Workshop on Higher Structures and Operadic Calculus (IRP Higher Homotopical Structures)

Higher structures in rational homotopy theory

Higher structures and operadic calculus are essential tools in modern approaches to rational homotopy theory and its applications. The aim of this lecture series is to give an introduction to rational homotopy theory through the lens of higher structures.

The rational homotopy type of a simply connected space is faithfully modeled by the collection of rational homotopy groups together with its minimal L-infinity algebra structure, or by the rational cohomology ring together with its minimal C-infinity algebra structure. These structures are related to higher order Whitehead products and Massey products, respectively. We will review these models and discuss their relation to the classical Sullivan and Quillen models. Then we will give a tour of the interplay between Koszul duality, formality and coformality, focusing on computational aspects. Finally, we will discuss classical and new results on the rational homotopy theory of classifying spaces for fibrations, highlighting connections to higher structures and operadic calculus.

An operadic view on associators

Associators, introduced by Drinfeld, became a central object in mathematics, with applications ranging from algebraic topology (formality of the little disks operad) to mathematical physics (deformation quantization). The aim of this lecture series is to explain the modern operadic approach to associators, and discuss generalizations of these. We will not discuss much of the homotopical aspects of the story.

Operadic deformation theory

A classical heuristic in deformation theory asserts that the infinitesimal deformations of an algebro-geometric object over a field of characteristic zero are controlled by a differential graded Lie algebra. In the setting of algebras over operads, various questions (e.g. formality) can be considered as deformation theoretic problems. In this case the operadic formalism produces very explicit and computable instances of such deformation Lie algebras in a universal way.

On the other hand, some algebraic problems admit a non-obvious deformation theoretic interpretation. In this lecture series we will introduce and give an overview of the recent developments concerning the use of algebraic operads in deformation theory. 

Weight structures and formality

I will discuss various formality results revolving around the idea of weight structure. A weight structure is a naturally occurring graduation on the cohomology groups of certain algebraic objects. If this weight structure can be lifted at the level of chains and if it satisfies a property called purity, then one can deduce formality. Algebraic geometry provides us with interesting weight structures as the cohomology of algebraic varieties tends to have additional structure (like a Hodge structure or a Galois action). Using some homotopy theoretic machinery, one can show that these weight structures can be lifted at the level of chains and obtain interesting formality results.

Infinity Modular Operads and the Grothendieck-Teichmüller group 

This three part series will discuss several models for infinity cyclic and modular operads and their applications to (profinite) Grothendieck-Teichmüller groups.

The first two talks in the series will cover basics on cyclic and modular operads and introduce models for their infinity counterparts. We will cover multiple models for infinity cyclic and modular operads, but our primary focus will be on a Segal model introduced in joint work with Hackney and Yau. The third talk will focus on applications and examples of infinity cyclic and modular operads with a focus on the applications to (profinite) Grothendieck-Teichmüller theory.