题目： On the structure of Calabi-Yau algebras and categories

报告人：Bernhard Keller 教授，巴黎西岱大学

地点：1318教室

暂定时间：

11月24日上午10:00-11:30

11月30日下午14:30-16:00

12月7日下午14:00-15:30

12月15日上午10:00-11:30

12月21日下午14:15-15:45

12月28日下午14:30-16:00

摘要：In this lecture series, we will present structure theorems for Calabi-Yau algebras and categories following mainly work by Michel Van den Bergh and recent joint work with Junyang Liu. Calabi-Yau algebras can be viewed as non-commutative analogues of symplectic varieties and the structure theorems we will present will be non-commutative analogues of Darboux' theorem, which states that locally, each symplectic variety is isomorphic to affine 2n-space with its standard symplectic structure. Of course, the locality condition is essential here. In the non-commutative context, it is replaced with the assumption that our algebras are (connective,complete augmented) pseudo-compact dg (=differential graded) algebras (and similarly for categories and for functors). The lecture series will consist of two main parts, the first one being devoted to the structure of Calabi-Yau dg algebras and morphisms and the second one to that of Calabi-Yau triangulated categories and stably Calabi-Yau Frobenius categories. In the first part, after an introduction with examples, we will first recall the necessary homological algebra and the Calabi-Yau conditions for algebras in the absolute and the relative (pseudo-compact) setting. We will then present the Darboux theorems in the absolute case (due to Michel Van den Bergh) and the relative case (obtained in joint work with Junyang Liu). In the second part, we will start with a reminder on cluster categories constructed from quivers with potential (following Amiot and Lingyan Guo) and Higgs categories constructed from ice quivers with potential (following Yilin Wu). We will then sketch how, via a dimension shift, the results of the first part allow to prove that algebraic d-Calabi-Yau categories with cluster-tilting object are cluster categories associated with (d+1)-Calabi-Yau algebras (absolute case) and that certain stably d-Calabi-Yau Frobenius exact categories are Higgs categories (relative case). These statements can be viewed as proofs of variants of Amiot's conjecture from 2010. 

报告人简介：Bernhard Keller是巴黎西岱大学教授、中国科学技术大学客座教授、著名代数学家，在微分分次理论、丛理论以及Hochschild同调理论中均做出了奠基性的学术成果。Bernhard Keller教授是法国科学院“索菲·热尔曼”2014年度大奖得主、法国大学研究院资深成员、挪威皇家科学通讯院士、比利时安特卫普大学荣誉博士、国际数学家大会ICM邀请报告人以及美国数学会会士，任Advances in Mathematics，Forum of Mathematics Pi 以及Journal of European Mathematical Society等国际知名杂志编委。