应伟德体育伟德体育官方网站邀请，比利时安特卫普大学(University of Antwerp)数学系Edward Bryden博士将于2025年11月3日—2025年11月7日、2025年11月12日—2025年11月14日和2025年11月17日—11月19日讲授微分几何领域的短期在线课程，欢迎广大师生参加。

时 间：11/3-11/7、11/12-11/14，11/17-11/19，16:30-18:00

Zoom会议：387 384 3920

会议密码：d8g6tm

课程网页：https://geometry-topology.github.io/course-convergence.html

课程介绍：This two-week online mini-course focuses on topics in differential geometry.It is designed to be self-contained and particularly suitable for young researchers and graduate students in related fields. Senior graduate students with interests in geometry and topology are also very welcome to participate.The following topics will be covered: Sobolve and Neumann constants, Gromov-Hausdorff convergence, convergence in the Dong-Song sense, stable systoles, harmonic representation of integral forms, scalar curvature and related theorems, torus rigidity theorem, Stern’s inequality, positive mass theorem rigidity using BKKS mass inequality, Dong-Song’s stability of positive mass theorem, stability of three-tori with small negative scalar curvature.

The main topic of this course is stability related to scalar curvature and positive mass theorem. Stability is also sometimes called almost rigidity results, arising from the study of geometric inequalities.

There are two ideas behind geometric inequalities.First, one wants to relate local analytic information such as curvature to geometric information such as volume, or area, or the isoperimetric inequality relating the two. Typically when one does this, there is a model space for which this relationship is particularly clear. When studying curvature, these models are usually the spaces of constant sectional curvature: the sphere, the plane, and the hyperbolic plane. Consider one such example: volume growth bounds arising from lower bounds on the Ricci curvature. This result says that if a space has Ricci curvature bounded below by a constant, then its volume growth is bounded by the volume growth present in the model space. This is the geometric inequality. The second idea is that the model space should be unique in the following sense: it should be the only space which saturates the geometric inequality. This called rigidity.

Although scalar curvature doesn't carry much information, it still has geometric consequences. There is a physical motivation coming from general relativity for this surprising fact: scalar curvature behaves a little bit like energy density. The clearest expression of this is the Positive Mass Theorem, which says that if the scalar curvature of a space is non-negative, then if we add up all of the mass present, the result will also be non-negative. This is the geometric inequality. The rigidity says that if the total mass is zero, and the scalar curvature is non-negative, then the space must be empty, and so isometric to the plane. The connection between scalar curvature and energy-density make these results very believable, but they are hard to be rigorously proved.

When we ask if the Positive Mass Theorem is stable, we are asking the following natural question. If the scalar curvature is non-negative, and the mass is small, must the space be nearly empty in the sense that it is close to being flat? These questions are so interesting because they are so natural to ask, but have many subtleties to them. For example, the first question one might ask is also perhaps the most interesting and difficult: what does it mean for a space to be close to flat? That is, what topology are we using, if any, to measure this closeness?

In this course we will study two cases of stability: that of the Positive Mass Theorem, and that of

the Torus. We will see the subtleties involved in all stages of the analysis, and hopefully it will become clear why these questions are so engaging.

Edward Bryden简介：

Edward Bryden,比利时安特卫普大学(University of Antwerp)博士后。他在美国石溪大学(Stony Brook University)数学系Marcus Khuri教授指导下取得博士学位，博士论文研究stability of the positive mass theorem in the presence of axisymmetry。Bryden博士现在的研究兴趣为微分几何，几何分析与广义相对论。