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Workshop Programme

for period 12 January - 17 April 2009

Instructional workshop

12 January - 17 April 2009

Timetable

Monday 12 January
08:30-10:00 Registration
10:00-11:00 Achar, P (Louisiana)
  Derived categories and perverse sheaves Sem 1
 

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.)

 
11:00-11:30 Coffee
11:30-12:30 Geck, M (Aberdeen)
  Hecke algebras at roots of unity Sem 1
 

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.

 
12:30-13:30 Lunch
14:00-15:00 Chuang, J (London)
  Categorification of sl(2)-modules Sem 1
 

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).

 
15:00-15:30 Tea
15:30-16:30 Discussion
16:30-17:30 Discussion
17:30-18:30 Wine Reception
18:45-19:30 Dinner
Tuesday 13 January
09:00-10:00 Ram, A (Melbourne)
  Symmetry, polynomials and quantisation Sem 1
10:00-11:00 Achar, P (Louisiana)
  Derived categories and perverse sheaves Sem 1
 

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.)

 
11:00-11:30 Coffee
11:30-12:30 Geck, M (Aberdeen)
  Hecke algebras at roots of unity Sem 1
 

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.

 
12:30-13:30 Lunch
14:00-15:00 Kleshchev, A (Oregon)
  W-algebras and Hecke algebras 1 Sem 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.

 
15:00-15:30 Tea
15:30-16:30 Discussion
16:30-17:30 Discussion
18:45-19:30 Dinner
Wednesday 14 January
09:00-10:00 Ram, A (Melbourne)
  Symmetry, polynomials and quantisation Sem 1
10:00-11:00 Achar, P (Louisiana)
  Derived categories and perverse sheaves Sem 1
11:00-11:30 Coffee
11:30-12:30 Geck, M (Aberdeen)
  Hecke algebras at roots of unity Sem 1
 

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.

 
12:30-13:30 Lunch
18:45-19:30 Dinner
Thursday 15 January
09:00-10:00 Ram, A (Melbourne)
  Symmetry, polynomials and quantisation Sem 1
10:00-11:00 Achar, P (Louisiana)
  Derived categories and perverse sheaves Sem 1
11:00-11:30 Coffee
11:30-12:30 Chuang, J (London)
  Categorification of sl(2)-modules Sem 1
12:30-13:30 Lunch
14:00-15:00 Kleshchev, A (Oregon)
  W-algebras and Hecke algebras 2 Sem 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.

 
15:00-15:30 Tea
15:30-16:30 Discussion
16:30-17:30 Discussion
18:45-19:30 Dinner
Friday 16 January
09:00-10:00 Ram, A (Melbourne)
  Symmetry, polynomials and quantisation Sem 1
10:00-11:00 Achar, P (Louisiana)
  Derived categories and perverse sheaves Sem 1
11:00-11:30 Coffee
11:30-12:30 Chuang, J (London)
  Categorification of sl(2)-modules Sem 1
12:30-13:30 Lunch
14:00-15:00 Kleshchev, A (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.

 
15:00-15:30 Tea
15:30-16:30 Discussion
16:30-17:30 Discussion
18:45-19:30 Dinner
Monday 19 January
09:00-10:00 Broué, M (Paris)
  Complex reflection groups and their associated braid groups and Hecke algebras Sem 1
10:00-11:00 Bonnafé, C (de Franche Comté)
  Introduction to Kazhdan-Lusztig theory with unequal parameters Sem 1
11:00-11:30 Coffee
11:30-12:30 Rouquier, R (Oxford)
  Higher representations of Lie algebras Sem 1
12:30-13:30 Lunch
14:00-15:00 Vogan, D (Massachusetts)
  Schubert varieties and representations of real reductive groups Sem 1
15:00-15:30 Tea
15:30-16:30 Discussion
16:30-17:30 Discussion
18:45-19:30 Dinner
Tuesday 20 January
09:00-10:00 Broué, M (Paris)
  Complex reflection groups and their associated braid groups and Hecke algebras Sem 1
10:00-11:00 Bonnafé, C (de Franche Comté)
  Introduction to Kazhdan-Lusztig theory with unequal parameters Sem 1
11:00-11:30 Coffee
11:30-12:30 Rouquier, R (Oxford)
  Higher representations of Lie algebras Sem 1
12:30-13:30 Lunch
14:00-15:00 Vogan, D (Massachusetts)
  Schubert varieties and representations of real reductive groups Sem 1
15:00-15:30 Tea
15:30-16:30 Discussion
16:30-17:30 Discussion
18:45-19:30 Dinner
Wednesday 21 January
09:00-10:00 Broué, M (Paris)
  Complex reflection groups and their associated braid groups and Hecke algebras Sem 1
10:00-11:00 Losev, I (Massachusetts)
  Finite W-algebras and their representations Sem 1
 

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.

 
11:00-11:30 Coffee
11:30-12:30 Rouquier, R (Oxford)
  Higher representations of Lie algebras Sem 1
12:30-13:30 Lunch
18:45-19:30 Dinner
Thursday 22 January
09:00-10:00 Losev, I (Massachusetts)
  Finite W-algebras and their representations Sem 1
 

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.

 
10:00-11:00 Bonnafé, C (de Franche Comté)
  Introduction to Kazhdan-Lusztig theory with unequal parameters Sem 1
11:00-11:30 Coffee
11:30-12:30 Rouquier, R (Oxford)
  Higher representations of Lie algebras Sem 1
12:30-13:30 Lunch
14:00-15:00 Vogan, D (Massachusetts)
  Schubert varieties and representations of real reductive groups Sem 1
15:00-15:30 Tea
15:30-16:30 Discussion
16:30-17:30 Discussion
18:45-19:30 Dinner
Friday 23 January
09:00-10:00 Losev, I (Massachusetts)
  Finite W-algebras and their representations Sem 1
 

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.

 
10:00-11:00 Bonnafé, C (de Franche Comté)
  Introduction to Kazhdan-Lusztig theory with unequal parameters Sem 1
11:00-11:30 Coffee
11:30-12:30 Rouquier, R (Oxford)
  Higher representations of Lie algebras Sem 1
12:30-13:30 Lunch
14:00-15:00 Vogan, D (Massachusetts)
  Schubert varieties and representations of real reductive groups Sem 1
15:00-15:30 Tea
15:30-16:30 Discussion
16:30-17:30 Discussion
18:45-19:30 Dinner
Tuesday 14 April
09:00-10:00 Cautis, S (MSRI)
  Geometry and categorification II Braid group actions in algebraic geometry
Friday 17 April
16:20-17:00 Masbaum, G (CNRS)
  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.

 

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