Plenary Speakers

Abstract: Given a curve, or more generally, a surface, in R^n, it is often important to understand its regularity properties, for example the degree of smoothness, the features of the set of tangent points, etc. If we knew that this curve (or surface) admitted an explicit parametrization, we would know how to determine these properties using tools from calculus and differential geometry. But, what if we do not know, a priori, an explicit parametrization of the curve? How do we then determine its smoothness? 

A tool that is helpful in this regard are geometric functions. These functions capture simple geometric information about the curve at each point and at each scale, e.g. how close the curve is to being a line. In the 1980's Carleson conjectured that the behavior of a specific geometric function characterizes the tangent points of certain planar curves. Since the recent resolution of this conjecture in 2021 by Jaye, Tolsa, and Villa, there has been a lot of activity in understanding the relationship between this geometric function and the smoothness of curves and surfaces.  

In this talk we will introduce two types of geometric functions and discuss their role in characterizing the regularity properties of curves and surfaces. We will also discuss recent joint works with Max Engelstein and Tatiana Toro, and Xavier Tolsa and Michele Villa.

Abstract: Exceptional orthogonal polynomials (XOPs) emerged in 2009 from the study of exactly and quasi-exactly solvable potentials in quantum mechanics. Following the Bochner Classification Theorem in 1929, it was widely believed that the only complete sets of eigenfunctions for rational Sturm-Liouville equations were the classical orthogonal polynomial (COP) families of Hermite, Laguerre, and Jacobi. However, the discovery of XOPs broadened the set of admissible solutions. Surprisingly, despite “missing” certain polynomial degrees, XOPs form a complete set in their associated Hilbert space.

Each XOP family is linked to a corresponding COP family by a sequence of Darboux transformations. While these transformations are often described as “state-deleting” since a finite number of degrees are removed from the set of eigenfunctions, this terminology can be misleading when studying the spectrum. The set of eigenvalues for an XOP system may remain identical to that of its associated COP or even include new points!

In this talk, we will discuss the formulation of the Laguerre XOPs through Maya diagrams, a combinatoric tool used to illustrate the Darboux transformation process. Additionally, we will examine the consequences of these transformations on self-adjoint extensions of the differential expressions and the resulting spectrum

Steve Sawin, Fairfield University

Title: A Deep Question From Physics, But Not An Answer!

Abstract: Analysis and Differential Equations as fields of mathematics have always involved a complicated dance between intuition and rigor, and translating powerful intuition into rigorous formalism has often driven these fields forward.  I will tell a little bit of that story, then talk about a beautiful and powerful intuition from physics, called the Path Integral or Feynman Integral, that has adamantly resisted this rigorous understanding even as it has become an essential tool in physics and a powerful one in mathematics.  I will introduce you to this question in an accessible and mostly nontechnical way, with a focus not on efforts to find the rigor, which so far are in their early stages, but on seeing this as part of a long tradition of translating intuition into rigor in these fields.