Speaker: Prof. Bernhard Keller 孔博恩 (University of Paris, formerly Paris 7)

Dates: July 17/22/ 29; Sep. 2/9/16/23, 2020

Time: Beijing time 8pm = Paris time 2pm

Duration: 2×45min

Zoom Room for

Sep. 16: Meeting ID: 931 3989 2056 (Password: 472553)

Sep. 23: Meeting ID: 941 4450 9174 (Password : 950830)

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Link to beer garden for informal discussions after the questions (CHROME/IREFOX ONLY!):

https://gather.town/WedPNeO4oTjyJb1x/YMSC-Beer-Garden

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Abstract: 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.

Downloads:

• Notes, Video and Refs (updated), and more Lecture Video