Abstract: 

We will introduce the general twistor space picture for moduli spaces of connections deforming to moduli spaces of Higgs bundles, initiated by Hitchin for his hyperkähler structure. The Deligne-Hitchin twistor space construction has a property analogous to weight 1 Hodge structures.

This may be extended to twistor structures of other weights. In the case of moduli theory of connections on quasiprojective varieties, this theory is still under development but it already provides a viewpoint on how parabolic structures enter into the nonabelian cohomology setup.

We'll then discuss more about parabolic logarithmic Higgs bundles, and in particular the problem of computation of higher direct images of these, in joint work with Donagi and Pantev.

This will be applied to the computation of Hecke operators in the Donagi-Pantev construction of spectral data for Hecke eigensheaves in the geometric Langlands correspondence, in our recent preprint arXiv:2403.17045. We will describe in detail the spectral data for geometric Langlands in the case of rank 2 bundles over genus 2 curves, using the classical theory of the quadric line complex.

The various questions and issues that would be involved in extending this to more general cases will be discussed.

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.

Notes:chern-lectures-notes.pdf

Videos: http://archive.ymsc.tsinghua.edu.cn/pacm_lecture?html=Twistor_families_of_connections_Higgs_bundle_calculations_and_applications_to_spectral_varieties_in_the_geometric_Langlands_correspondence.html