走向现代数学学术报告 - 殷浩教授（No. 969)

报告题目：Obstructions to Yang–Mills Bubbling in Dimension Four

报 告 人：殷浩 教授（中国科学技术大学）

邀 请 人：陈优民 副教授

报告时间：

腾讯会议ID：611-719-028

报告摘要：By Uhlenbeck's compactness theorem, a sequence of Yang–Mills connections with bounded energy on a Riemannian four-manifold converges, up to gauge transformations, to a weak limit together with a bubble tree. We study the reverse question: which bubble-tree configurations can actually arise as such limits? We present two types of obstructions. The first comes from the conservation law of the Yang–Mills stress–energy tensor combined with a sharp two-ended expansion of the connection in the neck region; it already rules out the "unit dipole", i.e. a sequence of SU(2) Yang–Mills connections on S^4 splitting into a charge-one instanton and a charge-one anti-instanton. The second, joint work with Alex Waldron, comes from deformation theory: the curvature of the exceptional bubble must be orthogonal to every infinitesimal deformation of the limit connection at the attaching point. As a corollary, an SU(2) bubble tree on S^4 containing at least two non-flat connections cannot consist of a single unit-charge self-dual instanton with all remaining pieces anti-self-dual. Applications include an improved energy-gap theorem—on an SU(2) bundle of charge κ over S^4, any Yang–Mills connection with energy below 4π²(|κ|+2)+ε_κ must be an instanton—and the discreteness of the critical values of the Yang–Mills functional below 16π² on the trivial SU(2) bundle over S^4.

报告人简介：殷浩，中国科学技术大学教授，主要研究方向为偏微分方程和几何分析。于2005年在北京大学获得博士学位，导师为丁伟岳院士。之后曾在华东师范大学和澳大利亚昆士兰大学从事博士后研究，2007年起在上海交通大学工作，2012年调入中国科学技术大学。在Math. Ann.、Comment. Math. Helv.、Ann. PDE、Trans. Amer. Math. Soc.、J. Funct. Anal.、Proc.London Math.Soc.、J. London Math. Soc.、Calc. Var. PDE、Math. Z.等国际著名期刊上发表多篇高水平论文。