Abstract: From P.A. Smith theory to the Connor conjecture to the present. Around 1940, P. A. Smith proved the remarkable result that if a finite $p$-group $G$ acts on a compact space $X$ that has the mod $p$ homology of a sphere, then the fixed point space $X^G$ also has the mod $p$ homology of a sphere.  Around 1960, Pierre Conner conjectured that if a compact Lie group $G$ acts on a space $X$, then under certain finiteness conditions the vanishing of the cohomology of $X$ implies the vanishing of the cohomology of the orbit space $X/G$. Equivariant algebraic topology has developed in fits and starts ever since. It has recently become one of the very most central areas of that subject. I'll give some glimpses of what equivariant cohomology is and how it applies to prove Smith theory and the Conner conjecture. I'll say just a little about current directions and questions.

Introduction of the speaker:

Professor J. Peter May is a professor at University of Chicago. He is a world famous algebraic topologist, and a member of the inaugural class of Fellows of the American Mathematical Society (AMS). He has made foundational and groundbreaking contributions to many subjects in algebraic topology and homotopy theory, such as the computations of stable homotopy groups of spheres, the May spectral sequences, theory of generalized cohomology operations, infinite loop space theory, equivariant homotopy theory, highly structured ring spectra theory, theory of operads, parametrized homotopy theory, theory of model categories, etc. Professor May has published more than 100 papers in leading mathematical journals, such as the Annals of Mathematics. He has also written many books in homotopy theory, which include a very influential undergraduate textbook "A Concise Course in Algebraic Topology".

Professor May had served for many years as the chair of the Department of Mathematics at University of Chicago and the graduate admission and study committees. Professor May values very highly of education in mathematics. Since the year 2000, he has been organizing every year the undergraduate summer research program at University of Chicago, also known as the REU; Until the year 2019, he has advised more than 60 doctoral students, many of which have held tenured positions in prestige universities all over the world. Professor May has made significant contributions to algebraic topology in China, especially in educating young Chinese algebraic topologists. Many students who got their bachelor's degrees from Peking University have got or are pursuing their Ph.D. degrees under the guidance of Professor May, such as Zhouli Xu, Foling Zou, Jia Kong and Weinan Lin, or have been greatly influenced by Professor May, such as Yifei Zhu, Guozhen Wang, Guchuan Li and Yunze Lu.