# Infinity-Categories, Infinity-Operads, and their Applications (18w5147)

Arriving in Oaxaca, Mexico Sunday, May 6 and departing Friday May 11, 2018

## Organizers

Gabriele Vezzosi (University of Florence)

Bertrand Toën (University of Toulouse)

David Gepner (Purdue University)

(University of Copenhagen)

## Objectives

We aim to gather mathematicians working both on more foundational aspects of the theory as well as on applications to other areas of mathematics. Because of the varied interests of the researchers involved, we expect the workshop to bring together many people who typically only have limited interactions with each other, which should lead to very fruitful discussions and collaborations and spur new developments both in foundations and applications. We are not aware that any similar conference has yet taken place, and we therefore believe that the proposed workshop would be particularly valuable. It is also clear that $\infty$-categories and $\infty$-operads are becoming increasingly interesting and useful to mathematicians in many different research communities, so the workshop would take place at a very opportune time for expanding and reimagining the foundations, envisioning new applications, and making the field more accessible to early career mathematicians.

Specifically, we currently envision talks and interactions on the following topics:
• various approaches to streamlining the foundations for
• $\infty$-categories and $\infty$-operads, making them more accessible to beginning researchers in the field as well as established researchers outside the field;
• higher or $(\infty,n)$-categorified'' versions of
• constructions from algebra such as algebraic K-theory and Brauer groups (there are hints of such in recent work of Hoyois--Scherotzke--Sibilla, Haugseng, and To\"{e}n--Vezzosi, but at the moment it is still very unclear what the correct definitions should be), and connections to chromatic homotopy theory (e.g. the redshift conjecture of Rognes);
• extensions of factorization homology to describe topological
• quantum field theories beyond the case of $E_{n}$-algebras, i.e. using enriched higher categories, related to ongoing work of Ayala--Francis--Rozenblyum;
• the appropriate $\infty$-categorical framework for describing
• categorical and algebraic structures in motivic and equivariant homotopy theory (in the case of finite groups the latter is clarified in ongoing work of Barwick--Dotto--Glasman--Nardin--Shah, but the case of compact Lie groups is still unclear);
• versions of Koszul duality that work more generally than just in
• characteristic $0$ (even in characteristic $p$ this seems to require a notion of enriched $\infty$-operads, and more generally one would hope to work integrally or even over the sphere spectrum);
• applications of linear $\infty$-categories to arithmetic
• geometry, such as vanishing cycles and matrix factorizations, trace formulas for dg-categories, and Bloch's conductor conjecture (cf. ongoing work of Blanc, Robalo, To\"en, and Vezzosi).