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Moduli spaces of (bi)algebra structures in topology and geometry

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, En-algebras and quantum group theory.
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Sinan Yalin. Moduli spaces of (bi)algebra structures in topology and geometry. 2016 MATRIX Annals, 1, Springer, pp.439-488, 2018, MATRIX Book Series. ⟨hal-01714224⟩

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