This is the innagural series of talks by distinguished Algebra and Number Theory aimed at a general public. It is organized by Thomas Haines.

2019 Ngô Bảo Châu

December 4, 2019

On The Hitchin Fibration
Monday, December 4 at 3:15pm
Ngo Bao Chau
Chicago

Abstract: Simpson's non-Abelian Hodge theory stipulates a diffeomorphism between the moduli space of flat connections on a smooth projective variety and the moduli space of semi-stable Higgs bundles with trivial Chern classes. The main feature of the moduli space of Higgs bundles is the Hitchin map calculating the characteristic polynomial of the Higgs field. Over a curve, the structure of the Hitchin map is fairly well understood as an abelian fibration with degeneration. When the base field is a finite field, counting points on the Hitchin fibration allows us to connect the geometry of the Hitchin fibration with orbital integrals and the trace formula. This interplay between geometry and harmonic analysis has been very fruitful for understanding both sides of the story, and in particular, it gave rise to a proof for the fundamental lemma. I will give an account of this interplay in my first lecture.

December 5, 2019

On The Hitchin Fibration II
Wednesday, December 5 at 2:00 pm
Ngô Bảo Châu
Chicago

Abstract: In the second lecture, I want to discuss the theory of non-archimedean integration on the Hitchin fibration due to Groechenig, Wyss and Ziegler. Surprisingly, calculating nonarchimedean integrals is not exactly the same as counting points and this approach gives another proof of the fundamental lemma, and this discrepancy sheds yet new lights on the theory of endoscopy. The proof is also more elementary in the sense that it does not use the theory of perverse sheaves. 

December 6, 2019

On The Hitchin Fibration III
Wednesday, December 6 at 2:00 pm
Ngô Bảo Châu
Chicago

Abstract: In my third lecture, I want to report on a completely different development on the moduli space of Higgs bundles. In joint work with T.H. Chen we started exploring the structure of the Hitchin map for the moduli space of Higgs bundles over higher-dimensional varieties, which raises interesting questions on the geometry of commuting varieties.