# Analysis Seminar

Current contact: Irina Mitrea

The seminar takes place Mondays 2:30 - 3:30 pm either in Wachman 617 or virtually via Zoom. Please see the announcment of the specific meeting. For virtual meetings, please write if you wish to join. Click on title for abstract.

• Monday January 30, 2023 at 14:30, Wachman 617

Gerardo Mendoza, Temple University

Abstract: Let $M$ be a closed $n$-manifold, $H^q(M)$ its de Rham cohomology groups, which are finite dimensional vector spaces. The Lefschetz number of a smooth map $f:M\to M$ is $L_f=\sum_{q=0}^n (-1)^q\mathrm{tr}(f_q^*)$ where $f^*_q:H^q(M)\to H^q(M)$ is the linear transformation induced by $f$ and $\mathrm{tr}(f_q^*)$ is its trace. A theorem of Lefschetz asserts that if $L_f\ne 0$ then $f$ has fixed points. A theorem of Atiyah and Bott gives a formula for $L_f$ under some condition on $f$. I plan to review this, then describe work in progress with L. Hartmann in a certain setting in which $M$ has singularities and the de Rham complex is replaced by a related complex.

• Monday February 6, 2023 at 14:30, Wachman 617
Radiation Conditions for Null-Solutions of the Helmholtz Operator

Irina Mitrea, Temple University

Abstract: The goal of this talk is to identify the broadest possible spectrum of radiation conditions for null-solutions of the vector Helmholtz operator. This contains, as particular cases, the Sommerfeld, Silver-Muller, and McIntosh-Mitrea radiation conditions corresponding to scattering by acoustic waves, electromagnetic waves, and null-solutions of perturbed Dirac operators, respectively. This is joint work with Dorina Mitrea and Marius Mitrea.

• Monday February 20, 2023 at 14:30, Wachman 617
Inverse Iteration for the Monge-Amp\ere Eigenvalue Problem

Farhan Abedin, Lafayette College

Abstract: I will present an iterative method for solving the Monge-Amp\ere eigenvalue problem: given a bounded, convex domain $\Omega \subset \mathbb{R}^n$, find a convex function $u \in C^2(\Omega) \cap C(\overline{\Omega})$ and a positive number $\lambda$ satisfying $$\begin{cases} \text{det} D^2u = \lambda |u|^n & \quad \text{in } \Omega,\\ u = 0 & \quad \text{on } \partial \Omega. \end{cases}$$ By a result of P.-L. Lions, there exists a unique eigenvalue $\lambda=\lambda_{MA}(\O)>0$ for which this problem has a solution. Furthermore, all eigenfunctions $u$ are positive multiples of each other. In recent work with Jun Kitagawa (Michigan State University), we develop an iterative method which generates a sequence of convex functions $\{u_k\}_{k = 0}^{\infty}$ converging to a non-trivial solution of the Monge-Amp\`ere eigenvalue problem. We also show that $\lim\limits_{k \to \infty} R(u_k, \O) = \lambda_{MA}(\O)$, where the Rayleigh quotient $R(v)$ is defined as $$R(v, \O) := \frac{\int_{\Omega} |v| \ \text{det} D^2v}{\int_{\Omega} |v|^{n+1}}.$$ Our method converges for a large class of initial choices $u_0$ that can be constructed explicitly, and does not rely on prior knowledge of the eigenvalue $\lambda_{MA}(\Omega)$. I will also discuss other relevant iterative methods in the literature that motivated our work.

• Monday February 27, 2023 at 14:30, Wachman 617
TBA

Jeongsu Kyeong, Temple University

Abstract: TBA

• Monday March 20, 2023 at 14:30, Wachman 617
TBA

Elie Abdo, Temple University

Abstract: TBA

• Monday April 17, 2023 at 14:30, Wachman 617
TBA

Irem Altiner, Temple University

Abstract: TBA

• Monday May 1, 2023 at 14:30, Wachman 617
TBA

Siqi Fu, Rutgers University, Camden

Abstract: TBA