This course is an introduction to differential graded (=dg) categories and their applications in representation theory and its links to algebraic geometry (commutative and non commutative). Much of our motivation and inspiration comes from the (additive) categorification of Fomin-Zelevinsky cluster algebras (with coefficients). We will begin with the study of dg algebras, their derived categories, derived Morita equivalence and Koszul duality. We will then introduce dg categories, their quasi-equivalences and Morita equivalences and describe the corresponding model categories with their closed monoidal structure after Tabuada and Toen. The construction and characterization of dg localizations (e.g. Drinfeld quotients) and homotopy pushouts will be particularly important. We will then examine various important invariants associated with dg categories, notably K-theory, Hochschild and cyclic homology and Hochschild cohomology. We will apply these in the construction of (relative) Calabi-Yau structures and Calabi-Yau completions following Ginzburg, Brav-Dyckerhoff and Yeung. The final part of the course will be an introduction to Bozec-Calaque-Scherotzke's recent work relating Calabi-Yau completions to shifted cotangent spaces.

        主持人：邱宇教授 （清华大学丘成桐数学科学中心）

        主讲老师：Bernhard Keller教授（中文名：孔博恩） 巴黎大学(原巴黎七大)

        孔老师是著名代数学家，在微分分次理论、cluster理论以及Hochschild同调理论中均做出了奠基性的学术成果。Keller教授是Sophie Germain奖得主，美国数学会会士，于2006年在国际数学家大会做邀请报告。

       

        日期:七月17/22/29号+九月2/9/16/23号（首次为周五，其余为周三）
时间：北京时间晚上8点--9点40分（45分钟*2，中间休息10分钟）

       

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Lecture notes (link：https://cloud.math.univ-paris-diderot.fr/s/xfAbBe3trzX7ndp)