| Date | SPEAKER and TITLE (No Seminar) | Host |
|---|---|---|
| January 19 (No seminar) | TITLE: ABSTRACT: |
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| January 26 | Kiril Datchev (Purdue University, USA)
TITLE:Low energy scattering asymptotics in dimension two ABSTRACT : Analysis of the Laplacian at low energy involves special challenges in dimension two. I will present a new technique, which is elementary and robust, for scattering theory in this setting, focusing on the fundamental example of obstacle scattering. The technique is based on a resolvent identity of Vodev. By an identity of Petkov and Zworski we deduce expansions for the scattering matrix and scattering phase, and similarly obtain expansions for the exterior Dirichlet-to-Neumann operator. The leading singularities at low energy are given in terms of the obstacle's logarithmic capacity or Robin constant. We expect these results to hold for more general compactly supported perturbations of the planar Laplacian, with the definition of the Robin constant suitably modified, under a generic assumption that the spectrum is regular at zero. This talk is based on joint work with Tanya Christiansen. |
Wang |
| February 2 | Stefania Spatrizi (University of Texas, Austin, USA) [Cancelled, to be rescheduled]
TITLE: Derivation of the 1-D Groma-Balogh equations from the Peierls-Nabarro model ABSTRACT : We consider a semi-linear integro-differential equation in dimension one associated to the half Laplacian whose solution represents the atom dislocation in a crystal. The equation comprises the evolutive version of the classical Peierls-Nabarro model. We show that for a large number of dislocations, the solution, properly rescaled, converges to the solution of a fully nonlinear integro-differential equation which is a model for the macroscopic crystal plasticity with density of dislocations. This leads to the formal derivation of the 1-D Groma-Balogh equations a popular model describing the evolution of the density of positive and negative oriented parallel straight dislocation lines. This is a joint work with Tharathep Sangsawang. |
Gao |
| February 9 |
Dorin Bucur (Universite de Savoie, France) Virtual TITLE: On polygonal nonlocal isoperimetric inequalities: Hardy, Riesz, Faber-Krahn ABSTRACT: The starting point is the Faber-Krahn inequality on the first eigenvalue of the Dirichlet Laplacian. Many refinements were obtained in the last years, mainly due to the use of recent techniques based on the analysis of vectorial free boundary problems. It turns out that the polygonal version of this inequality, very easy to state, is extremely hard to prove and remains open since 1947, when it was conjectured by Polya. I will connect this question to somehow easier problems, like polygonal versions of Hardy and Riesz inequalities and I will discuss the local minimality of regular polygons and the possibility to prove the conjecture by a mixed approach. This talk is based on joint works with Beniamin Bogosel and Ilaria Fragala. |
Indrei |
| February 16 | Aaron Yip (Purdue University, USA)
TITLE: Asymtotics of Anisotropic Stokeslets in Nematic Flows ABSTRACT : Stokeslets are solutions of the Stokes flow with a singular source term. They can be used as building blocks to construct solutions with distributed force and also to analyze the interactions between moving particles in a fluid. In this talk, we investigate such an object in a liquid crystal environment which necessarily produces anisotropic effects. We start from the Beris-Edwards model for nematic liquid crystal flows and consider the asymptotic regime (strong elastic effect and small particle) so that the far-field behavior of the Stokeslets can be clearly revealed. This is joint work in progress with Dmitry Golovaty. | Wang |
| February 23 | Tongseok Lim (Krannert Management School, Purdue University, USA)
TITLE: Hodge allocation for cooperative rewards: a generalization of Shapley’s cooperative value allocation theory via Hodge theory on graphs ABSTRACT: Lloyd Shapley’s cooperative value allocation theory is a central concept in game theory that is widely used in various fields to allocate resources, assess individual contributions, and determine fairness. The Shapley value formula and his four axioms that characterize it form the foundation of the theory. Shapley value can be assigned only when all cooperative game players are assumed to eventually form the grand coalition. The purpose of this talk is to extend Shapley’s theory to cover value allocation at every partial coalition state. To achieve this, we first extend Shapley axioms into a new set of five axioms that can characterize value allocation at every partial coalition state, where the allocation at the grand coalition coincides with the Shapley value. Second, we present a stochastic path integral formula, where each path now represents a general coalition process. This can be viewed as an extension of the Shapley formula. This generalization is made possible by taking into account Hodge calculus, stochastic processes, and path integration of edge flows on graphs. We recognize that such generalization is not limited to the coalition game graph. As a result, we define Hodge allocation, a general allocation scheme that can be applied to any cooperative multigraph and yield allocation values at any cooperative stage. |
Wang |
| March 2 | Giorgio Poggesi (University of West Australia, Australia) Virtual TITLE: ANALYSIS MEETS GEOMETRY: ON SOME GEOMETRIC PROPERTIES IN PDEs ABSTRACT : We consider Alexandrov–type, Serrin–type, and Gidas-Ni-Nirenberg–type symmetry results. We provide approaches to symmetry leading to several benefits and generalizations. The results presented are based on the works listed below. |
Indrei |
| March 9 | Qinfeng Li (Hunan University, P. R. China) Virtual
TITLE: Some variational problems related to thermal background ABSTRACT: In this talk, I will present classical and more recent results on two types of geometric variational problems related to thermal background. Particularly I will present recent work, joint with Yong Huang, Qiuqi Li, Hang Yang, and Ruofei Yao. Some unsolved questions will also be introduced. |
