# Seminars & Events for Algebraic Geometry Seminar

##### Categorified Duality in Boij-Soederberg Theory

The central idea in Boij-Soederberg Theory is that there is a connection between sheaf cohomology on projective space and free resolutions over the polynomial ring. I'll describe the construction of a duality pairing that provides a new foundation for this theory, and that greatly extends the reach of the theory. This is joint work with David Eisenbud.

##### Autoduality of Jacobians for singular curves

Let $C$ be a (smooth projective algebraic) curve. It is well known that the Jacobian $J$ of $C$ is a principally polarized abelian variety. In other words, $J$ is self-dual in the sense that $J$ is identified with the space of topologically trivial line bundles on itself.

Suppose now that $C$ is singular. The Jacobian $J$ of $C$ parametrizes topologically trivial line bundles on $C$; it is an algebraic group that is no longer compact. By considering torsion-free sheaves instead of line bundles, one obtains a natural singular compactification $J'$ of $J$.

##### Perverse coherent sheaves on the nilpotent cone in positive characteristic

In the context of geometric Langlands duality, it is a general principle that the "topological" aspects (e.g., intersection cohomology, perverse sheaves) of a given group G should correspond to "algebraic" aspects (e.g., representations, coherent sheaves) of its dual group G'. An archetypal instance of this idea is the "geometric Satake isomorphism" of Ginzburg and Mirkovic-Vilonen, but by now there are many results asserting an equivalence (often derived) between a topological category associated to G and an algebraic one associated to G'. In this talk, I will try to explain a few examples of this phenomenon, which can give rise to surprising kinds of objects: coherent sheaves (in characteristic 0) that behave as though they were perverse, or vice versa.

##### Comparison theorems in p-adic Hodge theory

A basic theorem in Hodge theory is the isomorphism between de Rham and Betti cohomology for complex manifolds; this follows directly from the Poincare lemma. The p-adic analogue of this comparison lies deeper, and was the subject of a series of extremely influential conjectures made by Fontaine in the early 80s (which have since been established by various mathematicians). In my talk, I will first discuss the geometric motivation behind Fontaine's conjectures, and then explain a simple new proof based on general principles in derived algebraic geometry --- specifically, derived de Rham cohomology --- and some classical geometry with curve fibrations. This work builds on ideas of Beilinson who proved the de Rham comparison conjecture this way.