太阳成集团tyc411(中国)有限公司-百度百科九十周年院庆系列活动之二十九

A series of talks on algebras and representations

1.      Title: The covering theory and its applications in representation theory (I, II)

Speaker: Professor Shiping Liu (The University of Sherbrooke, Canada)

   Time: (I) 16:00pm---, June 18 (Monday), 2018;

            (II) 16:00pm---, June 20 (Wednesday), 2018.

   Place: 玉泉校区工商楼200-9.

Abstract:

The covering theory was introduced by Bongartz, Gabriel and Riedtmann in the early eighties in order to study representations of finite dimensional algebras. More recently, it has been further developed by Asashiba, Bautista and Liu in order to study derived categories of algebras. It is also a power tool for the study of other types of linear categories such as cluster categories, singularity categories and representation categories of species of valued quivers.

   In this series of two talks, we shall start with the definitions of Galois covering for unvalued quivers, for valued translation quivers, and for linear categories with a link to the construction of cluster categories. We shall also introduce the Auslander-Reiten theory for a Krull-Schmidt additive category and study its behavior under a Galois covering functor. As application, we shall mainly speak about derived categories of algebras given by a finite or infinite quiver with relations. We shall how a Galois covering of algebras induces a Galois precovering between their bounded derived categories. In case an algebra has a radical squared zero, its bounded derived category of finite dimensional modules admits a Galois covering, which is the bounded derived category of finite dimensional representations of a gradable quiver. This enables us to apply our knowledge of the representations of a strongly locally finite quiver to obtain a complete description of the Auslander-Reiten components of the bounded derived category of finite dimensional modules over an algebra with radical squared zero. 

2.      Title: Reduction theory of arithmetic groups and some applications

Speaker: Professor Lizhen Ji (University of Michigan)

Time: 14:40pm---, June 20 (Wednesday), 2018

Place: 玉泉校区工商楼200-9.

Abstract:

Arithmetic subgroups Γof Lie groups G provide important examples of discrete subgroups of semisimple Lie groups. They act properly on symmetric spaces X associated with the Lie groups G. A basic example is Γ=SL(2, Z) contained in G=SL(2, R), and it acts on the upper half-plane. Reduction theory of arithmetic subgroups provide (coarse) fundamental domains for their actions on symmetric spaces, which are easy to describe and enjoy good properties. It plays an essential role in understanding group theoretic theoretic properties of arithmetic groups Γ and the geometry and topology of locally symmetric spaces Γ X.

In this talk, I will discuss its historical development starting from the work of Fermat, Lagrange and Gauss in number theory, functorial properties of fundamental domains and some applications to periods of compact Riemann surfaces.

联系人：李方（fangli@zju.edu.cn）