The Group Meeting will be held virtually unless the Covid 19 lockdown is over in which case the location will be L2.
The Group Meeting will be held virtually unless the Covid 19 lockdown is over in which case the location will be L3.
Oxford Mathematician Andre Guerra talks about quasiconvexity and its role in the Calculus of Variations:
Just as differential equations often boundary conditions of various types, so too do quantum field theories often admit boundary theories. I will explain these notions and then discuss a theorem proved with Constantin Teleman, essentially characterizing certain 3-dimensional topological field theories which admit nonzero boundary theories. One application is to gapped systems in condensed matter physics.
The classical Minkowski problem consists in finding a convex polyhedron from data consisting of normals to their faces and their surface areas. In the smooth case, the corresponding problem for convex bodies is to find the convex body given the Gauss curvature of its boundary, as a function of the unit normal. The proof consists of three parts: existence, uniqueness and regularity.
In this talk, we study a Minkowski problem for certain measure, called p-capacitary surface area measure, associated to a compact convex set $E$ with nonempty interior and its $p-$harmonic capacitary function (solution to the p-Laplace equation in the complement of $E$). If $\mu_p$ denotes this measure, then the Minkowski problem we consider in this setting is that; for a given finite Borel positive measure $\mu$ on $\mathbb{S}^{n-1}$, find necessary and sufficient conditions for which there exists a convex body $E$ with $\mu_p =\mu$. We will discuss the existence, uniqueness, and regularity of this problem which have deep connections with the Brunn-Minkowski inequality for p-capacity and Monge-Amp{\`e}re equation.
The parabolic Allen-Cahn equations is the gradient flow of phase transition energy and can be viewed as a diffused version of mean curvature flows of hypersurfaces. It has been known by the works of Ilmanen and Tonegawa that the energy densities of the Allen-Cahn flows converges to mean curvature flows in the sense of varifold and the limit varifold is integer rectifiable. It is not known in general whether the transition layers have higher regularity of convergence yet. In this talk, I will report on a joint work with Huy Nguyen that under the low entropy condition, the convergence of transition layers can be upgraded to C^{2,\alpha} sense. This is motivated by the work of Wang-Wei and Chodosh-Mantoulidis in elliptic case that under the condition of stability, one can upgrade the regularity of convergence.
