Events for 04/04/2025 from all calendars

Mathematical Physics and Harmonic Analysis Seminar

Time: 1:50PM - 2:50PM

Location: BLOC 302

Speaker: Markus Pflaum, University of Colorado

Title: Representations of energy landscapes by sublevel set persistent homology

Abstract: Given a molecular Hamiltonian, its first potential energy eigenfunction in the Born-Oppenheimer approximation, the electron density and the electron localization functions are functions from chemistry which provide crucial information about the conformations, dynamics and reactibility of the molecule. In the DELTA project (Descriptors of Energy Landscapes by Topological Analysis) the surfaces spanned by these functions are examined by methods stemming from topology and singularity theory. The talk will give an overview of methods applied and results achieved by the DELTA group. In the case of n-alkanes, the sublevel persistent homology of the energy landscape and a visual representation of its Morse-Smale complex for n small could be determined. Using methods from real algebraic geometry and statistics we also present in this talk a method for learning the underlying variety of a data set which in our scenario comes from a molecular potential energy surface or one of its reductions. We explain numerical methods how to find singularities and conclude the talk with an application to the conformational space of cyclooctane. The talk is on joint work with H. Adams, A. Clark, Y. Zhang, E. AlSai, H. Jordan, P. Gara, J. Mirth et al.

Algebra and Combinatorics Seminar

Time: 3:00PM - 3:50PM

Location: BLOC 302

Speaker: Xingting Wang, Louisiana State University

Title: Deformations of noncommutative spaces through their quantum symmetries

Abstract: The theory of quantum groups is essential to understanding quantum symmetries in noncommutative geometry and mathematical physics. We discuss a deformation theory to study various classes of algebras by lifting algebraic properties to categorical contexts via their universal quantum groups, where we can then apply the theory of tensor categories.

Free Probability and Operators

Time: 4:00PM - 5:00PM

Location: BLOC 306

Speaker: Daniel Perales, Texas A&M University

Title: Even Hypergeometric Polynomials and Finite Free Commutators

Abstract: The finite free convolutions are binary operations on polynomials that behave well with respect to the roots and can be understood as a discrete analogue of free probability. On the other hand, these operations can be realized as expected characteristic polynomials of adding (or multiplying) two randomly rotated matrices. We will focus on the class of even polynomials and their behavior with respect to these finite free convolutions. First, we will use rectangular finite free convolution to understand even real-rooted polynomials in terms of positive-rooted polynomials. Then, we will study some related operations, such as symmetrizations, and finite free commutators. We provide new examples using even hypergeometric polynomials, that include classical families of orthogonal polynomials (such as Laguerre, Hermite, and Jacobi). Finally, we relate the limiting root distributions of sequences of even polynomials with the corresponding symmetric measures that arise in free probability.