Speaker: 倪磊教授，加州大学圣地亚哥分校

Title: Lie Theory in Geometry: Symmetry, Holonomy, and Rigidity

Language: Chinese

Time & Venue: 2026.11.04 15:30-17:00 南楼N204 线上Zoom Meeting ID：898 1872 6621 密码：123456

Abstract: Lie theory provides a powerful bridge between differential geometry and finite-dimensional algebra. This talk explores that bridge through several classical developments and recent results, emphasizing how geometric structures give rise to Lie groups and Lie algebras, and how their algebraic properties lead to classification and rigidity. We begin with the passage from Klein’s homogeneous models to Cartan’s theory of connections, which brings the ideas of symmetry and parallelism into the study of curved spaces. We then discuss compact complex homogeneous manifolds through the work of H. C. Wang, Tits, and Borel, highlighting how integrability and the Kähler condition are reflected in the underlying Lie algebraic structure. Turning to Riemannian geometry, we examine Berger’s holonomy classification and Simons’ intrinsic approach through holonomy systems. These ideas also enter geometric evolution equations: Lie algebraic constructions yield curvature cones preserved by Ricci flow, while holonomy helps turn equality cases into rigidity theorems. Against this background, we discuss four recent developments: the vanishing of the first Ricci form under irreducible holonomy for non-Kähler Hermitian manifolds with a Hermitian connection of parallel torsion; the classification of Chern–Ambrose–Singer spaces, joint with Fangyang Zheng; a Lie algebraic approach to the Wang–Tits structure theory, joint with Nolan Wallach, featuring a canonical abelian algebra associated with an invariant integrable complex structure and applications to Bismut connections with parallel torsion and Kähler-like curvature; and a single-induction proof of Simons’ holonomy theorem. The unifying theme is how Lie theory converts differential identities and geometric constraints into algebraic structure—and then returns that structure to geometry as symmetry, splitting, or rigidity.