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Higher Structure 2013: Operads and Deformation Theory

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Workshop
2nd April 2013 to 5th April 2013

Organisers: Anton Alekseev (Université de Genève), John Jones (University of Warwick), Bruno Vallette (Université de Nice Sophia Antipolis) and Chenchang Zhu Bartholdi (Georg-August-Universität Göttingen).

Workshop Theme

The notion of an operad was introduced in the late 60's in algebraic topology as a tool to encode higher homotopies. It enjoyed a renaissance in the 90's when M. Kontsevich and others used algebraic operads in deformation theory.

The passage from classical mechanics to quantum mechanics prompted the general mathematical problem of deformation quantization. In Poisson geometry, such a problem was solved by D. Fedosov for symplectic manifolds, by V. Drinfeld for Poisson-Lie groups, and by M. Kontsevich for Poisson manifolds. In 1998, six months after Kontsevich's original proof, D. Tamarkin gave another but purely operadic proof of the deformation quantization of Poisson manifolds, using the formality of the little discs operad, the Deligne conjecture, and the deformation-quantization of Lie bialgebras due to P. Etingof and D. Kazhdan.

The introduction of operadic graph homology in 1991 by M. Kontsevich allowed V. Ginzburg and M. Kapranov, and E. Getzler and J. Jones to develop the Koszul duality theory on the level of algebraic operads. This theory gives a conceptual explanation of the duality between commutative algebras and Lie algebras in Rational Homotopy Theory, developed by D. Quillen and D. Sullivan. It was also shown by M. Kontsevich, E. Getzler, and Y.I. Manin to share nice relationships with moduli spaces of curves, i.e. quantum cohomology and Frobenius manifolds. The operadic calculus play a key role in Quantum Field Theory in mathematical physics since it provides an algebraic way to control higher structures.

The goal of this conference is to cover the most recent and interesting developments of these fields of research.

University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons