Higher homotopy algebras in topology 2

Hamilton Mathematics Institute, Trinity College Dublin

HMI Workshop: Higher homotopy algebras in topology 2

March 4-6, 2020

Speakers:

  • Manuel Amann (Universität Augsburg)
  • Urtzi Buijs (Universidad de Málaga)
  • Geoffroy Horel (Université Paris 13)
  • Andrey Lazarev (Lancaster University)
  • Paolo Salvatore (Università di Roma)
  • Andrew Tonks (University of Leicester)
  • Bruno Vallette (Université Paris 13)
  • Nathalie Wahl (University of Copenhagen)

Registration:

Location:

  • Synge Theatre (Wed. and Thur. 09:30-17:00, Friday 09:30-12:00), Hamilton Building

Organizers:



Wednesday 4 Speaker Title and abstract
9:30-10:00 Registration
10:00-11:00 Geoffroy Horel Homotopy transfer and formality. We use homotopy transfer techniques to prove formality results for algebras over certain operads (eg Ass, Com, Lie). Under the assumption that our algebra supports an automorphism that induces the grading action on homology (i.e. acts by multiplication by a^n in homological degree n for some fixed element a), we prove that the higher operations have to be trivial and therefore that the algebra is formal. This is very much in the spirit of the introduction of the Deligne-Griffiths-Morgan-Sullivan paper on the formality of Kähler manifold. The theory of etale cohomology provides us with such automorphisms on the cochains of algebraic varieties. As an application, we obtain partial formality and coformality results for configuration spaces with integral coefficients that generalize the known results with rational coefficients. This is joint work with Gabriel Drummond-Cole.
11:00-11:30 Coffee break
11:30-12:00 Pedro Tamaroff Derived Poincaré--Birkhoff--Witt theorems. We define derived Poincaré--Birkhoff--Witt maps between dg operads, or derived PBW maps, for short: these extend the definition of PBW maps between operads of V. Dotsenko and the author. We then show that the map from the homotopy Lie operad to the homotopy associative operad is derived PBW, and deduce from this results of V. Baranovsky and J. Moreno--Fernandez, as well as producing new results for universal envelopes of L-infinity algebras in the sense of Lada--Markl. We also apply our theory to obtain derived PBW theorems for associative envelopes. This is joint work with A. Khoroshkin.
12:00-14:30 Lunch
14:30-15:30 Bruno Vallette The properadic calculus. The operadic calculus developed over the past 20 years has now reached the status of a complete theory which is fruitfully used in many domains. It allows us to work out the homotopy properties of algebras, that is, algebraic structures with multiple inputs but one output. For instance, it encodes seminal notions like homotopy algebras and their infinity-morphisms, and it produces functorially such structures: homotopy transfer theorem, twisting procedure, and Koszul hierarchy. In this talk, I will survey the recent development of the properadic calculus which produces the similar tools and results but for algebraic structures made up of many inputs and many outputs, like the ones appearing in topological recursion, string topology, Poincaré duality and deformation theory. This is a joint work with Ricardo Campos, Eric Hoffbeck, and Johan Leray.
16:00-17:00 Andrew Tonks Cellular (and point-set) diagonal approximation for the associahedra. Stasheff's associahedron is a central object in homotopy algebra, as the cellular chain complexes of the associahedra define the d.g. A-infinity operad. They also appear in areas such as representation theory and theoretical physics. In this talk we use the theory of fibre polytopes to introduce a topological operad structure on (certain topological realizations of) the associahedra, and provide a compatible topological diagonal approximation. Joint work with Naruki Masuda, Hugh Thomas and Bruno Vallette.


Thursday 5 Speaker Title and abstract
9:30-10:30 Nathalie Wahl Infinity operads: an example. I'll present joint work (in progress) with Safia Chettih, Abigail Linton, Luciana Bonatto, Sophie Raynor and Marcy Robertson where we give an explicit description of the normalized cactus operad as an infinity operad.
10:30-11:00 Coffee break
11:00-12:00 Urtzi Buijs Quillen rational homotopy (revisited) and its applications. Starting from the study of the rational homotopy under Quillen’s approach, we define a new realization functor based on the construction of the "Eckmann-Hilton dual" of the classical differential forms on the standard simplices. We show a wide range of applications of the algebraic models involved. Joint work with Yves Félix, Aniceto Murillo and Daniel Tanré.
12:00-14:30 Lunch
14:30-15:30 Paolo Salvatore Small cellular decompositions of some geometric operads. We describe some small operadic cellular decompositions of familiar operads : the Fulton Mac Pherson operad, its framed version, and the Deligne-Mumford genus zero operad. A refinement of the cellular decomposition of the Fulton Mac Pherson operad satisfies a conjecture by Kontsevich and Soibelman, that motivated our work.
15:30-16:00 Coffee break
16:00-16:30 Anibal Medina Mardones A finitely presented E-infinity prop. The Comm operad in chain complexes admits a presentation in terms of finitely many generators and relations, but no such presentation can be given for a sigma-free resolution of it. By passing to the more general setting of props, we are able to describe finitely presented E-infinity props in the categories of chain complexes and of cellular spaces. We relate the operads associated to these to E-infinity operad models introduced by McClure-Smith, Berger-Fresse and Kaufmann, and describe novel actions on simplicial and cubical sets complementing these authors' work.
16:30-17:00 George Raptis Higher homotopy categories and K-theory. I will discuss the construction and the properties of the higher homotopy categories associated to a homotopy theory. These objects may be regarded as a sequence of refinements for the comparison between homotopy commutativity and homotopy coherence. Even though the idea of the higher homotopy category is not new, the study of these objects seems to have received less attention than the classical homotopy category. Then I will introduce K-theory for these objects and present some results on the comparison with Waldhausen K-theory


Friday 6 Speaker Title and abstract
9:30-10:30 Manuel Amann The Toral Rank Conjecture and variants of equivariant formality. An action of a compact Lie group is called “equivariantly formal”, if the Leray–Serre spectral sequence of its Borel fibration degenerates at the E2-term. This notion is as prominent as it is restrictive. In this talk, also motivated by the lack of junction between the notion of equivariant formality and the concept of formality of spaces (surging from rational homotopy theory) I shall present two new variations of equivariant formality: “MOD-formal actions” and “actions of formal core”. These new definitions admit different characterisations (in particular, also via A-infinity structures) and yield various applications ranging from the setting of manifolds with non-negative sectional curvature to the toral rank conjecture in this context. Indeed, an almost free action of an n-torus T on X possessing any of the two new properties satisfies that the total dimension of the rational cohomology of X is at least 2 to the power of n. This generalizes and proves the toral rank conjecture for actions with formal orbit spaces. The talk reports on joint work with Leopold Zoller.
10:30-11:00 Coffee break
11:00-12:00 Andrey Lazarev Koszul duality: a global approach. Koszul duality is a phenomenon that shows up in rational homotopy theory, deformation theory and other subfields of algebra and topology. Its modern formulation is due to the works of Hinich, Keller-Lefevre and Positselski. It is a certain correspondence between differential graded (dg) algebras and conilpotent dg coalgebras and modules and comodules. In this talk I explain what happens if one drops the condition of conilpotency on the coalgebra side; the consequences turn out to be quite dramatic. I will show how this non-conilpotent (or global) version of Koszul duality comes up naturally in the study of derived categories of complex algebraic manifolds and infinity local systems on topological spaces. This is joint work with Ai Guan.