2023-2024

We begin by introducing notions in the Theory of Holomorphic Invariants in the sense of Ecalle. For the purposes of resummation, it is often illuminating to consider families of distributions that separate points in a suitable space of testing functions (e.g. exponential distributions in Fourier Analysis), as well as , in a dual sense, to investigate zeros of particular integral operators (e.g. Fourier-Laplace transforms in analytic/Schwartz spaces). This approach leads naturally to classical gap theorems in Analysis like the Fabry Gap Theorem. And finally, by means of singular Poisson decompositions, we illustrate by example how to extract asymptotic information from partition functions in Conformal Field Theory.

We will discuss how factorizations of differential operators yield an elementary approach to classical inequalities of Hardy–Rellich type. In particular, we will first derive the classical Hardy and Rellich inequalities using factorizations. We will then turn to a refinement of Hardy’s inequality in one dimension, before turning to a general parameterized $n$-dimensional family of inequalities. All results will be proven directly with no prior specialized knowledge assumed. Some of the results discussed will include current active research involving multiple authors.

Topologically ordered quantum spin systems have become an area of great interest, in part because the ground state space for these systems is a quantum error correcting code. This was reflected in the axiomatization of topological order given by Bravyi, Hastings, and Michalakis. In this talk, we will describe new local topological order axioms given in recent joint work with Corey Jones, Pieter Naaijkens, and David Penneys. These axioms strengthen those of Bravyi, Hastings, and Michalakis, and they give rise to a 1-dimensional net of boundary algebras. We then provide an example satisfying these axioms, namely Kitaev’s quantum double model. We compute the boundary algebras for this model and show that they give nets of algebras either corresponding to Hilb(G) or Rep(G) depending on whether the boundary is rough or smooth. In either case, the inductive limit of the boundary algebras is $M_{|G|^\infty}$ and we have a canonical state on the boundary algebra which is tracial. This is joint work with Shuqi Wei, Chian Yeong Chuah, David Penneys, Mario Tomba, Brett Hungar, and Kyle Kawagoe.

In this talk we will discuss a basic model of extension operator: the half space extension, also known as the Seeley’s extension. It is an operator E that extends functions on R^n_+={x_n>0}, and has the form Ef(x’,x_n)=\sum_ja_jf(x’,-b_jx_n). We will talk about the basic properties of E, whose proofs require no more than one complex variables. If time permitted I will talk about a recent work of mine joint with Haowen Lu, and some follow up open problems which may be accessible for undergrads and masters students.

In this talk we will discuss the spectral stability of traveling waves in a thin-film model for two-fluid Couette flow. This model consists of a PDE which traveling wave solutions were studied in a previous work by D. T. Papageorgiou and S. Tanveer. We are going to discuss a strategy to determine spectral stability and instability of these solutions based on several conditional theorems where the conditions were checked with help of computer assist. This is joint work with S. Tanveer.

In this talk, we will introduce globally subanalytic sets and take about some of the basic results in subanalytic geometry such as their cellular decomposition, o-minimal structure, integration properties and the Finite Witness Theorem. We will also highlight the latter theorem in a mathematical data-science context.

In this talk, we discuss the utility of proving weak-type estimates to obtain the range of L^p boundedness for a singular integral operator from a harmonic analysis viewpoint. In addition, we show the failure of a weak-type estimate on a non-smooth domain in C^n using a fundamental integral operator from several complex variables.

TBD