The first half of the course is based on this paper (which is still in a preliminary version). The second half is based on The Hodge theory of Soergel bimodules.

The happenings of 1979. The miracle of KL polynomials. Arbitrary Coxeter groups. The miracle of the localisation proof. Soergel's dream of an algebraic explanation... the deepening mystery of positivity.

The Coxeter complex. The Hecke algebra of a Coxeter group. The presentation using standard generators. The standard basis. The Kazhdan-Lusztig basis and polynomials. The Kazhdan-Lusztig presentation. Products of Kazhdan-Lusztig generators and the defect formula. Slides.

Invariant theory for finite reflection groups. Bimodules and monoidal categories. The category of Soergel bimodules. Singular Soergel bimodules. First examples.

Higher algebra. Drawing adjunctions, cyclicity etc. Example: 2-groupoids. The Coxeter groupoid. The generalized Zamolodchikov relations.

Monday exercises

Standard bimodules. Support filtrations. Soergel's hom formula. Statement of Soergel's categorification theorem. Localization. Discussion.

Starting to draw Soergel bimodules. Soergel bimodules in rank 2. Jones-Wenzl projectors, connections to the Temperley-Lieb algebra and quantum groups. Categorification of the Kazhdan-Lusztig presentation.

Generators and relations in general. Light leaves morphisms as a categorification of the defect formula. Double leaves give a basis for morphisms.

to draw Bott-Samelson bimodules, Soergel bimodules. Intersection forms. Discussion.

Tuesday Exercises

The cellular structure. A discussion of idempotent lifting. Generators and relations proof of Soergel's categorification theorem. Examples of intersection forms and idempotents.

Review of the (real) Hodge theory of smooth projective algebraic varieties. A discussion of the weak and hard Lefschetz theorems. Lefschetz operators, Lefschetz forms and the Hodge-Riemann bilinear relations. Tricks establishing the Lefschetz package. The weak-Lefschetz substitute.

Statement of the results and outline of the methods. The embedding theorem, the limit argument. The absence of the weak Lefschetz theorem.

Wednesday Exercises

The homotopy category of Soergel bimodules. Minimal complexes. Rouquier complexes. Examples.

The perverse filtration on Soergel bimodules. The diagonal miracle. Factoring the Lefschetz operator. Hard Lefschetz.

Review of Verma modules, category $\OC$ and its block decomposition by central character. Statement of the Kazhdan-Lusztig conjecture. Soergel's functor $\mathbb{V}$. Soergel's conjecture implies the Kazhdan-Lusztig conjecture.

Thursday Exercises

(This talk ended up not getting given, as we discussed exercises instead!) Definition of Hecke algebras with unequal parameters. Equivariant

Lusztig's conjecture. Intersection forms. The

Categorifying the braid group. Example of Rouquier complexes. Generators and relations for strict braid group actions. Deligne's theorem and the EW version.

A discussion of ridiculous titles. Algebraizing the geometric Satake equivalence. Quantizing it in type

Some research problems