北京雁栖湖应用数学研究院

We will present an abstract topos-theoretic framework for building Galois-type theories in a variety of different mathematical contexts : this unifies and generalizes Grothendieck's theory of “Galois categories” and Fraïssé's construction of homogeneous structures in model theory. This theory notably allows one to construct fundamental groups in many classical contexts such as finite groups, finite graphs, motives and many more. We will in particular present an approach based on it for building “motivic toposes” and investigating the independence from l of l-adic cohomology.

We will explain how Olivia Caramello's proposed approach for constructing a Galois-type Theory of Motives can be implemented andfully verified in the case of 0-Motives. The models of this Theory - or equivalently the points of the associated Classifying Topos - are exactly cohomology functors of degree 0. As its Classifying Topos is Galois, this Theory is complete, which means that all its models share the same geometric-logic properties. In particular, their components at all different geometric objects all have the same dimensions and the same algebraic structures. These results are already non-trivial and can be considered as toy-models for cohomology in higher degrees, which is the objective of Caramello's proposed approach. This is joint work with Olivia Caramello and Goncalo Tabuada.

We will describe infinitesimal deformations of complex manifolds to the direction of something having possibly non-commutative structure sheaves by using Hochschild cohomology. We will also describe global non-commutative deformations of some surfaces.

I will first present basics of two adelic structures on relative elliptic surfaces over Spec of the ring of integers of a number field or a smooth projective irreducible curve over a finite field and of the higher adelic zeta integral. Then I will concentrate on the higher adelic program to prove the equality of the arithmetic and analytic ranks of the generic fibre.

In this talk, I will introduce a recent research progress on the following conjecture: any 3-fold of general type has the volume lower bound 1/420. I will also report some new results on the classification of 3-folds attaining minimal volumes. The context covers joint works with Jungkai Chen, Yong Hu and Chen Jiang.

In this talk, I will discuss some results about singularities in birational geometry and discuss their applications to relevant topics.

I will introduce an adelic variant of the Eisenstein classes of Beilinson, Kings and Levin and present applications to special values of partial zeta functions of totally real fields. This is joint work with Alexandros Galanakis.

This is a survey talk on the theory of Higgs-de Rham flow, its applications, and some recent developments.

Many geometric questions about K3-surfaces can be restated and solved in purely arithmetical terms, by means of an appropriately defined homological type. For example, this works well in the study of singular complex sextic curves or quartic surfaces, as well as in that of smooth real ones. However, when the two are combined (singular real curves or surfaces), the approach fails as the natural concept of homological type does not fully reflect the geometry. We show that the situation can be repaired if the curves in question have empty real part; then, one can confine oneself to the homological types consisting of the exceptional divisors, polarization, and real structure. The resulting arithmetical problem can be solved, and this leads to an equivariant equisingular deformation classification of real plane sextics with empty real part. Joint work with Alex Degtyarev.

In this talk we will introduce new birational invariants. Examples will be considered.

We discuss construction and computation of super-Gromov Witten invariants, which count super curves in non-super almost Kahler varieties. A major challenge in construction of supergeometric analogue of Gromov-Witten invariants is the existence of a suitable generalization of intersection theory in the context of super scheme theory. We propose to circumvent this difficulty by taking advantage of a virtual torus localization theorem for the odd directions. That is, after constructing the super moduli stack of super stable maps, we show that there exists an action of a torus on the super moduli stack, whose corresponding fixed locus amounts to the regular stable maps defined by Maxim Kontsevich. We then compute the super virtual normal bundle to the torus-fixed loci of the super moduli stack, and compute the super Gromov-Witten invariants, via dividing the appropriate cohomology classes (primary insertions) by the equivariant Euler class of the super virtual normal bundle, and intersecting with the virtual class of the torus fixed components. We define the super Gromov-Witten invariants of genus zero super curves embedded in non-super projective varieties, and show that these satisfy generalized Kontsevich-Manin axioms for Gromov-Witten classes. Furthermore, we provide computation of super Gromov-Witten invariants of projective space. This is joint work with Enno Keßler and Shing-Tung Yau.

We expose an approach based on semi-local power series expansion of cyclotomic rationals gamma( theta) arising from presumed solutions of equations of the type (x^p+y^p)/(x+y) = p^e z^q, with p, q odd primes, p > 3, (x,y,z)= 1 and e = 1 if p| x+y and 0 otherwise. The gamma depends on the choice of elements theta in I, the Stickelberger ideal, and the crux of the approach contains in proving the existence of theta, for which the power series sums up (locally) to gamma -- rather than some local sum differing from gamma by local roots of unity. Having found such theta, one obtains powerful upper bounds, which suffices for showing there are no solutions for p = q or q > 3p, and strongly limit the set of possible solutions otherwise. Further combinations of results will be mentioned in the talk.

A smooth projective variety X is called rigid if any deformation of X is isomorphic to itself. A first example is the projective space, but in general it is a subtle and difficult problem to prove the deformation rigidity. I'll report some recent progress in this problem. Part of this talk is based on joint works with Yifei Chen and Qifeng Li (Shandong Univ).