Monday, January 23
18:00-19:00
Speaker: Benoit Fresse (Université de Lille)
Title: Rational homotopy of operads. Models of mapping spaces and applications
Abstract: I will survey research on the definition of models for the rational homotopy of operads, an application of graph complexes to the computation of mapping spaces of operads, and the applications of these results to the embedding calculus and the theory of Grothendieck-Teichmüller groups (joint work with Victor Turchin and Thomas Willwacher).
To be more specific, the main objects of this study are E_n-operads, a class of operads, defined by a reference model, the operads of little n-discs (or n-cubes), and which can be used to govern a hierarchy of homotopy commutative structures, from fully homotopy associative but non-commutative (n=1) up to fully homotopy associative and commutative (n=infinity). The definition of rational models relies on an interpretation of formality results, which I am going to explain first. Then I will explain that certain combinatorial operads, defined in terms of graphs, can be used to get a combinatorial description (a graph complex description) of mapping spaces associated to these objects.
The applications to the study of embeddings goes through the Goodwillie-Weiss embedding calculus, which relates the spaces of embeddings of Euclidean spaces to these operadic mapping spaces. The spaces of homotopy automorphisms of E_n-operads can also be identified with the Grothendieck-Teichmüller group in the case n=2, and hence, with higher dimensional generalizations of these objects in the case n>2. Depending on time and on the interest of the audience, I will address one or the other of these connections with more details.
Tuesday, January 24
9:30-10:30
9:30-10:30
Speaker: Felix Wierstra (University of Amsterdam)
Title: A recognition principle for iterated suspensions as coalgebras over the little cubes operad
Abstract: I will discuss and prove a recognition principle for iterated suspensions as coalgebras over the little disks operad. This is based on joint work with Oisín Flynn-Connolly and José Moreno-Fernández.
12:00 - 13:00
Speaker: Víctor Carmona (Universidad de Sevilla)
Title: A title containing the words "localization", "operads" and "Thompson-groups" or "quantum field theories"
Abstract: Localization, thought as inverting operations, is a fundamental tool in mathematics. We are exposed to it from basic commutative algebra, localization of rings, to advanced topics as chromatic homotopy theory, where localization plays a fundamental role to focus on phenomena over a prime at a time. In this talk we will revisit what localization is, how to localize operads and similar algebraic structures and we will apply this technique to obtain interesting results in other contexts. In particular, we will use localization of operads in group theory to generate Thompson-like groups (based on joint work with J.Aramayona-F.Cantero-J.J.Gutiérrez) or in mathematical physics to understand phenomena in quantum field theory (based on joint work with M.Benini-A.Schenkel and with D.Calaque). The assistants will decide what application is going to be exposed thoroughly.
Wednesday, January 25
11:00-12:00
Speaker: Fernando Muro (Universidad de Sevilla)
Title: Massey products and higher operations
Abstract: It was long believed that Massey products in the cohomology of a differential graded associative algebra were determined by the higher operations on a minimal A-infinity model. This was disproved by Buijs, Moreno-Fernández, and Murillo. They also gave sufficient conditions for it to hold. In this talk, we will present two further conditions:
- Associative algebras with sparse cohomology.
- Algebras over Koszul operads.
In the latter case, we define an analogue of triple Massey products which extends the cases previously considered in the literature (associative and Lie algebras).
12:00-13:00
Speaker: Imma Gálvez ( Universitat Politècnica de Catalunya)
Title: On a canonical B∞ structure
Abstract: The motivating example of a B∞-algebra was given by Baues in his work on iterated loop spaces, when he endowed the bar construction (a free coalgebra) with a compatible differential graded algebra structure. Another famous example is the B∞ structure on the Hochschild cochain complex: the relation of B∞ and G∞ algebras was central in the resolutions of Deligne's Hochschild cohomology conjecture.
We present work in progress (joint with A Tonks and M O Ronco) investigating the appearance of canonical B∞-algebra structures on any algebra endowed with a not-necessarily compatible dga structure, recovering previous results by Markl on A∞-algebras and by Loday-Ronco on multibrace algebras.
Application to a new Milnor-Moore type theorem will be given in tomorrow's talk by Andy Tonks.
Thursday
11:00-12:00
Speaker: Pedro Tamaroff (Humboldt-Universität zu Berlin)
Title: Generalized Cohomological Field Theories in the Higher Order Formalism
Abstract: In the classical Batalin—Vilkovisky formalism, the BV operator is a differential operator of order two with respect to a commutative product; in the differential graded setting, it is known that if the BV operator is homotopically trivial, then there is a genus zero level cohomological field theory induced on homology. In this talk, we will explore generalisations of (non-commutative) Batalin–Vilkovisky algebras for differential operators of arbitrary order, showing that homotopically trivial operators of higher order also lead to interesting algebraic structures on the homology. This is joint work with V. Dotsenko and S. Shadrin.
12:00-13:00
Speaker: Andy Tonks (Universidad de Málaga)
Title: A new Milnor-Moore type theorem
Abstract: We present work in progress (with I Gálvez and M Ronco) investigating an extension of the Milnor-Moore theorem to non-cocommutative differential graded Hopf algebras, generalising the previous work of Loday-Ronco taking into account the construction of canonical B∞-algebra structures discussed in yesterday's talk by Imma Gálvez.