YMSC course

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algebraic geometry category theory K-theory and homology rings and algebras representation theory

Université de Paris / Yau Mathematical Sciences Center

Audience: Researchers in the topic
Seminar series times: No fixed schedule
Organizer: Yu Qiu
Curator: Bernhard Keller*
*contact for this listing

This course consists of 7 lectures of 2 times 45 minutes with a break of 10 minutes. The 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 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 bundles. Here is a link to the Notes, references and recordings and a link to an alternative home page.

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