# Manchester Geometry Seminar - Andrey Lazarev

Dates: | 4 December 2023 |
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Times: | 13:30 - 14:30 |

What is it: | Seminar |

Organiser: | Department of Mathematics |

Who is it for: | University staff, External researchers, Current University students |

Speaker: Professor Andrey Lazarev (Lancaster)

Title: Homotopy gauge equivalence of Maurer-Cartan elements

Abstract: A Maurer-Cartan (MC) element in a differential graded algebra (dga) A is an element x satisfying the equation of flat connection dx+x^2=0. Two MC elements x and y are gauge equivalent if there is an invertible element a in A such that x=aya^{-1}-daa^{-1}. The set of MC elements in A modulo gauge equivalence is called the MC moduli set of A. These are well-known and classical notions familiar to experts in differential geometry and deformation theory. It is also well-known that the moduli set of MC elements is not a quasi-isomorphism invariant of a dga. In this talk I will explain how one can usefully weaken the notion of a gauge equivalence so that it leads to the MC moduli set becoming a homotopy invariant (in a certain precise sense). This is the beginning of a long story, with many interesting ramifications of which I will attempt to outline a few. Nontrivial examples come from de Rham and Dolbeault algebras.