Wang |
| March 23 | Joshua Kortum (Universitat of Winrzburg, Germany) Virtual TITLE: Global Existence and Uniqueness Results for Nematic Liquid Crystal and Magnetoviscoelastic Flows ABSTRACT : We investigate the existence and weak stability of global weak solutions for certain complex fluid models. These comprise a suitably adapted version of the Navier-Stokes equations and a harmonic mapheat flow-like equation. The Ericksen-Leslie model for liquid crystals is the simpliest non-trivial system of such a type. A second, more general example represents the flow of ferromagnetic thin films and involves a variant of the Landau-Lifshitz-Gilbert equation. The construction of weak solutions for the Ericksen-Leslie system relies on the Ginzburg-Landau approximation where the main problem consists of the limit passage in the Navier-Stokes equation. This issue is handled by invoking partial regularity techniques and the method of concentration-cancellation originally introduced for the incompressible Euler equations. For the ferromagnetic system, the relations to the harmonic map heat flow allows to use adapted versions of the technique introduced byStruwe to prove the unique existence of a global-in-time solution which is regular away from finitely many times. |
Wang |
| March 30 | Isaac Harris (Purdue University, USA) TITLE: Asymptotic Analysis Applied to Small Volume Inverse Shape Problems 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. |
Wang |
| April 6 | Franz Gmeineder (Universitat Konstanz, Germany) Virtual TITLE:Quasiconvexity, $(p,q)$-growth and partial regularity ABSTRACT : We display new results on the regularity properties of relaxed minimizers of quasiconvex functionals of $(p,q)$-growth. These results apply to the natural range for which the functionals can be meaningfully extended (or relaxed) and apply to signed integrands as well. This extends previously known exponent ranges of Schmidt in a basically optimal way. Moreover, despite being natural in view of coercivity, signed, quasiconvex allow for different phenomena that are invisible in the convex situation. Specifically, some focus will be put on the non-availability of measure representations a la Fonseca & Maly for the relaxed functionals and, more importantly, why they are not really required for partial regularity. Based on joint work with Jan Kristensen. |
Torres |
| April 13 | Luca Capogna (Smith College, USA) (SPEICIAL TIME and LOCATION: 3:30pm, BRNG B222) TITLE: Parallel Parking, Unicycles, and Visual Illusions: PDE and Geometry in Sub-Riemannian manifolds and beyond ABSTRACT : In this talk we will describe some mathematical models for various phenomena (parallel parking, motion of robot arms, unicycles, visual illusions, ...) that can be expressed in terms of certain non-Euclidean geometries structures called sub-Riemannian geometries. In each model there are natural ordinary differential equations (ODE) and partial differential equations (PDE) that arise in connection with optimization problems. We will describe how the properties of solutions of these equations are affected by the geometry of the ambient space. I will focus on problems I have personally worked on, since my time as a graduate student at Purdue, and hint at possible future directions. The talk is meant to be accessible to advanced undergraduate and to graduate students. |
Yip and Wang |
| April 19 (Probability Seminar/PDE talk) | Jessica Lin (McGill University), LILY G401 (1:30-2:30pm)
TITLE: Quantitative Homogenization of the Invariant Measure for Nondivergence Form Elliptic Equations ABSTRACT : In this talk, I will first give an overview of stochastic homogenization for nondivergence form elliptic equations (from the PDE perspective) and quenched invariance principles for nonreversible diffusion processes (from the probability perspective). I will then present new quantitative homogenization results for the parabolic Green Function (fundamental solution) and for the unique ergodic invariant measure. This invariant measure is a solution of the adjoint equation in doubly divergence form, satisfying certain integrability conditions. I will discuss the implications of these homogenization results, such as heat kernel bounds on the parabolic Green function and quantitative ergodicity for the environmental process. This talk is based on joint work with Scott Armstrong (NYU) and Benjamin Fehrman (Oxford). |
Janjigian |
| April 20 | Tim Laux (Universitat Bonn, Germany) Virtual TITLE: Local minimizers of the interface length functional based on a concept of local paired calibrations ABSTRACT : Interfacial energy functionals are ubiquitous in nature. However, some of the most basic questions are still open. In this talk, I will address one of these questions and characterize local minimizers of the interface energy. We'll establish that regular flat partitions are locally minimizing the interface energy with respect to L^1 perturbations of the phases. Regular flat partitions are partitions of open sets in the plane whose network of interfaces consists of finitely many straight segments with a singular set made up of finitely many triple junctions at which the Herring angle condition is satisfied. The proof relies on a localized version of the paired calibration method which was introduced by Lawlor and Morgan (Pac. J. Appl. Math., 166(1), 1994) in conjunction with a relative energy functional that precisely captures the suboptimality of classical calibration estimates. Vice versa, we show that any stationary point of the length functional (in a sense of metric spaces) must be a regular flat partition. This is joint work with J. Fischer, S. Hensel, and T. Simon. |
Yip |
| April 27 | Wojciech Ozanski (Florida State University, USA) TITLE: ABSTRACT : |
Novack |
| May 4 | Peter Morfe (MPI, Leipzig, Germany) TITLE: ABSTRACT : |
Yip |