Homotopical Algebra Summer Day in Barcelona 2012

Transcription

1 Homotopical Algebra Summer Day in Barcelona 2012 MTM Barcelona, 13 July,

2 Homotopical Algebra Summer Day in Barcelona A day of talks and informal exchange held at the IMUB research centre in the Historical Building of the Universitat de Barcelona, Plaça Universitat, consisting of lectures on current research and free time for discussion. Speakers Joana Cirici (Universitat de Barcelona and Universitat Pompeu Fabra) Joseph Hirsh (City University of New York) Leandro Lombardi (Universidad de Buenos Aires) Fernando Muro (Universidad de Sevilla) Andrew Tonks (London Metropolitan University) Frank Neumann (University of Leicester) Urtzi Buijs (Universitat de Barcelona) Organising Committee Imma Gálvez - Departament de Matemàtica Aplicada III, Escola d Enginyeria de Terrassa, Universitat Politècnica de Catalunya Javier J. Gutiérrez - Departament de Barcelona d Àlgebra i Geometria, Universitat 2

3 Schedule Friday 13th July Talks in the Aula de l IMUB 9:30-10:10 Andrew Tonks Unital associahedra 10:15-11:05 Fernando Muro On the uniqueness of homotopy units for A algebras 11:05-11:45 Coffee Break+Discussion 11:45-12:35 Joseph Hirsh Deformation theory with noncommutative parameters :20 Leandro Lombardi An explicit up-to-homotopy associative algebra structure in the cactus operad 13:20-15:30 Lunch at La Llavor dels Orígens+Discussion 15:30-16:20 Joana Cirici Homotopy theory of mixed Hodge diagrams 16:25-17:15 Frank Neumann Rational homotopy types of moduli stacks of principal G-bundles 17:15-17:45 Coffee+ Discussion 17:45-18:30 Urtzi Buijs The Lawrence-Sullivan construction and algebraic models for non-connected spaces 18:30-20:30 Discussion 21:00 - Dinner at El Mussol, les Arenes. 3

4 Abstracts The Lawrence-Sullivan construction and algebraic models for non-connected spaces Urtzi Buijs (Universitat de Barcelona) Abstract: In this talk we analyze in detail the complete differential graded Lie algebra given by the Lawrence-Sullivan construction. We show its geometric meaning in terms of rational homotopy theory and its applications and interactions with number theory and the Baues-Lemaire cylinder, and deformation theory and the Fiorenza-Manetti mapping cone. This construction plays a fundamental role in order to develop a consistent homotopy theory of L algebras. This is a joint work with Aniceto Murillo. Homotopy theory of mixed Hodge diagrams Joana Cirici (Universitat de Barcelona and Universitat Pompeu Fabra) Abstract: I will study the homotopy theory of mixed Hodge diagrams of cdga s via the construction of cofibrant minimal models. As an application, and extending the Formality Theorem of compact Kähler varieties, I will show that every complex algebraic variety (possibly open and singular) is filtered formal: the rational homotopy type is determined by the first term of the spectral sequence associated with the weight filtration. Deformation theory with noncommutative parameters Joseph Hirsh (City University of New York) Abstract: In this talk I will describe the classical yoga of deformation theory that commutative moduli problems are locally described by dglie algebras in characteristic zero and provide some examples. Then, importing tools from operad theory, most notably Koszul Duality, I will describe a generalization that classifies the local structure of noncommutative moduli problems in arbitrary characteristic. I will conclude by describing some ongoing work to exploit the relationship between noncommutative deformation theory and Goodwillie s calculus of functors. 4

5 An explicit up-to-homotopy associative algebra structure In the cactus operad Leandro Lombardi (Universidad de Buenos Aires) Abstract: It is well known that the Hochschild cohomology of an algebra A, HH(A) carries a structure of a Gerstenhaber algebra. Deligne s conjecture states that this structure can be lifted to an action of the Little Disks operads. This conjecture has been solved introducing different operads, one of them being the Cactus operad. In this talk we will show how these operads interplay and show an explicit construction of an up-to-homotopy structure in the Cactus operad. This is joint work in progress with Imma Gálvez (UPC) and Andrew Tonks (Londonmet). On the uniqueness of homotopy units for A algebras Fernando Muro (Universidad de Sevilla) Abstract: We all know that associative algebras may have at most one unit, therefore being unital can be regarded as a property rather than a structure. Although this fact admits an elementary proof, it also follows from the sophisticated observation that the natural map from the associative operad to the unital associative operad is an epimorphism, in the sense of category theory. In this talk we show that this operad map is also a homotopy epimorphism. Using explicit operad resolutions given by (unital) associahedra, we obtain the optimal homotopical generalization of the aforementioned classical result. We consider the moduli space of unital A structures on a given complex, given by mapping spaces in the model category of non-symmetric operads, and show it is homotopy equivalent to certain subspace, formed by some connected components of the moduli space of (possibly non-unital) A-infinity structures on that complex. If time allows, I will also comment on the identification of the unital components in connection with a result of Lyubashenko and Manzyuk, and/or the applications of the result to the geometry of moduli spaces of algebras in the context of homotopical algebraic geometry. 5

6 Rational homotopy types of moduli stacks of principal G-bundles Frank Neumann (University of Leicester) Abstract: We will discuss ideas on how to determine the rational homotopy type of some moduli stacks of principal G-bundles on complex algebraic varieties. This uses Noohi s theory of topological stacks and relies on computations with the Haefliger-Brown-Szczarba model for the rational homotopy type of mapping spaces. This is work in progress with U. Buijs (UB Barcelona). Unital associahedra Andrew Tonks (London Metropolitan University) Abstract: The classical theory of Koszul duality for (homogeneous) quadratic operads provides an explicit definition of the operad A governing up-tohomotopy DG algebras. In the topological setting, Stasheff s classical associahedra are polytopes that form a cellular operad Ass, from which the operad A may be recovered on taking the cellular chains. In classical applications it was not necessary to consider units for the algebra multiplication, or it was assumed units were strict. The introduction of non-strict units into the picture was considerably harder: Fukaya-Oh-Ohta- Ono introduced homotopy units for A algebras in their work on Lagrangian intersection Floer theory, and equivalent descriptions of the corresponding dg operad ua have now been given, for example, by Lyubashenko and by Hirsh-Millès. In this talk we present the missing link : a new cellular topological operad uass of unital associahedra, providing a resolution for the operad uass governing topological monoids, from which the operad ua may be recovered on taking the cellular chains. This is joint work with Fernando Muro. 6