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Timetable (ALTW01)

Instructional workshop

Monday 12th January 2009 to Friday 23rd January 2009

Monday 12th January 2009
08:30 to 10:00 Registration
10:00 to 11:00 P Achar ([Louisiana])
Derived categories and perverse sheaves
Perverse sheaves are a very important and powerful tool in representation theory. In these talks, we will begin with the formalism of derived categories and t-structures, needed to define perverse sheaves. Next, we will go through the list of remarkable properties of perverse sheaves that make them "better" than ordinary sheaves, and are the source of their usefulness. Lastly, we will look at a couple of representation theory applications. (Some basic familiarity with ordinary sheaves may be a helpful prerequisite.)
INI 1
11:00 to 11:30 Coffee
11:30 to 12:30 Hecke algebras at roots of unity
We discuss various issues around non-semisimple Hecke algebras: cellular structure, James' conjecture, application to modular Harish-Chandra series for finite groups of Lie type.
INI 1
12:30 to 13:30 Lunch
14:00 to 15:00 Categorification of sl(2)-modules
The aim of this course is to introduce the Kazhdan-Lusztig theory, in the general framework of unequal parameters. We plan to talk about Lusztig's Conjectures, the asymptotic algebra, and to give a detailed account on some examples (symmetric group, type B,...). If time is left, we might talk about the connections with Cherednik algebras, or with representations of finite reductive groups, or with the geometry of the flag variety, or with the computation of decomposition matrices using the canonical bases of the Fock space (at most one of these four topics will be treated).
INI 1
15:00 to 15:30 Tea
15:30 to 16:30 Discussion INI 1
16:30 to 17:30 Discussion INI 1
17:30 to 18:30 Wine Reception
18:45 to 19:30 Dinner
Tuesday 13th January 2009
09:00 to 10:00 Symmetry, polynomials and quantisation INI 1
10:00 to 11:00 P Achar ([Louisiana])
Derived categories and perverse sheaves
Perverse sheaves are a very important and powerful tool in representation theory. In these talks, we will begin with the formalism of derived categories and t-structures, needed to define perverse sheaves. Next, we will go through the list of remarkable properties of perverse sheaves that make them "better" than ordinary sheaves, and are the source of their usefulness. Lastly, we will look at a couple of representation theory applications. (Some basic familiarity with ordinary sheaves may be a helpful prerequisite.)
INI 1
11:00 to 11:30 Coffee
11:30 to 12:30 Hecke algebras at roots of unity
We discuss various issues around non-semisimple Hecke algebras: cellular structure, James' conjecture, application to modular Harish-Chandra series for finite groups of Lie type.
INI 1
12:30 to 13:30 Lunch
14:00 to 15:00 A Kleshchev ([Oregon])
W-algebras and Hecke algebras 1
We discuss a new presentation for blocks of cyclotomic Hecke algebras which allows to grade these blocks. This opens up a way for studying graded representation theory of cyclotomic Hecke algebras. We give emphasis to the recent result of Khovanov-Lauda, Rouquier, Brundan-Kleshchev and Kleshchev-Ram.
INI 1
15:00 to 15:30 Tea
15:30 to 16:30 Discussion INI 1
16:30 to 17:30 Discussion INI 1
18:45 to 19:30 Dinner
Wednesday 14th January 2009
09:00 to 10:00 Symmetry, polynomials and quantisation INI 1
10:00 to 11:00 P Achar ([Louisiana])
Derived categories and perverse sheaves
INI 1
11:00 to 11:30 Coffee
11:30 to 12:30 Hecke algebras at roots of unity
We discuss various issues around non-semisimple Hecke algebras: cellular structure, James' conjecture, application to modular Harish-Chandra series for finite groups of Lie type.
INI 1
12:30 to 13:30 Lunch
18:45 to 19:30 Dinner
Thursday 15th January 2009
09:00 to 10:00 Symmetry, polynomials and quantisation INI 1
10:00 to 11:00 P Achar ([Louisiana])
Derived categories and perverse sheaves
INI 1
11:00 to 11:30 Coffee
11:30 to 12:30 Categorification of sl(2)-modules INI 1
12:30 to 13:30 Lunch
14:00 to 15:00 A Kleshchev ([Oregon])
W-algebras and Hecke algebras 2
We discuss a new presentation for blocks of cyclotomic Hecke algebras which allows to grade these blocks. This opens up a way for studying graded representation theory of cyclotomic Hecke algebras. We give emphasis to the recent result of Khovanov-Lauda, Rouquier, Brundan-Kleshchev and Kleshchev-Ram.
INI 1
15:00 to 15:30 Tea
15:30 to 16:30 Discussion INI 1
16:30 to 17:30 Discussion INI 1
18:45 to 19:30 Dinner
Friday 16th January 2009
09:00 to 10:00 Symmetry, polynomials and quantisation INI 1
10:00 to 11:00 P Achar ([Louisiana])
Derived categories and perverse sheaves
INI 1
11:00 to 11:30 Coffee
11:30 to 12:30 Categorification of sl(2)-modules INI 1
12:30 to 13:30 Lunch
14:00 to 15:00 A Kleshchev ([Oregon])
W-algebras and Hecke algebras 3
We discuss a new presentation for blocks of cyclotomic Hecke algebras which allows to grade these blocks. This opens up a way for studying graded representation theory of cyclotomic Hecke algebras. We give emphasis to the recent result of Khovanov-Lauda, Rouquier, Brundan-Kleshchev and Kleshchev-Ram.
15:00 to 15:30 Tea
15:30 to 16:30 Discussion INI 1
16:30 to 17:30 Discussion INI 1
18:45 to 19:30 Dinner
Monday 19th January 2009
09:00 to 10:00 M Broué ([Paris])
Complex reflection groups and their associated braid groups and Hecke algebras
INI 1
10:00 to 11:00 Introduction to Kazhdan-Lusztig theory with unequal parameters INI 1
11:00 to 11:30 Coffee
11:30 to 12:30 Higher representations of Lie algebras INI 1
12:30 to 13:30 Lunch
14:00 to 15:00 D Vogan ([Massachusetts])
Schubert varieties and representations of real reductive groups
INI 1
15:00 to 15:30 Tea
15:30 to 16:30 Discussion INI 1
16:30 to 17:30 Discussion INI 1
18:45 to 19:30 Dinner
Tuesday 20th January 2009
09:00 to 10:00 M Broué ([Paris])
Complex reflection groups and their associated braid groups and Hecke algebras
INI 1
10:00 to 11:00 Introduction to Kazhdan-Lusztig theory with unequal parameters INI 1
11:00 to 11:30 Coffee
11:30 to 12:30 Higher representations of Lie algebras INI 1
12:30 to 13:30 Lunch
14:00 to 15:00 D Vogan ([Massachusetts])
Schubert varieties and representations of real reductive groups
INI 1
15:00 to 15:30 Tea
15:30 to 16:30 Discussion INI 1
16:30 to 17:30 Discussion INI 1
18:45 to 19:30 Dinner
Wednesday 21st January 2009
09:00 to 10:00 M Broué ([Paris])
Complex reflection groups and their associated braid groups and Hecke algebras
INI 1
10:00 to 11:00 I Losev ([Massachusetts])
Finite W-algebras and their representations

