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Summer Graduate School

New Directions in Representation Theory (AMSI and U. of Hawaii, Hilo) June 19, 2022 - July 01, 2022
Parent Program: --
Location: University of Hawaii, Hilo
Organizers Angela Coughlin (Australian Mathematical Sciences Institute), Joseph Grotowski (University of Queensland), Tim Marchant (Australian Mathematical Sciences Institute), Ole Warnaar (University of Queensland), Geordie Williamson (University of Sydney)
Lecturer(s)

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Speaker(s)

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Description

This school is offered in partnership with the Australian Mathematical Sciences Institute and the University of Hawaii, Hilo.

MSRI-supported students will participate from the University of Hawaii in Hilo.

Representation Theory has undergone a revolution in recent years, with the development of what is now known as higher representation theory. In particular, the notion of categorification has led to the resolution of many problems previously considered to be intractable.

The school will begin by providing students with a brief but thorough introduction to what could be termed the “bread and butter of modern representation theory”, i.e., compact Lie groups and their representation theory; character theory; structure theory of algebraic groups.

We will then continue on to a number of more specialized topics. The final mix will depend on discussions with the prospective lecturers, but we envisage such topics as:

• modular representation theory of finite groups (blocks, defect groups, Broué’s conjecture);

• perverse sheaves and the geometric Satake correspondence;

• the representation theory of real Lie groups.

Suggested prerequisites

Prerequisites in algebra and representation theory is the material covered in the following texts (or equivalent):

Dummit and Foote, Abstract Algebra
In particular:

  • Part I (Group theory)
  • Part II (Ring theory)
  • Part III (Modules and vector spaces)
  • Part V (Introduction to commutative rings, algebraic geometry, and homological algebra)
  • Part VI (Introduction to the representation theory of finite groups)


Atiyah and Macdonald, Introduction to Commutative Algebra
In particular:

  • Chapter 1 (Rings and ideals)
  • Chapter 2 (Modules)
  • Chapter 3 (Rings and modules of fractions)
  • Chapter 4 (Primary decomposition)
  • Chapter 5 (Integral dependence and valuations)
  • Chapter 6 (Chain conditions)
  • Chapter 7 (Noetherian rings)


James and Liebeck, Representations and Characters of Groups
In particular: 

  • Chapter 13 (Characters)
  • Chapter 16 (Character tables and orthogonality relations)
  • Chapter 19 (Tensor products)
  • Chapter 29 (Permutations and characters)

 

Useful, but non-essential additional reading material may be found in

  • James and Kerber, The Representation Theory of the Symmetric Group
  • Mathas, Iwahori-Hecke Algebras and Schur Algebras of the Symmetric Group
  • Alperin, Local representation theory
  • Humphreys, Introduction to Lie Algebras and Representation Theory
  • Springer, Linear algebraic groups


Any material needed from these additional sources during the lectures and tutorials will be recalled, and is not assumed knowledge.

For eligibility and how to apply, see the Summer Graduate Schools homepage

Due to the small number of students supported by MSRI, only one student per institution will be funded by MSRI.

Keywords and Mathematics Subject Classification (MSC)
Tags/Keywords
  • modular representation theory

