| Date | SPEAKER | Host |
|---|---|---|
| January 22 | No Seminar
TITLE : ABSTRACT: |
|
| January 29 | Changyou Wang, Purdue University
TITLE : Heat flow of harmonic maps into CAT(0)-metric spaces ABSTRACT: In this talk, we will describe a new approach to construct the existence of a unique suitable weak solution of the heat flow of harmonic maps into CAT(0)-metric space. The target space is the non-smooth version of nonpositively curved smooth manifolds in the sense of Alexandrov. The approach is variational and based on the minimization of a Weighted Energy Dissipation (WED) energy functional, which can be viewed as an elliptic regularization of parabolic problems proposed by De Giorgi back in 1990’s. By introducing a parabolic frequency function in the spirit of Frederick Almgren, we are able to show the existence of a unique global weak solution of the heat flow into a CAT(0) space that is Lipschitz in spatial variable and halfH\”older continuous in time variable. As a byproduct, this provides a new proof of the celebrated Eells-Sampson theorem on heat flow into non-positively curved manifolds. This is a joint work with F. Lin, A. Segatti, and Y. Sire. |
Wang |
| February 5 | Nick Gismondi, Purdue University
TITLE: Integrable Weak Solutions of Stationary Active Scalar Equations ABSTRACT: In this talk, I will discuss the construction of nontrivial integrable weak solutions for certain classes of stationary active scalar equations using a convex integration framework. We focus in particular on stationary active scalar equations with non-odd drift, as well as the stationary surface quasi-geostrophic equation. A central feature of the scheme is measuring the error in a negative-regularity homogeneous Sobolev norm, which enables the use of generic intermittent building blocks and allows us to consider arbitrary dissipation exponents. This talk is based on joint work with Alexandru Radu. |
Wang |
| February 12 | Howen Chuah, Purdue University TITLE : On a Nonlinear Model for Long Range Segregation ABSTRACT : We consider a system of fully nonlinear elliptic equations, depending on a small parameter, that models long-range segregation in population dynamics. The diffusion is governed by the negative nonlinear Pucci operator. We establish the existence of solutions and prove convergence, as the parameter goes to zero, to a free boundary problem. In the limt, high competition forces the species to segregate at a positive distance. Geometric properties of the free boundaries will be discussed, including directions for future research. This talk is based on joint work with Professors Monica Torres and Stefania Patrizi. | Torres |
| February 19 | Peter Morfe, Penn. State University
TITLE: Diffuse Interface Energies with Microscopic Heterogeneities: Homogenization and Rare Events ABSTRACT : I will revisit the scaling limit of the van der Waals-Cahn-Hilliard energy in the context when the energy has stationary, ergodic random coefficients, modeling a heterogeneous medium. The van der Waals-Cahn-Hilliard (WCH) energy provides a mesoscopic-scale, phenomenological description of interfacial energy in physics and materials science. In the 1970s, Modica and Mortola proved that, in the limit in which the characteristic interface width goes to zero, the WCH energy converges to the surface area functional. In this talk, I will discuss what is known in the case when the coefficients of the WCH energy are stationary, ergodic random fields. In this setting, interesting challenges arise due to the interplay between homogenization (averaging), energy minimization, and geometry. |
Yip |
| February 26 | Iassc Harris, Purdue University
TITLE: Transmission Eigenvalue Problems for a Scatterer with a Conductive Boundary ABSTRACT: : In this talk, we will investigate the acoustic transmission eigenvalue problem associated with an inhomogeneous medium with a conductive boundary. These are a new class of eigenvalue problems that are not elliptic, not self-adjoint, and non-linear, which gives the possibility of complex eigenvalues. The talk will consider the case of an Isotropic and Anisotropic scatterer. We will discuss the existence of the eigenvalues as well as their dependence on the material parameters. Because this is a non-standard eigenvalue problem, a discussion of the numerical calculations will also be highlighted. Lastly, we will discuss recovering the scatterer using a monotonicity method that is independent of the transmission eigenvalues. This is joint work with O. Bondarenko, V. Hughes, A. Kleefeld, H. Lee, and J. Sun. | Wang |
| March 5 | Raghav Venkatraman, University of Utah
TITLE: Eigenvalue problems for "Epsilon-Near-Zero" devices ABSTRACT: We consider novel eigenvalue problems for Laplace-type operators and Maxwell operators in core-shell geometries, where the dielectric permittivity epsilon is close to zero in the shell region, while being fixed in the core (representing a dielectric inclusion). These represent resonators made from "Epsilon-Near-Zero" materials with a dielectric inclusion and are of interest in the photonics community for possessing resonances that are robust to losses and are sometimes, at leading order, independent of the geometric features of the shell. We will investigate this class of problems through the lens of partial differential equations and spectral perturbation theory and elucidate the character of the geometry independence. This portion of the talk is joint work with Bob Kohn. Along the way we are naturally led to the following PDE question: given a C^1 bounded, connected domain D in \mathbb{R}^d, among the first N eigenvalues of the Dirichlet Laplacian on D, what fraction of the associated eigenfunctions have nonzero mean? Generically one expects (and one can prove) that the fraction is asymptotically 1, as N \to \infty, but the ball in R^d shows that the fraction of such eigenvalues can be small (\sim N^{1/d}). We prove a lower bound which is sharp up to a log factor, that for any domain D, among the first N eigenvalues, at least N^{1/d}/ \sqrt{log N} of them have eigenfunctions with nonzero mean. This portion of the talk is joint work with Stefan Steinerberger. |
Yip |
| March 12 | Mathew George, Purdue University | Wang |
| March 19 | (Spring Break, No Seminar) | |
| March 26 | Monica Torres, Purdue University TITLE: ABSTRACT : |
Wang |
| April 2 | Antonios Zitridis, University of Michigan TITLE: ABSTRACT : |
Han |
| April 9 | Christopher Irving, Georgetown University TITLE: ABSTRACT : |
Torres |
| April 16 | Agnid Banerjee, ASU TITLE: ABSTRACT : | Wang |
| April 23 | Roman Shvydkoy, UIC
TITLE: ABSTRACT : | Novack |
| April 30 | Anuj Kumar, UC Davis
TITLE: ABSTRACT : | Novack |
| May 7 | Junyuan Fang, UTK
TITLE: ABSTRACT : | Wang |