Higher Categories and Applications

The inaugural two-week program of IAS / South America, Higher Categories and Applications, will take place at two sites: the Universidad de Buenos Aires (July 5–9, 2027) and the Universidad Nacional de La Plata (July 12–16, 2027).

Higher category theory gives a flexible language for working with objects "up to equivalence," generalizing the notion of homotopy in topology. Over the past two decades, it has featured in significant applications in many areas of mathematics, including homotopy theory, representation theory, algebraic and symplectic geometry, and number theory. This program brings together leading researchers, senior and junior, who have both laid the foundations of the subject and applied it creatively elsewhere in mathematics, and aims to introduce participants to its core concepts and modern research directions. 

Format at a Glance

Online Prequel Lecture Series

In June 2027, before the in-person minicourse series in Argentina begins, accepted participants will complete three courses online to acquire the necessary background. This will comprise four 60-minute lectures per minicourse, accessed live via Zoom or viewed asynchronously.

Lecturers and Minicourses

• Guido Arnone — IMAS, Universidad de Buenos Aires
  Homological Algebra and Infinity Categories
• Carlos Di Fiore — Universidad de Buenos Aires
  Background on Sheaf Theory
• Devarshi Mukherjee — University of Aberdeen
  Homotopical Algebra in Functional Analytic Settings

In-Person Lecture Series

The program runs for two weeks, Monday to Friday. Unscheduled time is available on the afternoon of Friday, July 9—Argentina’s Día de la Independencia—and on the afternoon of Wednesday, July 14. Weekends are also unscheduled.

Each minicourse includes four lectures concentrated within a single week.

• Monday–Friday Teaching Blocks: 9:00–10:30, 11:00–12:30, 14:00–15:30, 16:00–17:30
• Coffee Breaks: 10:30–11:00, 15:30–16:00
• Lunch: 12:30–14:00

Week 1: July 5–9, 2027, Universidad de Buenos Aires

• Benjamin Antieau — Northwestern University
  Filtrations and D-Modules: This course will explain the role of filtrations in derived algebraic geometry, touching on connections to spectral sequences, de Rham cohomology and infinitesimal cohomology and their universal properties, Gauss-Manin connections, derived foliations (after Toën-Vezzosi), and the theory of D-modules.
• Julie Bergner — University of Virginia
  Higher Categories and Higher Segal Spaces: Segal spaces have become a foundational framework for modeling the structure of a category up to homotopy. The addition of a completeness condition allows them to model $(\infty,1)$-categories, and this approach can be generalized to multisimplicial and $\Theta_n$-models for $(\infty,n)$-categories. On the other hand, they can be generalized to 2-Segal spaces, in which composition need not always exist or be unique, even up to homotopy, yet are still associative. These structures end up appearing in a surprising number of different areas of mathematics. In this minicourse, we'll introduce Segal spaces and these two different kinds of generalization, as well as some of their applications.
• Kobi Kremnitzer — University of Oxford
  A Bornological Approach to Derived Analytic and Differential Geometry (abstract pending)
• Ishan Levy — Institute for Advanced Study, Princeton, NJ
  Periodic Phenomena in Stable Homotopy Theory: Stable homotopy theory plays a major role in many parts of mathematics ranging from geometric topology to algebraic K-theory. This course will describe a modern perspective on the subject as a universal setting for linear algebra, then delve into periodic phenomena in stable homotopy theory, focusing on the first nontrivial periodicity, which is captured by topological K-theory.

Week 2: July 12–16, 2027, Universidad Nacional de La Plata

• Alexander Efimov — Steklov Mathematical Institute, Moscow
  Localizing Invariants and Dualizable Categories: This is an overview of the theory of localizing invariants of dualizable categories based on recent papers and including various results on the algebra of dualizable categories (including the analysis of inverse limits and internal Hom), applications of these results to the computation of localizing invariants in interesting examples, and the study of the category of localizing (noncommutative) motives. The examples include the categories of sheaves on locally compact Hausdorff spaces, their non-standard versions such as completed cosheaves, and also (two versions of) the categories of nuclear solid modules associated with an adic ring. Will include how this theory allows for proving the strongest version of pro-cdh descent for an arbitrary \omega_1-finitary localizing invariant.
• Angélica Osorno — Reed College
  Applications of 2-Categorical Techniques to (Equivariant) Homotopy Theory: It is common to take categorical structures as input for constructions of topological spaces with desired homotopical properties, for example, when constructing K-theory. Since (small) categories, functors and natural transformations form a 2-category, one often needs to use 2-categorical techniques. In this lecture series we will discuss some standard 2-categorical techniques often used in homotopy theory. We will see these techniques applied to equivariant homotopy theory, in particular to constructions of equivariant spectra with desired properties.
• Maxime Ramzi — University of Münster
  Localizing Motives and Geometric Topology: Localizing invariants, defined by Blumberg, Gepner, and Tabuada, axiomatize certain key properties of the K-theory or Hochschild homology of stable categories. There is a universal localizing invariant, with values in localizing motives; this category serves as a useful formal tool to study K-theory and organize many of its structural features. In this lecture course, I will introduce localizing motives, discuss their basic features, and explain some applications to geometric topology when specializing to sheaves of categories. 
• Germán Stefanich — Max-Planck-Institut for Mathematics
  Higher Algebraic Geometry: The goal of this minicourse will be to explain an extension of algebraic geometry which is built not out of spectra of commutative rings, but spectra of symmetric monoidal higher categories. The resulting world of geometry contains classical algebraic varieties, but also provides a home for various new and interesting objects which are not available through more classical means. The course will feature a mixture of general theory and examples; in particular, we will discuss how this framework allows one to geometrize general six-functor formalisms, and discuss a categorified form of Galois theory with connections to topological field theory and the stable homotopy groups of spheres. Based on joint work with Peter Scholze.