Operads, deformation quantization and higher structures

(MPIM seminar, Fall 2010)

Theme:  Operads (Koszul duality theory, generalized operads), deformation quantization (deformation theory, Poisson manifolds) and higher structures (homotopy algebras, higher category theory)

Programme: The purpose of this seminar is to provide courses on the aforementioned fields. In the early 90's, operads enjoyed a renaissance under the impulse of the proofs by Kontsevich and Tamakin of the deformation quantization of Poisson manifolds and the Koszul duality theory of operads by Ginzburg-Kapranov and Getzler-Jones. These discoveries have opened new doors in mathematics by relating various fields such as homotopy theory, graph homology, field theories and higher categories. Mathematics is often made of alternating breakthroughts and periods of sedimentation of ideas. The spirit of this seminar is to explain the ideas on these fields that are now well understood after 15 years. The ultimate hope is to prepare the audience for new results. For instance, very recent progress have been made in relationship with Grothendieck-Teichmüller groups and Lie algebra.

The talks will take place every Wednesday at 10.30 am in the main lecture hall (Hörsaal) of the MPIM.

Talks:

     September 29: Organization meeting [exceptionally at 4.30 pm !]
     October 6: Definitions, examples and first properties of operads, by Bruno Vallette
     October 13: Koszul duality theory for associative algebras, by Bruno Vallette
     October 20: Koszul duality theory for operads, by Bruno Vallette
     October 27: Homotopy theory of algebras, by Bruno Vallette
     November 3: From Kontsevich's formality to Deformation quantization, by Carlo Rossi
     November 9 [10-11 am]: Model categories, by Arturo Prat-Waldron
     November 10: Proof of the formality conjecture (after Kontsevich), by Carlo Rossi
     November 16 [2-3.30 pm]: Model categories on algebras/operads and Tamarkin Proof, by Bruno Vallette
     November 17: Proof of the formality conjecture (after Kontsevich) II, by Carlo Rossi
     November 24: Tangent structures on Kontsevich formality, by Carlo Rossi
     December 1: Props, cyclic operads, modular operads, etc., by Dennis Borisov
     December 8: Operads: the general framework, by Dennis Borisov
     December 14 [4.30-6 pm]: Higher dimensional operads I, by Dennis Borisov
     December 16 [4.30-6 pm; seminar room]: Higher dimensional operads II, by Dennis Borisov
     December 22 [2-3.30 pm]: Higher dimensional operads III, by Dennis Borisov

References:

Books

     Loday Jean-Louis; Vallette Bruno, Algebraic operads. Avaible to download here. link

Articles

     Kontsevich Maxim, Deformation quantization of Poisson manifolds, Lett. Math. Phys. 66 (2003), no. 3, 157–216.   link
     Tamarkin, Dmitry E., Another proof of M. Kontsevich formality theorem,  arXiv:math/9803025.   link
     Kontsevich, Maxim, Operads and motives in deformation quantization, Lett. Math. Phys. 48 (1999), no. 1, 35–72.   link

Surveys

     Keller, B., Deformation quantization after Kontsevich and Tamarkin. Déformation, quantification, théorie de Lie, 19–62, Panor. Synthèses, 20, Soc. Math. France, Paris, 2005.   link
     Hinich, Vladimir, Tamarkin's proof of Kontsevich formality theorem, Forum Math. 15 (2003), no. 4, 591–614.   link

Related seminar

     Graduate Seminar Topology S4D2 "Model categories" organized by Stefan Schwede  Link

Speakers:

     Dennis Borisov
     Arturo Prat-Waldron
    Carlo Rossi
     Bruno Vallette

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Last update: December 8, 2010.