve? This is a classical and still unsolved problem in characteristic zero, but today we study this problem after reducing modulo a prime number p. We see that this gives a rich additional structure based on the p-torsion group scheme, which provides new tools for checking if an Abelian variety can be isomorphic to a Jacobian of a curve.