Here is the planned schedule for Spring 2023
Here is the planned schedule for Spring 2023
January 26th, 2023
Title: Microlocal Stability of the X-ray transform
Abstract: The X-ray transform is a map that associates to a function f its integral over a line. The motivation for this study of this operator comes from medicine: if we think of the lines as rays trough the patient, by inverting X-ray transform one can understand the body of a patient (the function f). This is known as the X-ray computed tomography problem. Another interesting problem is to obtain "stability results", that is, do “small” errors in the data lead to “small ”errors in f? Mathematically, we look for bounds of the norm of f in terms of the norm of the X-ray. On this talk we are going to show how to obtain a stability result to this operator by using methods of microlocal analysis. First, we are going to review basic facts of pseudodifferential operators, and then, we are going to apply this theory to the normal operator X*X to obtain the desire bound.
February 2nd, 2023
Title: Analysis of the Transmission Eigenvalue Problem with Two Conductivity Parameters.
Abstract: In this talk, we show the existence and discreetness of the transmission eigenvalue problem with two conductivity parameters. In previous studies, this problem was analyzed with one conductive boundary parameter, while in this study we consider the case of two parameters. The underlying physical model is given by the scattering of a plane wave for an isotropic scatterer. We will study the dependence on the physical parameters and the monotonicity of the first transmission eigenvalue with respect to the parameters. Lastly, we will consider the limiting procedure as the second boundary parameter vanishes and present numerical results.
February 16th, 2023
Title: Reciprocity gap functional methods for potentials/sources with small volume support for two elliptic equations.
Abstract: We consider two inverse shape problems coming from diffuse optical tomography and inverse scattering. For both problems, we assume that there are small volume subregions that we wish to recover using the measured Cauchy data. We will derive an asymptotic expansion involving their respective fields. Using the asymptotic expansion we derive a MUSIC-type algorithm for the Reciprocity Gap Functional, which we prove can recover the subregion(s) with a finite amount of Cauchy data. Numerical examples will be presented for both problems in two dimensions in the unit circle.
February 23rd, 2023
Title: An Introduction to Quadrature Domains and the X Marks the Spot Problem
Abstract: We define quadrature domains in the complex plane using only integrals and finite sums. We then discuss the X Marks the Spot problem for quadrature domains: Suppose we bury treasure on an island with smooth boundary. It is known that we can encrypt the location of the treasure as a point in a nearby quadrature domain. Can we now decrypt the location of the treasure using properties of quadrature domains? We also expand on the X Marks the Spot problem to study other functions associated with quadrature domains.
March 2nd, 2023
Title: The Factorization Method in Inverse Scattering
Abstract: Inverse problems concern discerning underlying causes from a set of observations. In many situations the mathematical modeling of these problems leads to the study of inverse boundary problems of partial differential equations that are highly non-linear and ill-posed: small errors in the data may cause uncontrollable errors in the solution. Of particular interest is the study of inverse time-harmonic wave scattering in modeling sound wave propagation in the frequency domain. Applications range from offshore oil exploration and underwater communication to nondestructive testing. This talk will introduce the factorization method for for time-harmonic inverse obstacle scattering problems where the task is to determine the support of the scatterer from multi-static far field measurements at fixed frequency. We will discuss how the factorization method helps us obtain a mathematically rigorous characterization of the scatterer’s support.
March 9th, 2023
Title: Minimal networks on Riemannian spheres
Abstract: A network is a finite union of curves whose endpoints meet at junctions. In a Riemannian manifold, a network is said to be minimal if it is a critical point of the length functional. In this talk I will first review some of the literature concerning minimal networks on surfaces, with a focus on the case of the 2-sphere. I will then consider certain minimal networks (θ-networks) on the sphere of any dimension endowed with a Riemannian metric which is close to the standard one. We obtain an existence and multiplicity result using a finite dimensional reduction method jointly with the Lusternik-Schnirelmann category.
March 30th, 2023
Title: An introduction to quasidiagonal operators
Abstract: The notion of quasidiagonality was first introduced by Halmos for a bounded operator on a Hilbert space. More specifically, an operator is quasidiagonal iff it can be written as a sum of a compact and a block diagonal operator with finite blocks. The definition was later extended for a family of bounded operators and for C^∗-algebras. In the case of C^∗-algebras, quasidiagonality has played an important role in the classification program. In this talk, we will see examples of quasidiagonal operators, permanence properties, as well as obstructions for quasidiagonality. Only basic knowledge regarding Hilbert spaces will be assumed.
April 4th, 2023
Title: Scalar curvatures estimates from Seiberg Witten Theory and Applications
Abstract: Seiberg Witten theory is a version of gauge theory that provides powerful tools in the study of smooth 4-manifolds. In this talk we will provide a gentle introduction to Seiberg Witten theory and discuss work of Witten and LeBrun on apriori estimates for the total scalar curvature of some closed 4-manifolds. These estimates have many interesting applications, such as uniqueness and nonexistence results for Einstein metrics. Time permitting, I will talk about some extensions of these estimates that I am working on.