HSAGA Seminar

    In this talk, we will cover several aspects: Firstly, we will delve into the higher-dimensional Auslander-Reiten theory over an infinite-dimensional coalgebra; the finite-dimensional case by duality reduces to that of finite-dimensional algebra. We will introduce the n-transpose of a finitely n-copresented comodule and n-Auslander-Reiten translations, and then prove the n-Auslander-Reiten formula on n-cluster-tilting subcategories of comodule categories. Secondly, we will introduce the Gorenstein transpose via a minimal Gorenstein injective copresentation of a quasi-finite comodule, and explore the relations between the Gorenstein transpose of comodules and the transpose of the same comodule. As an application, we will construct the almost split sequences in terms of Gorenstein transpose. Finally, we will generalize the notion of the Auslander transpose of comodules to that of the transpose with respect to a semidualizing bicomodule T and showcase some nice homological properties of T-transpose. Additionally, we will provide a Foxby equivalence of comodule categories and investigate a characterization of T-reflexive comodules. This work is joint with Prof. Hailou Yao.

In 1996, A. Tamagawa discovered a surprising phenomenon: anabelian geometry also exists for curves over algebraically closed fields of characteristic p>0 (i.e., curves in positive characteristic can possibly be determined by their geometric fundamental groups without relying on Galois actions). However, after 28 years, only a few results have emerged in this field. In this talk, I want to explain the following insight of the speaker about fundamental groups of curves in positive characteristic: 

The (admissible or geometric log etale) fundamental groups of pointed stable curves over algebraically closed fields of characteristic p can be regarded as an analogue of local moduli spaces of the curves. 

This observation led to the speaker discovering some new kinds of anabelian phenomena of curves in characteristic p and to formulated numerious new conjectures. For example, the following highly non-trivial anbelian results of the speaker provide strong evidence supporting this insight:

• The homeomorphism conjecture holds for 1-dimensional moduli spaces (roughly speaking, this conjecture means that the moduli spaces of curves can be reconstructed group-theoretically as topological spaces).

• A new proof of Mochizuki’s famous result concerning (Isom-version) Grothendieck’s anabelian conjecture for curves over sub-p-adic fields without using Faltings’ p-adic Hodge theory.

• The group-theoretical specialization conjecture holds (roughly speaking, this conjecture means that the topological and group-theoretical degeneration of curves can be completely determined by open continuous homomorphisms of dmissible fundamental groups).

    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.

    For any reductive group we find a stacky interpretation of the stability condition for principal bundles over a curve: it is the unique maximal open locus that admits a schematic moduli space. Some applications and further progress will be discussed. This is a joint work with Dario Weissmann and Andres Fernandez Herrero.

    I shall start from a classical result on Brauer-Manin obstruction of a product of two varieties, by introducing Yang Cao's formulas of Kunneth type. Then we will generalize it to a class of algebraic stacks.