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Moduli Spaces of (Bi)algebra Structures in Topology and Geometry

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2016 MATRIX Annals

Part of the book series: MATRIX Book Series ((MXBS,volume 1))

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Abstract

After introducing some motivations for this survey, we describe a formalism to parametrize a wide class of algebraic structures occurring naturally in various problems of topology, geometry and mathematical physics. This allows us to define an “up to homotopy version” of algebraic structures which is coherent (in the sense of ∞-category theory) at a high level of generality. To understand the classification and deformation theory of these structures on a given object, a relevant idea inspired by geometry is to gather them in a moduli space with nice homotopical and geometric properties. Derived geometry provides the appropriate framework to describe moduli spaces classifying objects up to weak equivalences and encoding in a geometrically meaningful way their deformation and obstruction theory. As an instance of the power of such methods, I will describe several results of a joint work with Gregory Ginot related to longstanding conjectures in deformation theory of bialgebras, E n -algebras and quantum group theory.

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Acknowledgements

The idea of writing such a survey originates in the inaugural 2-week program at the mathematical research institute MATRIX in Australia called Higher Structures in Geometry and Physics, which took place in June 2016. The author gave a talk at this program about moduli spaces of algebraic structures and their application to the recent paper [30]. The present article is somehow a (largely) extended version of his talk, which will be eventually part of a Proceedings Volume devoted to this workshop. The author would like to thank the MATRIX institute for supporting this program, the organizers of this programme for inviting him, and all the participants for their interest and for the very enjoyable atmosphere during the 2 weeks spent there. Last but not least, kangaroos are very much thanked for their natural awesomeness.

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  1. Laboratoire angevin de recherche en mathématiques (LAREMA), University of Angers, 2 Boulevard Lavoisier, 49045, Angers, France

    Sinan Yalin

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Editors and Affiliations

  1. University of Melbourne, Melbourne, Australia

    Jan de Gier

  2. University of Western Australia, Crawley, Australia

    Cheryl E. Praeger

  3. UCLA, Los Angeles, USA

    Terence Tao

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Yalin, S. (2018). Moduli Spaces of (Bi)algebra Structures in Topology and Geometry. In: de Gier, J., Praeger, C., Tao, T. (eds) 2016 MATRIX Annals. MATRIX Book Series, vol 1. Springer, Cham. https://doi.org/10.1007/978-3-319-72299-3_20

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