Masterclass: Infinity Operads and Applications to Geometry

• Rune Haugseng (NTNU Trondheim): Introduction to Infinity Operads

Abstract: The theory of ∞-operads gives a useful language and toolkit for working with homotopy-coherent algebraic structures. I will start by recalling ordinary operads and their description via categories of operators as motivation for Lurie’s definition of ∞-operads. Then I will discuss a number of important constructions in this framework, such as symmetric monoidal envelopes, free algebras and operadic Kan extensions, Day convolution, and the Boardman-Vogt tensor product. I will then mention some other descriptions of ∞-operads, such as via analytic monads and dendroidal Segal spaces. If there is any time left, I might also talk about some descriptions of (bi)modules, their relative tensor products, and Morita (∞,2)-categories.

• Marcy Robertson (University of Melbourne): Infinity Operads and Variations on Grothendieck-Teichmüller Theory

Abstract: This class introduces the theory of cyclic and modular ∞-operads and explores their applications in topology, geometry, and the Grothendieck–Teichmüller program. The first two talks develop the foundations of cyclic and modular ∞-operads, following the framework introduced in joint work with Hackney and Yau. No prior knowledge is required of cyclic and modular operads, as we will first introduce the classical definitions, highlighting examples from low-dimensional topology, before passing to homotopical refinements. The main goal is to introduce examples of cyclic and modular ∞-operads which arise in profinite homotopy theory and algebraic geometry.

The final two talks focus on applications of modular and cyclic ∞-operads to the Grothendieck–Teichmüller program. We review some basics of the mapping class groups, the Hatcher–Thurston complex, Oda’s work on étale homotopy types, and the operadic and modular descriptions of the Grothendieck–Teichmüller group and related subgroups. The final talk will describe the rational, or prounipotent, versions of this story,  mentioning connections to the Kashiwara–Vergne problem and knot theory.

• Tashi Walde (University of Regensburg): Assembly of Constructible Factorization Algebras

Abstract: The aim of this course is to provide a mostly self-contained introduction to the theory of (constructible) factorization algebras on stratified manifolds. Factorization algebras are multiplicative versions of (co)sheaves that appear both in quantum physics and in the study of higher algebraic structures. We employ the language of infinity-operads, which you will learn in parallel from Rune Haugseng's course. I will introduce a collection of basic tools for working with factorization algebras. The final goal is to understand gluing and assembly, answering the question of how a (constructible) factorization algebra on a stratified manifold can be reconstructed from its restriction to simpler pieces.

This course is based on joint work with Eilind Karlsson and Claudia Scheimbauer.

The talks are in HCØ Auditorium 4. 

Registration has closed.

Organisers: Jonathan Clivio, Isaac Moselle, Azélie Picot, Jan Steinebrunner, Nathalie Wahl

For inquiries, please contact Azélie Picot <azpi@math.ku.dk>.

Admin: Jan Tapdrup <jt@math.ku.dk>