Colloquium
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Posted January 15, 2017

Last modified January 17, 2017

Vu Hoang, Rice University, Department of Mathematics

Singularity formation for equations of fluid dynamics

The basic equations of fluid mechanics were written down about 200 years

ago by Euler. To this day, they present a challenge for mathematical analysis and

many basic questions are still unsolved. One of these basic concerns the issue

of finite-time singularity formation versus global regularity. A great obstacle

for mathematical analysis is the fact that these equations involve both nonlinear

and non-local interactions. In my talk, I will describe recent efforts to understand the

mechanisms that are behind the singularity formation in fluid equations, starting from simple

model equations.

Applied Analysis Seminar
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Posted December 2, 2016

Last modified January 15, 2017

Malcolm Brown, Department of Computer Science & Informatics, Cardiff University

Scattering and inverse scattering for a left-definite Sturm-Liouville problem

This talk reports on recent work which develops a scattering and an inverse scattering theory for the Sturm-Liouville equation u'' + qu = λ w u, where w may change sign but q is positive. Thus the left-hand side of the equation gives rise to a positive quadratic form, and one is led to a left-definite spectral problem. The crucial ingredient of the approach is a generalised transform built on the Jost solutions of the problem and hence termed the "Jost transform" and the associated Paley-Wiener theorem linking growth properties of transforms with support properties of functions. One motivation for this investigation comes from the Camassa-Holm equation for which the solution of the Cauchy problem can be achieved by the inverse scattering transform for u'' + qu = λ w u.

This is joint work with Christer Bennewitz (Lund, Sweden) and Rudi Weikard (Birmingham, AL).

Colloquium
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Posted January 16, 2017

Last modified January 20, 2017

Aynur Bulut, Princeton University, Department of Mathematics

Recent developments on deterministic and probabilistic well-posedness for nonlinear Schrödinger and wave equations.

Abstract: Dispersive equations such as nonlinear Schrödinger and wave equations arise as mathematical models in a variety of physical settings, including models of plasma physics, the propagation of laser beams,
water waves, and the study of many-body quantum mechanics. They also serve as model equations for studying fundamental issues in many aspects of nonlinear partial differential equations. Key questions in
the analysis of these equations include issues of well-posedness (for instance, existence of solutions, uniqueness of these solutions, and their continuous dependence on initial data in appropriate topologies) locally in time, long-time existence and behavior of solutions, and, conversely, the possible existence of solutions which blow-up in finite time.

In this talk, we will give an overview of several recent results concerning the local and global (long-time) theory, including some results where probabilistic tools are used to obtain estimates for randomly chosen initial data which are not available in deterministic settings. A recurring theme (and oftentimes obstacle) is the notion of supercriticality arising from the natural scaling of the equation - seeking to characterize long-time behavior of solutions when the relevant scale-invariant norms are not controlled by the conserved energy, or for initial data of very low regularity. The techniques involved include input from several areas of mathematics, including ideas arising in many areas of PDE, harmonic analysis, and probability.

Colloquium
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Posted January 17, 2017

Last modified January 19, 2017

Davit Harutyunyan, EPFL

From buckling to rigidity of shells: Recent mathematical progress

Abstract:

It is known that the rigidity of a shell (for instance under compression) is closely related to the optimal Korn's constant in the nonlinear Korn's first inequality (geometric rigidity estimate) for H^1 fields under the appropriate conditions (with no or with Dirichlet type boundary conditions arising from the nature of the compression). In their celebrated work, Frisecke, James and Mueller (2002, 2006) derived an asymptotically sharp nonlinear geometric rigidity estimate for plates, which gave rise to a derivation of a hierarchy of nonlinear plate theories for different scaling regimes of the elastic energy depending on the thickness h of the plate (the optimal constant scales like h^2). Frisecke-James-Mueller type theories have been derived by Gamma-convergence and rely on LpLp compactness arguments and of course the underlying nonlinear Korn's inequality. While plate deformations have been understood almost completely, the rigidity, in particular the buckling of shells is less well understood. This is first of all due to the luck of sharp rigidity estimates for shells. In our recent work we derive linear sharp geometric estimates for shells by classifying them according to the Gaussian curvature. It turns out, that for zero Gaussian curvature (when one principal curvature is zero, the other one never vanishes) the amount of rigidity is h^{3/2}, for negative curvature it is h^{4/3} and for positive curvature it is h. These are new classical results in both shell buckling and nonlinear shell theories. All three estimates have completely new optimal constant scaling for any sharp geometric rigidity estimates to have appeared. This is partially joint work with Yury Grabovsky (Temple University)

Colloquium
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Posted January 16, 2017

Last modified January 18, 2017

Marcel Bischoff, Vanderbilt

Subfactors, Fusion Categories and Conformal Nets

Abstract: Von Neumann algebras were mainly introduced to understand quantum theory and group representations. A factor is a von Neumann algebra with trivial center and an inclusion of two factors is called a subfactor. Finite index subfactors, in some sense, describe quantum symmetries which generalize finite groups. Similarly, fusion categories generalize the representation categories of finite groups. One can use von Neumann algebras to study chiral conformal field theory via so-called conformal nets. It turns out that conformal nets are a natural source of subfactors and fusion categories. It is an exciting open question if all fusion categories and subfactors come from conformal nets. I will introduce these three concepts and their interaction and discuss some recent results on the structure of inclusions of conformal nets and their representation theory.