Lecture Notes
Lecture 1 (23-08-2019)
Ordinary differential equation over the complex plane: ordinary and Fuchsian points, fundamental solutions, associator.
Lecture 2 (04-09-2019)
Ordinary differential equation over the complex plane: some examples. Hypergeometric series.
Lecture 3 (06-09-2019)
Ordinary differential equation over the complex plane: irregular singularities, asymptotic expansions, Stokes' theory.
Lecture 4 (09-09-2019)
Several variables. Consistency equation. Differential forms. Kohno's lemma.
Lecture 5 (11-09-2019)
Example of configuration space of ordered points. Root systems - definition, rank 2 classification.
Lecture 6 (13-09-2019)
Root systems. Positive/negative roots. Simple roots. Cartan matrix and Dynkin diagram.
Lecture 7 (16-09-2019)
Root systems. Classification theorem.
Lecture 8 (18-09-2019)
Weyl group of a root system. Definition, generators and relations, exchange property.
Lecture 9 (20-09-2019)
A presentation of Weyl group. Braid groups.
Lecture 10 (23-09-2019)
Statement of Brieskorn's theorem. Towards De Concini--Procesi theory.
Lecture 11 (25-09-2019)
Maximal nested sets. Adapted bases. Affine charts labeled by maximal nested sets.
Lecture 12 (27-09-2019)
DCP associators. De Concini-Procesi wonderful model (statements only).
Lecture 13 (30-09-2019)
Lie algebras and representations. Definitions. Schur's lemma. sl(2)-representations.
Lecture 14 (02-10-2019)
sl(2) representation theory continued. Complete reducibility theorem.
Lecture 15 (04-10-2019)
Weyl group action. Simple Lie algebras from root systems.
Lecture 16 (07-10-2019)
Simple Lie algebras from root systems continued.
Lecture 17 (09-10-2019)
Finite-dimensional representations of simple Lie algebras. Harish-Chandra's theorem. Hermann Weyl's character formula. (mostly statements, some proofs).
Lecture 18 (14-10-2019)
Knizhnik-Zamolodchikov equations. Casimir equations.
Lecture 19 (16-10-2019)
Braided tensor categories.
Lecture 20 (18-10-2019)
Braided tensor categories cntd. Quasi-triangular quasi-bialgebras.
Lecture 21 (21-10-2019)
Braided tensor categories via KZ equations I (pentagon axiom).
Lecture 22 (23-10-2019)
Braided tensor categories via KZ equations II (hexagon axiom).
Lecture 23 (25-10-2019)
Quasi-triangular Hopf algebras. Quantum group of sl(2) - definition.
Lecture 24 (28-10-2019)
R-matrix of the quantum group of sl(2). Explicit formula.
Lecture 25 (30-10-2019)
Quantum group of sl(2) - proof of the intertwining equation.
Lecture 26 (01-11-2019)
Quantum group associated to root data. Definition via invariant form.
Lecture 27 (04-11-2019)
q-Serre relations. Quantum Weyl group operators.
Lecture 28 (06-11-2019)
Root vectors. Kirillov-Reshetikhin formula (statement only).
Lecture 29 (08-11-2019)
Dynamical constructions - fusion operators, dynamical Weyl group.
Lecture 30 (13-11-2019)
Deformation theory. Hochschild cohomology.
Lecture 31 (15-11-2019)
Chevalley-Eilenberg complex. Whitehead's lemma.
Lecture 32 (18-11-2019)
Towards Kohno-Drinfeld theorem: matching of the coproduct and the R-matrix.
Lecture 33 (20-11-2019)
Cobar complex. Matching the associators - end of the proof of Kohno-Drinfeld theorem.
Lecture 34 (22-11-2019)
Cohomology of cobar complex of the symmetric algebra.
Lecture 35 (25-11-2019)
Drinfeld element. Proof of the ribbon relation.