# 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.