Harvard Center of Mathematical Sciences and Applications — Math Science Lectures in Honor of Raoul Bott

Non-abelian Hodge theory, developed by Corlette, Simpson and others, equips representation spaces of the fundamental group of a complex variety with structures analogous to those on its cohomology. Taking that analogy further suggests that every cohomological structure should admit a counterpart in the non-abelian setting. Daniel Litt presents work toward this programme, related conjectures, and new structures whose analogues in the abelian setting remain less clear.

When can an algebraic differential equation be solved by algebraic functions? Daniel Litt discusses a question dating to Fuchs in 1875 and the number-theoretic perspective supplied by conjectures of Grothendieck and Katz for linear equations, and of Ekedahl, Shepherd-Barron, Taylor and Bost more generally. Joint work with Josh Lam provides progress for isomonodromy differential equations, a class that includes Painlevé VI. These results also motivate the study of motivic structures in the space of algebraic differential equations.