The seminar usually takes place in Astro-Math 202 at 4-5PM every Tuesday, unless noted otherwise.
Organizers: Yng-Ing Lee 李瑩英 (NTU), Ulrich Menne 孟悟理 (NTNU), Chung-Jun Tsai 蔡忠潤 (NTU), Mao-Pei Tsui 崔茂培 (NTU)
Hung-Yu Yeh 葉弘裕 (Academia Sinica): Stabilities on normal projective surfaces over arbitrary fields
Stability, first introduced by Mumford in the 1960's, is used as a tool to construct moduli space of sheaves on algebraic varieties. Motivated by homological mirror symmetry conjecture, Bridgeland introduces stability conditions on triangulated categories which depend on the existence of Harder-Narasimhan (HN) filtration and central charges on the relevant K group of associated triangulated categories. The existence of Bridgeland stability is still an open problem in general varieties even in normal projective surfaces. Therefore, I introduce a notion of stability filtration in arbitrary categories which is equivalent to the existence of HN sequence on objects. Indeed, it is equivalent to existences of a zero morphism, a partial order on objects, and a collection of some universal sequences. In this talk, I would focus on normal projective surface over a field, and present the result of the existence of stability filtration (of degree 1) which is equivalent to Bridgeland stability on normal projective surfaces. As an application this result implies new effective restriction theorems of semistable sheaves on normal surfaces over any field. Moreover, over an algebraically closed field we can recover Hodge Index theorem and Bogomolov Inequality.
Mario Santilli (University of Augsburg): Heintze-Karcher-Ros inequalities and the soap bubble theorems in geometric measure and convex geometry
One of the most celebrated result in differential geometry asserts that a compact, embedded, smooth hypersursurface in $\mathbb{R}^n+1$ is a sphere, provided it has constant scalar mean curvature. This is the famous Alexandrov soap bubble theorem. It was later discovered by Ros that this result (as well as its generalization to constant higher-order mean curvatures) is a special case of an isoperimetric-type result, which asserts that if $\Omega$ is a bounded open set with smooth boundary and $\partial\Omega$ has non-negative scalar mean curvature $h$ (i.e. $\Omega$ is a mean convex set), then
$$ \mathcal{L}^{n+1}(\Omega) \leq \frac{n}{n+1} \int_{\partial\Omega} \frac{1}{h} d\mathcal{H}^n ~, $$
with equality if and only if Ω is a sphere.
In this talk I will present several generalizations of this isoperimetric result to classes of mean convex sets arising from different areas of geometric analysis, where the smoothness assumption is no longer guaranteed. They include all convex bodies, the mean convex sets introduced by White in the regularity theory of the mean curvature flow, and the sets of finite perimeter whose reduced boundary is a varifold with bounded anisotropic distributional mean curvature. In particular, we partially answer a conjecture of Maggi, generalizing the Alexandrov soap bubble theorem to stationary points of the anisotropic isoperimetric problem for smooth elliptic integrands.
Part of the results are a joint work with S lawomir Kolasinski (Warsaw) and Antonio De Rosa (Courant Institute).
