Clemson Analysis and PDE Seminar

Abstract: We establish local well-posedness for the Einstein-Euler system with a physical vacuum boundary in spherical symmetry. Our proof relies on a new way of thinking about Einstein’s equations in spherical symmetry that is well-adapted to the fluid’s characteristics on the free boundary. We also exploit the Einstein constraint equations in spherical symmetry in a new way, as a tool to understand the evolution problem. This is joint work with Jared Speck.

Abstract: It is known in physics that steady state of fluids under the influence of uniform gravity is stable if and only if the convection is absent. In the context of incompressible fluids, convection happens when heavier fluids is on top of lighter fluids, known as Rayleigh-Taylor instability. However, in real world, heat transfer plays an important role in convection of fluids, such as the weather changes, and or cooking a meal. In this context, the compressibility of the fluids becomes important. Indeed, using the more realistic model of compressible flow with heat transfer, the behavior of solutions is much closer to the real world and more complicated. We will discuss these topics in this lecture, including some on-going research projects. The lecture will be accessible to audiences with basic knowledge on multivariable calculus, and little of differential equations.

Abstract: In the context of the classical central limit theorem, let Y_n denote the n-th normalized sums of i.i.d copies of a random variable X with mean 0 and variance 1. Following Shannon's work in information theory, Lieb conjectured in 1978 that the differential entropy of Y_n increases monotonically in n. This conjecture was finally settled by Artstein, Ball, Barthe and Naor (ABBN) in 2004. In fact, the ABBN article proved more general results and tied the so-called entropy power inequalities into this framework. These inequalities are extremely useful in proving several coding theorems in information theory. 

On the non-commutative side of the story, Cushen and Hudson, in 1971, proved a quantum probability analog of the classical central limit theorem. The monotonicity of von Neumann entropy under the Cushen-Hudson central limit theorem remains an open problem in this area. Guha in 2008 showed that certain quantum analogs of entropy power inequalities, if proved, will produce several coding theorems in quantum information theory, but these problems also remain open to this day. In this talk, we discuss an overview of this area of research and show a new result on the monotonicity of the entropy of the distribution of observables.

Abstract: Surprising asymmetric transport phenomena along interfaces separating insulating bulks have been observed in many areas of applied sciences, e.g., electronics, photonics, and geophysics. Such transport is quite unusual in that it displays extremely strong robustness to perturbations. In fact, this phenomenon affords a topological origin: systems in the same topological class, i.e., in the same topological phase of matter, display the same robust, quantized, edge/interface transport. Perturbations may then be interpreted as continuous deformations that preserve the topological class.

This talk considers such systems modeled by general elliptic partial differential operators on the Euclidean plane. We review a recent simple topological classification, which provides an explicit, reasonably straightforward, computation of a topological invariant, technically the index of a Fredholm operator obtained by means of confining domain walls. We next introduce a physical observable that allows us to quantify the asymmetry of the edge transport. The evaluation of such an observable is challenging in practice. We then present a bulk-edge correspondence, a pillar of topological phases of matter in the physical literature, which shows that the interface current observable is in fact equal to the aforementioned simple topological invariant.

The theoretical findings are illustrated with examples ranging from electronics applications to geophysical fluid flows.

Abstract: The existence of global-in-time solutions to the 2 1/2 - dimensional Hall-magnetohydrodynamics system for any smooth initial data remains an open problem. This talk presents several global regularity results along with the result that requires 11+epsilon for any epsilon>0 while the analogous sum for the open problem is 12, concerning the terms -\Delta u and -\Delta b. This is the joint work with Prof. Kazuo Yamazaki.

Abstract: The aim of this talk is to discuss a family of extremal problems that arise from estimating the operator norm of certain embeddings between weighted Paley-Wiener spaces. In general, we study the asymptotic behavior of this norm and, in some particular cases, we can determine the sharp constants through the theory of reproducing kernel Hilbert spaces. Also, we will present some connections with other extremal problems and a direct connection with Poincaré inequalities.