A (finite) W-algebra is a certain associative algebra constructed from a semisimple Lie algebra and its nilpotent element. The main reason why they are interesting is their relation to the representation theory of universal enveloping algebras.

In this course I am going to explain two different definitions of W-algebras: by Hamiltonian reduction (Premet, Gan-Ginzburg) and deformation quantization (I.L). Then I am going to explain category equivalence theorems relating representations of W-algebras and universal enveloping algebras and describe relation between primitive ideals.

INI 1
11:00 to 11:30 Coffee
11:30 to 12:30 Higher representations of Lie algebras INI 1
12:30 to 13:30 Lunch
18:45 to 19:30 Dinner
Thursday 22nd January 2009
09:00 to 10:00 I Losev ([Massachusetts])
Finite W-algebras and their representations

A (finite) W-algebra is a certain associative algebra constructed from a semisimple Lie algebra and its nilpotent element. The main reason why they are interesting is their relation to the representation theory of universal enveloping algebras.

In this course I am going to explain two different definitions of W-algebras: by Hamiltonian reduction (Premet, Gan-Ginzburg) and deformation quantization (I.L). Then I am going to explain category equivalence theorems relating representations of W-algebras and universal enveloping algebras and describe relation between primitive ideals.

INI 1
10:00 to 11:00 Introduction to Kazhdan-Lusztig theory with unequal parameters INI 1
11:00 to 11:30 Coffee
11:30 to 12:30 Higher representations of Lie algebras INI 1
12:30 to 13:30 Lunch
14:00 to 15:00 D Vogan ([Massachusetts])
Schubert varieties and representations of real reductive groups
INI 1
15:00 to 15:30 Tea
15:30 to 16:30 Discussion INI 1
16:30 to 17:30 Discussion INI 1
18:45 to 19:30 Dinner
Friday 23rd January 2009
09:00 to 10:00 I Losev ([Massachusetts])
Finite W-algebras and their representations

A (finite) W-algebra is a certain associative algebra constructed from a semisimple Lie algebra and its nilpotent element. The main reason why they are interesting is their relation to the representation theory of universal enveloping algebras.

In this course I am going to explain two different definitions of W-algebras: by Hamiltonian reduction (Premet, Gan-Ginzburg) and deformation quantization (I.L). Then I am going to explain category equivalence theorems relating representations of W-algebras and universal enveloping algebras and describe relation between primitive ideals.

INI 1
10:00 to 11:00 Introduction to Kazhdan-Lusztig theory with unequal parameters INI 1
11:00 to 11:30 Coffee
11:30 to 12:30 Higher representations of Lie algebras INI 1
12:30 to 13:30 Lunch
14:00 to 15:00 D Vogan ([Massachusetts])
Schubert varieties and representations of real reductive groups
INI 1
15:00 to 15:30 Tea
15:30 to 16:30 Discussion INI 1
16:30 to 17:30 Discussion INI 1
18:45 to 19:30 Dinner
Tuesday 14th April 2009
09:00 to 10:00 Geometry and categorification II Braid group actions in algebraic geometry
Friday 17th April 2009
16:20 to 17:00 Integral lattices in TQFT and integral modular categories
The SO(3)-TQFT at an odd prime has a natural integral structure. It means that all matrix coefficients of the mapping class group representations in this theory are algebraic integers. I will discuss the structure of the resulting "integral modular functor". This is joint work with Patrick Gilmer.
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons