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The stable reduction theorem was proved by Grothendieck for Abelian varieties and subsequently by Deligne and Mumford for projective curves.
A difference field is a field with a distinguished automorphism. Automorphisms of a finite field are powers of the Frobenius map. In this talk, I will discuss how to estimate the number of rational po...
In this talk, I will discuss arithmetic purity of strong approximation : motivations and related results. Finally, for complete toric varieties, I will briefly explain my work on this topic.
Shimura varieties are a class of algebraic geometric spaces that play a very important role in arithmetic geometry, especially in the Langlands program. In recent years, methods from the geometric rep...
We expect that the study of pluricanonical systems of varieties of general type with many global forms can be reduced to the study of pluricanonical systems of varieties of lower dimensions. We show t...
The mod p points of a Shimura variety have a conjectural description called the Langlands-Rapoport conjecture. In accordance with the conjecture, Rapoport defined (generalized) affine Deligne-Lusztig ...
In this talk I will introduce special and generalized cycles on Shimura varieties and discuss how to use them to construct geometric and arithmetic theta series. Then I will briefly discuss the connec...
We study Mumford-Tate groups of families of Calabi-Yau manifolds in this notes, for example, we have interest in the relations between the geometry of families of Calabi-Yau manifolds and their Mumfor...
The research on Fano variety is always a central topic in mathematics. In the recent years, there has been vast progress in the the problems on boundedness,stability and moduli space.The primary goal ...
Let F be the complete °ag variety over SpecZ with the tautological ˉltration 0 ½ E1 ½ E2 ½ ¢ ¢ ¢ ½ En = E of the trivial bundle E of rank n over F. The trivial hermitian metric o...
We use classicalSchubert calculus to evaluate the integral formula of Kaiser and Kohler [KK] for the Faltings height of certain homogeneous varieties in terms of combinatorial data, and verify thei...
These notes are based on lectures given at Zhejiang University, July 14–20, 2008. My goal was to present some analytic techniques that can be used to study the action of the mapping class group on the...
We determine the action of the Torelli group on the equivariant cohomology of the space of flat SL(2, C) connections on a closed Riemann surface. We show that the trivial part of the action contains t...
Let G be a semisimple complex algebraic group and P a parabolic subgroup of G. The homogeneous space X = G=P is a projective complex manifold. My aim in this lecture is to survey what is known about...
In this paper we use the Morse theory of the Yang-Mills-Higgs functional on the singular space of Higgs bundles on Riemann surfaces to compute the equivariant cohomology of the space of semistable U(2...

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