Analysis Seminar

Fall 2025

For the Fall 2025 semester we will meet virtually on zoom at 11:00 am on Fridays. For more information contact Mourad Ismail or Xiang-Sheng Wang.

• 5 September 2025 (on Zoom: https://ullafayette.zoom.us/j/2022002220)
  Al-Salam–Chihara polynomials and limits of random Motzkin paths
  Alexey Kuznetsov
  York University, Canada
  Abstract: The central theme for this talk is the interplay between probability and analysis. We begin by discussing Motzkin paths with general weights and their connection with orthogonal polynomials. Next, we examine the limiting behavior of the initial and final segments of a random Motzkin path, as well as the macroscopic limits of the resulting processes. These results rely on the behavior of the Al-Salam–Chihara polynomials near the right endpoint of their orthogonality interval, along with the limiting properties of the q-Pochhammer and q-Gamma functions. The significance of these findings lies in the fact that these limiting processes also arise in the description of the stationary measure for the KPZ equation on the half-line and of the conjectural stationary measure of the hypothetical KPZ fixed point on the half-line.
• 12 September 2025 (on Zoom: https://ullafayette.zoom.us/j/2022002220)
  Nonlinear extension of the J-matrix method of scattering
  Abdulaziz Alhaidari
  Saudi Center for Theoretical Physics
  Jeddah, Saudi Arabia
  Abstract: The J-matrix method was developed in the mid 1970s by a group of physicists at Harvard University that included: Heller, Yamani, Reinhardt, et. al. The method describes quantum scattering due to short-range linear interaction potentials. It compares favorably to other well established scattering methods with enhanced accuracy and convergence. It was applied successfully in atomic, molecular, and nuclear physics. The method was turned into a rigorous mathematical technique by Ismail, Koelink, et. al. Here, we introduce an extension of the method to nonlinear self-interaction potentials. The extension relies predominantly on the linearization of products of orthogonal polynomials.
• 19 September 2025 (on Zoom: https://ullafayette.zoom.us/j/2022002220)
  Hypergeometric Functions and Modular Forms
  Ling Long
  Louisiana State University
  Abstract: The theories of hypergeometric functions and modular forms are highly intertwined. In this talk, we will give an overview of the theories leading to an explicit “Hypergeometric-Modularity” method for associating a modular form to a given hypergeometric datum. It is based on joint papers with Michael Allen, Brian Grove and Fang-Ting Tu, as well as recent papers by Esme Rosen.
• 26 September 2025 (on Zoom: https://ullafayette.zoom.us/j/2022002220)
  Finite asymptotic expansion for the energy of greedy sequences on the unit circle, and density of limit points
  Abey López-García
  University of Central Florida
  Abstract: We consider sequences on the unit circle obtained by a greedy algorithm which optimizes at each step the Riesz potential generated by previously selected points. Using an asymptotic series expansion due to Brauchart, Hardin, and Saff for the Riesz energy of equally spaced points on the unit circle, we give a finite asymptotic expansion for the energy of the first N points of a greedy sequence. We also show that for all values of the Riesz parameter s>-1, the normalized energy has limit points that fill out an interval, as it was expected from numerical experiments. The density of the limit points follows from the continuity of certain extensions of arithmetic functions defined on the interval [1/2, 1]. This talk is based on joint work with Erwin Mina-Diaz.
• 3 October 2025 (on Zoom: https://ullafayette.zoom.us/j/2022002220)
  Riesz Interpolation Formula for Askey-Wilson Operators and Beyond
  Dr. Xin Li
  University of Central Florida
  Abstract: This talk presents recent work with Rajitha Ranasinghe and Seok-Young Chung, tracing a path from classical interpolation formulas to modern generalizations in $q$-calculus. We begin with the Riesz Interpolation Formula for the derivatives of trigonometric functions and its generalization to functions of exponential type by Boas. We then detail our extension of this classical result to the Askey-Wilson operator, an important difference operator in $q$-orthogonal polynomials. A compelling consequence of this new formula is its equivalence to the classical Sampling Theorem, a connection revealed by the freedom afforded by the parameter $q$. The talk will conclude with a preview of ongoing work extending these ideas to a broad class of difference operators.