Speaker: Dr. Xiaolong Li (Wichita State University) 

 

Title: Geometry and Analysis of Einstein Four-manifolds 

 

 

Abstract: Einstein manifolds are fundamental objects in differential geometry, with deep connections to general relativity and mathematical physics. Their natural generalizations—quasi-Einstein manifolds or Ricci solitons—play a crucial role in the study of geometric flows, particularly Ricci flow. While Einstein metrics in low dimensions (two and three) are well understood, their classification in dimensions higher than four appears intractable. However, dimension four occupies a unique and intriguing position: it is rich enough to exhibit intricate geometric and topological phenomena, yet still accessible to classification efforts under suitable conditions. In this talk, I will survey some key developments in the study of Einstein metrics on four-manifolds, highlighting known examples and classification results. I will also mention some outstanding open problems and promising directions for future research.

 

The talk is based on a joint survey with Xiaodong Cao (Cornell) and Hung Tran (Texas Tech).