Algebraic Geometry Seminar

In this work, we study étale cohomology of p-adic rigid-analytic spaces with F_p coefficients. For general (smooth) spaces, this cohomology theory does not behave so well. For example, F_p-cohomology groups of the 1-dimensional closed unit ball are infinite. Nevertheless, Scholze showed that H^i(X, F_p) are finite-dimensional for proper X. He further conjectured that these cohomology groups should satisfy Poincaré Duality when X is both smooth and proper. I will explain the proof of this conjecture using the concept of almost coherent sheaves that provides tools to "localize" the question in an appropriate sense and eventually reduce it to computations in group cohomology.

Log Gromov-Witten invariants, introduced by Abramovich-Chen-Gross-Siebert, are counts of curves in pairs (X,D) consisting of a smooth projective variety X together with a normal crossing divisor D, with prescribed tangency conditions along D. These invariants play a key role in mirror symmetry for log Calabi-Yau pairs (X,D), in which case D is an anticanonical divisor. After briefly reviewing log Gromov-Witten theory, I will explain a combinatorial recipe based on tropical geometry and wall-crossing algorithms to calculate such curve counts when (X,D) is obtained as a blow-up of a toric variety along hypersurfaces in the toric boundary divisor. This is based on joint work with Mark Gross.

In this talk, I will discuss my joint work with Sam Raskin on the derived geometric Satake equivalence. As time permits, I will explain how this work has been applied in the proof of the geometric Langlands conjecture and in my work with Hayash. The main point is that when combined with local-to-global methods, my results with Raskin give information about deformations of reducible local systems on a compact Riemann surface.