Here is the planned schedule for Fall 2023
Here is the planned schedule for Fall 2023
November 13th, 2023 (In Person)
Title: Comparing and Integrating the Probe Method and Method of Singular Sources.
Abstract: Inverse obstacle scattering problems governed by wave equations are concerned with the recovery of unknown objects from the knowledge of scattered waves. Several so-called direct methods have been proposed to detect the unknown scatterer in the cases when the physical properties are not known or not given in a parametrized model. The main theme of these methods is in developing an indicator functional defined outside an unknown obstacle that blows up on the surface of the obstacle. This talk will give a comparison of two such methods: the probe method and the singular source method. I will first provide a theoretical framework for the two methods, emphasizing the advantages and disadvantages of each. I will then show that these two methods are closely related and basically form one unit with two different realizations.
November 6th, 2023 (In person)
Title: A Taste of Statistical Inverse Problems and the EM algorithm.
Abstract: In this talk, I will introduce the expectation maximization (EM) algorithm, a powerful tool for solving statistical inverse problems.
October 30th, 2023 (In person)
Title: How to decompose a normal operator as a sum of a compact and a diagonal operator.
Abstract: In 1909 Weyl proved that a bounded Hermitian operator on a separable Hilbert space can be written as the sum of a diagonal operator and a compact operator. This was generalized by von-Neumann who showed that boundedness is unnecessary, and the compact operator can be made Hilbert-Schmidt. This raises the natural question to ask whether similar decomposition results can be proved for normal operators. In this talk I will present the proof of a result of Berg, who showed that every normal operator on a separable Hilbert space can be written as a sum of a compact and a diagonal operator.
If time permits, I will mention modern generalizations and analogues of Berg's Theorem.
October 23rd, 2023 (In person)
Title: Bilinear Hilbert Transform and Beyond.
Abstract: The story starts with Thiele and Lacey’s resolution of Calderon’s Conjecture for the Bilinear Hilbert Transform (B.H.T.). Their techniques—“Time-Frequency Analysis” opens up new ways to look at problems in Harmonic Analysis. Soon afterwards, a surge of progress has been made regarding several forms of multilinear singular integrals and Carleson-type operators. At the current stage, we’re reaching the boundary capability of the technique. “What comes next?” is the big question. I’ll go over the development of relating topics on B.H.T. and if time permits, I’ll also talk about two of the related ongoing projects I’m working on.
October 16th, 2023 (In person)
Title: A Gentle Introduction to the Circle Method.
Abstract: The Hardy-Littlewood circle method, initially developed in the early 20th century for investigating the partition function $P(n)$, has since evolved into a pivotal tool for modern analytic number theorists and even some harmonic analysts. In this talk, we aim to introduce the Hardy-Littlewood circle method to a general audience with a foundational understanding of Fourier Series. We will delve into the evolution of the circle method and highlight the diverse problems it has resolved over the past century, not all of which are number theoretical in nature.
October 2nd, 2023 (Online)
Title: Recovering coefficients in a system of semilinear Helmholtz equations from internal data
Abstract: We study an inverse problem for a coupled system of semilinear Helmholtz equations where we are interested in reconstructing multiple coefficients in the system from internal data measured in applications such as thermoacoustic imaging. We derive results on the uniqueness and stability of the inverse problem in the case of small boundary data based on the technique of first- and higher-order linearization. Numerical simulations are provided to illustrate the quality of reconstructions that can be expected from noisy data.
September 25th, 2023 (Online)
Title: Stable determination of time-dependent collision kernel in the nonlinear Boltzmann equation.
Abstract: In this talk, we consider an inverse problem for the nonlinear Boltzmann equation with a time-dependent kernel in dimensions $n\ge 2$. We establish a logarithm-type stability result for the collision kernel from measurements under certain additional conditions. A uniqueness result is derived as an immediate consequence of the stability result. Our approach relies on second-order linearization, multivariate finite differences, as well as the stability of the light-ray transform.
September 11th, 2023
Title: Boundary rigidity problem under the presence of a magnetic field and a potential.
Abstract: The boundary rigidity problem asks if it is possible to determine the Riemannian metric on a compact manifold, up to a boundary fixing isometry, from the knowledge of the boundary distance function. In this talk I will discuss the solution to this problem in dimension 2, and what is known for the magnetic case, and the magnetic case with potential. Finally, and if time allows it, I will briefly discuss some open problems and recent results.