The proposed project is to support participation by USA-based mathematicians and physicists, graduate students, and post-docs, in the conference on representation theory, quantization, and supergeometry to be held in Bialowieza, Poland, during the week of June 26 - July 2, 2011. The topics of the conference include representations of local current groups and other infinite-dimensional groups and Lie algebras, together with supergroups and Lie superalgebras, formality theorems for Hochschild cochains via transfer, construction of differential graded categories for compact symplectic manifolds using the microlocal theory of sheaves by Kashiwara-Schapira, "monstrous moonshine" and vertex operator algebras, quantization of three-dimensional quantum gravity based on the quantum universal Teichmueller space, odd symplectic geometry and the Batalin-Vilkovissky formalism, and relations between the chiral de Rham complex and generalized complex structures, vertex algebroids and supergeometry.

The conference brings together researchers who have made significant contributions to representation theory, quantization, and super-mathematics. The interaction between these three fields has been one of the major driving forces of the modern mathematical physics. The format of the conference, its location and facilities provide an intimate research environment for the interaction of established researchers with each other and with graduate students and post-docs, and give an opportunity to younger researchers to present their work to an international audience. Additional information can be found on the conference web page

Project Report

; Quantum Groups and Operator Algebras; and Foundations of Quantum Mechanics. There were 112 participants from 16 countries from four continents. The conference on Representations, Quantization, and Supergeometry was dedicated to the 80th anniversary of the famous Soviet Russian mathematician and physicist Felix Berezin known for his contributions to the topics of the conference, and especially for the creation of what is now known as supermathematics. Nine senior and four early career participants of this conference from the United States were supported by NSF. Five senior participants from the United States, Vasily Dolgushev (Philadelphia, PA), Alexander Karabegov (Abilene, TX), Vera Serganova (Berkeley, CA), Dmitry Tamarkin (Evanston, IL), and Arkady Vaintrob (Oregon), were among the 17 plenary speakers of the workshop. Vasily Dolgushev gave a talk "On stable formality quasi-isomorphisms for Hochschild cochains", where he introduced stable formality quasi-isomorphisms for Hochschild cochains of a polynomial algebra. The zeroth cohomology H_0(GC) of Kontsevich's graph complex GC acts naturally on homotopy classes of stable formality quasi-isomorphisms. In his talk Vasily Dolgushev proved that this action turns the set of homotopy classes of stable formality quasi-isomorphisms into a torsor over the group exp(H_0(GC)). In his talk "Infinitesimal deformations of a formal symplectic groupoid" Alexander Karabegov introduced the notion of an infinitesimal deformation of a formal symplectic groupoid. To each pair of natural star products with the same formal symplectic groupoid G he associated an infinitesimal deformation of G. For a given star product with separation of variables * he constructed another star product with separation of variables and used the infinitesimal deformation of the formal symplectic groupoid with separation of variables corresponding to this pair of star products to express the principal symbols of the components of the logarithm of the formal Berezin transform of the star product *. Vera Serganova gave a talk "Geometric methods in representation theory of Lie supergroups", where she discussed two geometric approaches for representation theory of classical Lie supergroups: associated variety and Borel-Weil-Bott theory. Her results included superdimension and character formula for irreducible finite-dimensional representations. In his talk "Action of Sp(2n) on sheaves" Dmitry Tamarkin defined for a given smooth manifold X a certain full sub-category D_>0(X x R) in the derived category of sheaves of vector spaces on X x R; this sub-category has close ties with the cotangent bundle T*X. One can define the notion of microsupport for objects of this category as a closed subset of T*X. Each Hamiltonian symplectomorphism F of T*X can be quantized: there exists an endofunctor QF of D_>0(X x R) which transforms microsupports of objects according to F. He applied this formalism to the quantization of the symplectic action of Sp(2n) on T*R^n. It turns out that it is naturally defined on the universal cover G of Sp(2n). The kernel of the covering map G -> Sp(2n) acts by homological shifts; the restriction to the sub-group GL(n) differs from the action by coordinate changes by a twist via a local system. Arkady Vaintrob discussed several applications of the supermanifold theory to the study of generalized complex and Kaehler structures in his talk "Generalized complex geometry and supermanifolds". During the conference researchers in the fields of representation theory, quantization, and supergeometry were actively interacting with other mathematicians and physicists participating in the workshop. There were many informal discussions between and after talks, involving senior and early career participants. Parallel sessions and a poster session were organized during the workshop, where the early career researchers could present their work. The annual workshop traditionally takes place in the heart of the Bialowieza primeval forest - a site of great natural beauty. Among the activities of the workshop is an excursion to the Bialowieza preserve. A volume of proceedings of the workshop will be published by Birkhauser in 2012. Besides research papers, the volume will contain the articles "The Bialowieza Workshop on Geometric Methods in Physics: An Impression of Three Extraordinary Decades" by Gerald A. Goldin (Piscataway, NJ), and "Felix Alexandrovich Berezin and his work" by Alexander Karabegov, Yuri Neretin, and Theodore Voronov.

National Science Foundation (NSF)
Division of Mathematical Sciences (DMS)
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Joanna Kania-Bartoszynska
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Abilene Christian University
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