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Invited Paper Session Abstracts - Harmonic Analysis and Applications to Complex Analysis and Partial Differential Equations

Thursday, July 30, 8:00 a.m. - 10:50 a.m., Philadelphia Marriott Downtown, Grand Ballroom C

This invited paper session focuses on research problems at the interface between Harmonic Analysis, Complex Analysis, and Partial Differential Equations. This choice is motivated by the fact that combinations of techniques originating in these fields has proved to be extremely potent when dealing with a host of difficult and important problems in analysis. Indeed, there are many recent notable achievements in this direction whose degree of technical sophistication is truly breathtaking. The main scientific aims of this effort are to introduce young mathematicians (advanced undergraduate students, graduate students and postdoctoral fellows) to problems of interest in Harmonic Analysis, Complex Analysis and Partial Differential Equations, to strengthen their background in these areas, and to make them aware of possible new avenues of research and collaboration.

Irina Mitrea, Temple University
Jeongsu Kyeong, Temple University

Inverse Problems: Determining the Equation from the Solution

8:00 a.m. - 8:20 a.m.
Shari Moskow, Drexel University


Mathematicians and scientist are often looking for the solution to a known partial differential equation. However, sometimes rather than solving the equation, we are working backwards from data to find the equation itself. This kind of inverse thinking shows up in areas like medical imaging, remote sensing, nondestructive testing, and many other areas of science. Specialists may be using this approach to find a tumor in a breast or to locate oil in a geological reservoir. However, the smallest miscalculations in the acquired data can lead to large errors in defining the equation’s parameters. We will discuss the challenges facing scientist and mathematicians in solving these ”inverse problems.”


Geometrically Stable Oscillatory Integral Operators

8:30 a.m. - 8:50 a.m.
Ellen Urheim, University of Pennsylvania


One group of objects studied in harmonic analysis are oscillatory integrals, an example of which is the Fourier transform. An oscillatory integral operator usually takes an input function, multiplies by a cutoff function and a function oscillating at a rate given by a parameter \(λ\), and then integrates. In general, we expect that the more oscillation we have (the bigger \(λ\) is), the smaller the output should be, due to cancellation happening in the integral. We often seek to quantify this relationship via a decay rate that depends on \(λ\). In this talk, we will discuss some particular examples of oscillatory integral operators, including examples from recent work with P. T. Gressman of operators for which the decay rates do not change when we make small enough modifications to the oscillatory function and the cutoff function.


A Sharp Divergence Theorem and Applications to Complex Analysis

9:00 a.m. - 9:20 a.m.
Dorina Mitrea, Baylor University


In this talk I will discuss a version of the Divergence Theorem for vector fields defined in rough domains satisfying minimal conditions. As applications of this theorem I will prove a very general version of the Cauchy-Pompeiu representation formula and of the classical Residue Theorem in complex analysis. This is joint work with Irina Mitrea and Marius Mitrea.


Mellin Analysis Techniques for Boundary Value Problems

9:30 a.m. - 9:50 a.m.
Katharine Ott, Bates College


In this talk I will discuss the use of Mellin analysis techniques to prove sharp invertibility results for layer potential operators acting on \(L^p(∂Ω)\), \(1 < p < ∞\), whenever \(Ω\) is an infinite sector in \(\mathbb{R}^2\). This analysis is relevant to the layer potential treatment of boundary value problems in domains that are infinite sectors in the plane.


An Interplay between Fuglede Conjecture and Gabor Analysis

10:00 a.m. - 10:20 a.m.
Azita Mayeli, City University of New York


The Fuglede Conjecture states that a bounded set \(Ω\) in \(\mathbb{R}^d\) tiles the entire space with the translations if and only if the set is spectral, that is \(L^2(Ω)\) admits an orthogonal basis of exponentials. In this talk, we present some results obtained from our investigation on the Fuglede-Gabor Problem which links the conjecture to the existence of Gabor orthonormal bases for \(L^2(\mathbb{R}^d)\).


Singular Integral Operators for Elliptic Boundary Value Problems

10:30 a.m. - 10:50 a.m.
Jeongsu Keyong, Temple University


The plan is to discuss the employment of singular integral operators in the study of second order elliptic partial differential equations in Lipschitz domains. More specifically, the presentationwill be centered around the relationship between the geometry of the domain and the availability of a Fredholm theory for the boundary to boundary singular integral operators associated with the given PDE. The talk will be accessible to any graduate student who has completed a first year graduate course in measure theory.