| Date | SPEAKER | Host |
|---|---|---|
| January 22 | No Seminar
TITLE : ABSTRACT: |
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| 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 TITLE Complex Monge-Ampere equations for positive (p,p) forms on Kähler manifolds ABSTRACT The Monge-Ampere equation has been studied extensively on real and complex manifolds for several decades. On Kahler manifolds, this equation is used to prescribe the Ricci curvature of the manifold. In this talk, I will introduce an extension of the complex Monge-Ampere equation to (p,p) forms for $p > 1$. This equation has been previously studied for the case $p = n-1$ in connection with the Gauduchon conjecture. We show the existence of solutions for all $p$ on compact Kahler manifolds by deriving second-order estimates for the solution. |
Wang |
| March 19 | (Spring Break, No Seminar) | |
| March 26 | Cina Nolan, Purdue University TITLE: On the Tangential Trace of Curl Measure Fields ABSTRACT : Curl measure fields are p-integrable vector fields whose distributional curl isa vector-valued Radon measure with finite total variation. They have been used to establish a framework for problems involving vorticities under lower regularity conditions than often considered. In this talk, we discuss two approaches to establishing trace theorems for curl measure fields on sets of finite perimeter. We then compare the two approaches to obtain the tangential property of the trace. This talk is based on joint work with Professor Monica Torres. |
Torres |
| April 2 | Antonios Zitridis, University of Michigan TITLE: Quantitative homogenization of convex Hamilton-Jacobi equations in the Wasserstein space ABSTRACT : We study a homogenization problem for first-order Hamilton-Jacobi equations in the Wasserstein space with a convex Hamiltonian. We show that the solution , which is the value function of a mean field control problem, converges uniformly as to the solution of a limiting Hamilton-Jacobi equation whose Hamiltonian is obtained through a suitable cell problem. Furthermore, we establish quantitative rates of convergence. Under general assumptions, we prove that the rate of convergence is, while for a particular class of Hamiltonians depending only on the fast variable and the momentum, we obtain the optimal rate. Finally, we show that our homogenization result extends to mean field planning problems, where the terminal condition imposes a constraint on the final distribution. Joint ongoing work with Zhiyan Ding, Ibrahim Ekren and Yuxi Han. |
Han |
| April 9 | Christopher Irving, Georgetown University TITLE: Regularity for variational problems under constant rank constraints ABSTRACT : I will discuss the problem of regularity for variational problems governed by differential constraints satisfying a constant rank condition. Our focus will be on the linear growth setting, which in contrast to the superlinear growth setting presents significant challenges due to the absence of Korn-type inequalities, which prevents any form of reduction to the familiar gradient setting. To overcome these difficulties, as an essential tool we will establish certain Poincaré-Sobolev type inequalities for constant rank operators, which takes an unexpected detour through the theory of Ehrenpreis-Palamodov. This is joint work with Zhuolin Li (MPI Leipzig) and Bodgan Raiţă (Georgetown). |
Torres |
| April 16 | Agnid Banerjee, ASU TITLE: Quantitative uniqueness for parabolic equations with applications ABSTRACT : I will talk about a new sharp estimate of the order of vanishing of solutions to parabolic equations with variable coefficients. I will then show that an application of such an estimate for real-analytic leading coefficients leads to a parabolic generalization of the well known Donnelly-Fefferman nodal set estimate. I will also talk about applications to Landis type decay results in the parabolic setting and extensions to fractional powers of the heat operator. This is based on joint works with Vedansh Arya, Nicola Garofalo and Abhishek Ghosh. | Wang |
| April 23 | Roman Shvydkoy, UIC
TITLE: On regularity and asymptotics of kinetic alignment models ABSTRACT : In this talk we discuss wellposedness, regularization, and relaxation for a broad class of Fokker-Planck-Alignment models which play a central role in collective dynamics. The main feature of these results, as opposed to previously known ones, is the lack of smoothness or no-vacuum requirements on the initial data. With a particular application to the classical kinetic Cucker-Smale model, we demonstrate that any bounded data with finite high moment, $(1+|d|^q)f_0\in L^1$, $f_0\in L^\infty$, $q\gg 2$, gives rise to a globally instantly smooth solution, satisfying entropy equality and relaxing exponentially fast. The proof is based on hypocoercivity techniques and DiPerna-Lions renormalization. | Novack |
| April 30 | Anuj Kumar, UC Davis
TITLE: Nonuniqueness of solutions to the two-dimensional Euler equations with integrable vorticity ABSTRACT : Yudovich established the well-posedness of the two-dimensional incompressible Euler equations for solutions with bounded vorticity. DiPerna and Majda proved the existence of weak solutions with vorticity in L^p ( p > 1). A celebrated open question is whether the uniqueness result can be generalized to solutions with L^p vorticity. In this talk, we resolve this question in negative for some p > 1. To prove nonuniqueness, we devise a new convex integration scheme that employs non-periodic, spatially-anisotropic perturbations, an idea that was inspired by our recent work on the transport equation. To construct the perturbation, we introduce a new family of building blocks based on the Lamb-Chaplygin dipole. This is a joint work with Elia Brue and Maria Colombo. | Novack |
| May 7 | Junyuan Fang, UTK
TITLE: ABSTRACT : | Wang |