Abstract:

This will be a survey of some recent directions and trends in nonabelian Hodge theory.

Biography:

Carlos Simpson grew up in Eugene, Oregon, and was an undergraduate and then a graduate student at Harvard University. He wrote his thesis under the supervision of Wilfried Schmid, on the generalization of Donaldson-Uhlenbeck-Yau theory to variations of Hodge structure.

Based on Hitchin's definition of Higgs bundles and the Hitchin equations, and integrating the work of Donaldson and Corlette, Simpson began a program of developing nonabelian Hodge theory for representations of fundamental groups of algebraic varieties.

Simpson moved to France in 1991 and entered the CNRS in 1992 as a Directeur de Recherche, first in Toulouse and then in Nice from 1999 onwards.

Motivated by the idea of generalizing NAH to higher nonabelian cohomology, and following work of his first thesis student Zouhair Tamsamani, Simpson went on to work on higher category theory. This led to the theory of descent for n-stacks in work with André Hirschowitz. Subsequent discussions with Hirschowitz led to the topic of computer formalization of mathematical proofs using the Coq system in the early 2000's.

Recent topics include work with Ron Donagi and Tony Pantev on higher direct images of Higgs bundles and on the Donagi-Pantev program for understanding the geometric Langlands correspondence. Simpson has also been working on the use of AI/machine-learning techniques for guiding mathematical proofs with particular attention to the subject of classification proofs for combinatorial structures. 

Video：http://archive.ymsc.tsinghua.edu.cn/pacm_lecture?html=Current_topics_in_nonabelian_Hodge_theory.html