Mini-courses

Pramod Achar: Infinity categories

This course will present an overview of the basic definitions and theorems in the theory of infinity-categories, in the incarnation developed by Lurie following Boardman-Vogt and Joyal.

Expected background: Familiarity with the concepts listed below.
1. From ordinary category theory: categories, functors, natural transformations, essentially surjective functors, fully faithful functors, Yoneda's lemma, limits and colimits. Optional (but useful for lecture 3): monoidal categories.
2. From topology: homotopy for continuous maps, fundamental group, higher homotopy groups.
3. From homological algebra (optional, but useful for lecture 4): abelian categories, short exact sequences, exact functors, derived categories and derived functors.

Approximate outline of the course:
1. Simplicial sets, Kan complexes, examples of infinity-categories
2. Mapping spaces; fully faithful and essentially surjective functors
3. The straightening-unstraightening theorem and applications: monoidal categories; Yoneda's lemma
4. Limits and colimits; stable infinity-categories; Lurie tensor product

Exercise Sheet (for exercise session on Wednesday 16:45 - 17:30 in the lecture room).

Simon Pepin Lehalleur: Motivic Sheaves

This course will be an introduction to categories of motivic sheaves and their functoriality, with a view towards the ongoing applications to geometric representation theory and the Langlands program which are at the heart of ReMoLD. Motives and motivic sheaves are algebro-geometric objects which refine many cohomological and sheaf-theoretic invariants of algebraic varieties, and clarify their relation with algebraic cycles and algebraic K-theory. The main tool for working with motivic sheaves is their rich functoriality ("six-functor formalism") which parallels the functoriality and duality properties of ordinary sheaves on complex algebraic varieties. In the modern formulation of the subject, pioneered by Morel and Voevodsky, motives embed into the wider world of motivic homotopy theory. This gives rise to many "flavours" of motivic sheaf theories, some of which have already found applications to representation theory.

Essential background:
- Basic category theory (including monoidal categories)
- Homological algebra (including derived categories); I will briefly review triangulated categories in Lecture 1.
- Basic homotopy theory
- Sheaf theory and sheaf cohomology
- Étale topology and étale cohomology
Other useful but not essential topics:
- stable homotopy theory
- Hodge theory
- algebraic cycles/Chow groups
- algebraic K-theory
- algebraic stacks

I will not assume familiarity with infinity-categories since they will be introduced in parallel in Achar's lectures. I will largely stick to triangulated categories (while pointing out some of their insufficiencies along the way) and only in the last lecture explain how infinity-categories are useful - indeed essential - to the latest developments of the subject.

Lecture 1: Overview and constructions
- Motivation from geometric representation theory
- Some key properties of the singular (co)homology of complex varieties
- Construction of two central examples of motivic sheaf categories: DA^et(S) (étale motivic sheaves) and SH(S) (stable motivic homotopy types).
- Motives over a field and some basic computations

Lecture 2: Six-functor formalism for motivic sheaves
- Six-functor for ordinary sheaves on complex varieties
- Six-functor formalism for motivic sheaves
- Constructibility and Verdier duality
- Realization functors
- Orientations, Chern classes and intersection theory

Lecture 3: Motivic t-structures and weight structures
- Conjectures on algebraic cycles and the motivic t-structure
- Motivic weight structures
- Stratifications and gluing
- Stratified Tate motives and geometric representation theory

Lecture 4: Infinity-categorical techniques for motivic sheaves
- Motivic ring spectra, module categories and realizations
- Descent and equivariance
- Motivic sheaves on stacks
- Six-functor formalisms and correspondences

References:
- Ayoub's ICM 2014 survey.
- Cisinski-Deglise's book.
- Cisinski's lecture notes on h-motives.
- Ayoub's survey of motivic conjectures.
- Gallauer's survey on six-functor formalisms.
- Scholze's lecture notes on six functor formalisms.
- Richarz-Scholbach's paper The intersection motive of the moduli stack of shtukas, especially Section 2.

Simon Riche: Representation theory of reductive algebraic groups

In these lectures I will explain the most central results in the representation theory of reductive algebraic groups over algebraically closed fields. The most relevant references for this topic are:

- J.C. Jantzen, "Representations of algebraic groups - Second edition", AMS.

- J.S. Milne, "Algebraic groups - The theory of group schemes of finite type over a field", Cambridge University Press

I will assume that the audience is familiar with the structure theory of reductive groups over algebraically closed fields (Borel subgroups, maximal tori, roots, etc.), although I will recall the properties I will use. For a quick overview of this topic, illustrated by examples, see these notes by Makisumi.

I will follow the following plan:

- Lecture 1: Reminder on representations of algebraic groups over fields

- Lecture 2: Classification of simple representations of reductive algebraic groups

- Lecture 3: General results: Kempf's vanishing theorem, Borel-Weil-Bott theorem, semisimplicity, Weyl's character formula.

- Lecture 4: The case of positive characteristic: Steinberg's theorem, linkage principle, translation functors.

Olivier Taïbi: Shimura varieties

This course will be an introduction to Shimura varieties from the point of view of the Langlands program:
- Shimura varieties over C, modular forms and relation with automorphic forms
- canonical models over number fields
- cohomology of "automorphic" local systems on Shimura varieties: Hecke operators, Matsushima's formula, Kottwitz' conjecture on the cohomology of l-adic local systems

Abstracts

Jens Eberhardt: Gluing together representations and flag manifolds

This talk is about the fascinating interplay of representation theory, geometry and combinatorics. We will study the intricate ways in which representations and flag manifolds (of reductive groups and Lie algebras) are glued together. Our main result is a combinatorial description of this gluing data in terms of certain pictures, which we call Fukaya diagrams. We will illustrate this with many explicit examples and invite the audience experiment with these diagrams themselves. The talk is based on joint work with Catharina Stroppel.

Jakob Scholbach: Dualizability

In this purely expository talk, I will recall the definition and basic examples and properties of dualizable objects and traces. I will then illustrate how this concept can be exploited in the context of various cohomology theories.