## Mathematical Sciences Department

The mathematical sciences department graduate seminar is a venue for expository talks and presentations on research by undergraduate students, graduate students, and faculty members. Anyone interested is welcome to attend and participate.

The seminar is currently organized by Naveed Iqbal and Adam Prinkey. Please send an email to nchaudhrob@my.fit.edu with your suggestions for future talks, as well as any questions or comments.
Date and Time Speaker Title & Abstract
Jan. 30
Mon. 9-10am
MAC
Evan Cosgrove
Math Sciences Department
Florida Tech
Title: On the Frechet Differentiability in Optimal Control of Coefficients in Parabolic Free Boundary Problems
Abstract:

We consider the inverse Stefan type free boundary problem, where the coefficients, boundary heat flux, and density of the sources are missing and must be found along with the temperature and the free boundary. We pursue an optimal control framework where boundary heat flux, density of sources, and free boundary are components of the control vector. The optimality criteria consists of the minimization of the $L_2$-norm declinations of the temperature measurements at the final moment, phase transition temperature, and final position of the free boundary. We prove the Frechet differentiability in Besov-Hölder spaces, and derive the formula for the Frechet differential under minimal regularity assumptions on the data. The result implies a necessary condition for optimal control and opens the way to the application of projective gradient methods in Besov-Hölder spaces for the numerical solution of the inverse Stefan problem.

Feb. 06
Mon. 9-10am
MAC
Dr. Jonathan Goldfarb
Math Sciences Department
Florida Tech
Title: Some details of the proof of Frechet Differentiabilty in Optimal Control of Coefficients in Parabolic Free Boundary Problems
Abstract:

We provide some details on certain technical results from our joint work on Frechet Differentiabilty in Optimal Control of Coefficients in Parabolic Free Boundary Problems.

Feb. 27
Mon. 9-10am
MAC
Tsz Chung Ho
Math Sciences Department
Florida Tech
Title: Variational methods for nonlinear PDEs
Abstract:

We apply basic tools from critical point theory to prove existence results for nonlinear PDEs.

Mar. 13
Mon. 9-10am
MAC
Saleheh Seif
Math Sciences Department
Florida Tech
Title: Breast Cancer Detection via Electrical Impedance Tomography and Optimal Control Theory
Abstract:

Electrical Impedance Tomography (EIT) is a rapidly developing non-invasive medical imaging technique with various applications for cancer detection and screening purposes. Electrical conductivity of the cancerous tumor is significantly different from healthy tissue. Mathematical formulation of recovering electrical conductivity of the body from the surface electrode measurements of electrical current is a nonlinear inverse problem for the second order elliptic partial differential equation, referred to as Calderon's inverse problem. In this project we use a variational approach and formulate the breast cancer detection as an optimal control problem for the second order elliptic PDE with bounded measurable coefficients. In this talk I will demonstrate proof of the existence and uniqueness of the solution to mixed problem for elliptic PDE through application of the Lax-Milgram Theorem. Next I will discuss Frechet differentiability of the cost functional in Hilbert-Sobolev spaces.

Mar. 20
Mon. 9-10am
MAC
Shiqiu Fu
Math Sciences Department
Florida Tech
Title: Variational methods for nonlinear PDEs
Abstract:

In this talk, I will speak about the Krasnoselskii Genus of an Index Theory. We will discuss some definitions, propositions, theorems and some proofs for the lemmas. Moreover, I will present some applications of Genus in PDE