*10th Meeting, September 29, 2018, Tsinghua*

Ahmed Abbes (CNRS & IHÉS)

*The p-adic Simpson correspondence*

The p-adic Simpson correspondence, initiated by Gerd Faltings in 2005, aims at describing all p-adic representations of the fundamental group of a proper smooth variety over a p-adic field in terms of linear algebra - namely Higgs bundles. My lecture will be an introduction to this topic. I will focus on the approach that I developed with Michel Gros relying on a new family of period rings built from the torsor of deformations of the variety over a universal p-adic thickening defined by J.-M. Fontaine.

Daxin Xu (Caltech)

*Generalized Kloosterman sheaves and their p-adic variants*

I will first review the relationship between the classical Bessel equation and the classical Kloosterman sum. Such a relation can be regarded as an instance of the geometric Langlands correspondence for GL2. I will then explain the recent generalizations of this story for arbitrary reductive groups, based on the works by Frenkel-Gross, Heinloth-Ngo-Yun, and X. Zhu. In the end, I will discuss some joint work in progress with X. Zhu, where we study the p-adic aspect of this theory.

*11th Meeting, October 17, 2018, CAS*

Constantin Shramov (Steklov Institute)

*Finite groups of birational selfmaps of surfaces*

I will speak about finite groups acting by birational automorphisms of surfaces. In particular, we will consider surfaces over function fields. One of our important observations here is that for a smooth geometrically rational surface S, either it is birational to a product of a projective line and a conic (so that S is rational provided that it has a point), or finite subgroups of its birational automorphism group are bounded. I will also discuss some particular types of surfaces with interesting automorphism groups, including Severi-Brauer surfaces.

Jian Xiao (YMSC, Tsinghua)

*Local positivity for curves*

One of the most important invariants measuring the local positivity of a nef line bundle is the Seshadri constant. We first give a brief introduction to this invariant. Then using the duality of positive cones, we show that applying the polar transform to local positivity invariants for divisors gives interesting and new local positivity invariants for curves. These new invariants, studied also independently by M. Fulger, have nice properties similar to those for divisors. In particular, this enables us to obtain a Seshadri type ampleness criterion for movable curves, and give a characterization of the divisorial components of the non-ample locus of a big class. (Joint work with N. McCleerey.)

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