  • character theory

  • homological algebra

  • symmetric groups

  • Algebraic groups

  • root systems

  • Lusztig’s conjecture

  • higher representation theory

Primary Mathematics Subject Classification
Secondary Mathematics Subject Classification
Schedule, Notes/Handouts & Videos
Show Schedule, Notes/Handouts & Videos
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Jun 19, 2022
Sunday
12:30 PM - 01:30 PM
  Registration and Opening Ceremony
01:30 PM - 02:30 PM
  Introduction to Linear Algebraic Groups
Ting Xue (University of Melbourne)
02:30 PM - 03:00 PM
  Tea Break
03:00 PM - 04:00 PM
  Representation Zeta Functions
Uri Onn (The Australian National University)
04:00 PM - 05:00 PM
  Problem Session 1
05:00 PM - 06:00 PM
  Office Hour
06:00 PM - 07:00 PM
  Dinner
07:00 PM - 07:30 PM
  Tea Break
07:30 PM - 09:00 PM
  Problem Session 2
Jun 20, 2022
Monday
01:30 PM - 02:30 PM
  Introduction to Linear Algebraic Groups
Ting Xue (University of Melbourne)
02:30 PM - 03:00 PM
  Tea Break
03:00 PM - 04:00 PM
  Representation Zeta Functions
Uri Onn (The Australian National University)
04:00 PM - 05:00 PM
  Problem Session 1
05:00 PM - 06:00 PM
  Office Hour
06:00 PM - 07:00 PM
  Dinner
07:00 PM - 07:30 PM
  Tea Break
07:30 PM - 09:00 PM
  Problem Session 2
09:00 PM - 09:30 PM
  Careers in Maths Presentation
Jun 21, 2022
Tuesday
01:30 PM - 02:30 PM
  Introduction to Linear Algebraic Groups
Ting Xue (University of Melbourne)
02:30 PM - 03:00 PM
  Tea Break & Group Photo
03:00 PM - 04:00 PM
  Representation Zeta Functions
Uri Onn (The Australian National University)
04:00 PM - 05:00 PM
  Problem Session 1
05:00 PM - 06:00 PM
  Office Hour
06:00 PM - 07:00 PM
  Dinner
07:00 PM - 07:30 PM
  Tea Break
07:30 PM - 09:00 PM
  Problem Session 2
Jun 22, 2022
Wednesday
01:30 PM - 02:30 PM
  Introduction to Linear Algebraic Groups
Ting Xue (University of Melbourne)
02:30 PM - 03:00 PM
  Tea Break
03:00 PM - 04:00 PM
  Representation Zeta Functions
Uri Onn (The Australian National University)
04:00 PM - 05:00 PM
  Problem Session 1
05:00 PM - 06:00 PM
  Office Hour
06:00 PM - 07:00 PM
  Dinner
07:00 PM - 07:30 PM
  Tea Break
Jun 23, 2022
Thursday
01:30 PM - 02:30 PM
  Introduction to Linear Algebraic Groups
Ting Xue (University of Melbourne)
02:30 PM - 03:00 PM
  Tea Break
03:00 PM - 04:00 PM
  Representation Zeta Functions
Uri Onn (The Australian National University)
04:00 PM - 05:00 PM
  Problem Session 1
05:00 PM - 06:00 PM
  Office Hour
06:00 PM - 07:00 PM
  Dinner
07:00 PM - 07:30 PM
  Tea Break
07:30 PM - 09:00 PM
  Problem Session 2
Jun 26, 2022
Sunday
01:30 PM - 02:30 PM
  Kazhdan-Lusztig Polynomials: Representation, Geometry and Combinatorics
Geordie Williamson (University of Sydney)
02:30 PM - 03:00 PM
  Tea Break
03:00 PM - 04:00 PM
  Triangulations, Rigid Motions and Applications to Representation Theory
Asilata Bapat (University of Chicago)
04:00 PM - 05:00 PM
  Problem Session 1
05:00 PM - 06:00 PM
  Office Hour
06:00 PM - 07:00 PM
  Dinner
07:00 PM - 07:30 PM
  Tea Break
07:30 PM - 09:00 PM
  Problem Session 2
Jun 27, 2022
Monday
01:30 PM - 02:30 PM
  Kazhdan-Lusztig Polynomials: Representation, Geometry and Combinatorics
Geordie Williamson (University of Sydney)
02:30 PM - 03:00 PM
  Tea Break
03:00 PM - 04:00 PM
  Triangulations, Rigid Motions and Applications to Representation Theory
Asilata Bapat (University of Chicago)
04:00 PM - 05:00 PM
  Problem Session 1
05:00 PM - 06:00 PM
  Office Hour
06:00 PM - 07:00 PM
  Dinner
07:00 PM - 07:30 PM
  Tea Break
07:30 PM - 09:00 PM
  Problem Session 2
Jun 28, 2022
Tuesday
01:30 PM - 02:30 PM
  Kazhdan-Lusztig Polynomials: Representation, Geometry and Combinatorics
Geordie Williamson (University of Sydney)
02:30 PM - 03:00 PM
  Tea Break
03:00 PM - 04:00 PM
  Triangulations, Rigid Motions and Applications to Representation Theory
Asilata Bapat (University of Chicago)
04:00 PM - 05:00 PM
  Problem Session 1
05:00 PM - 06:00 PM
  Office Hour
06:00 PM - 07:00 PM
  Dinner
07:00 PM - 07:30 PM
  Tea Break
07:30 PM - 09:00 PM
  Problem Session 2
Jun 29, 2022
Wednesday
01:30 PM - 02:30 PM
  Kazhdan-Lusztig Polynomials: Representation, Geometry and Combinatorics
Geordie Williamson (University of Sydney)
02:30 PM - 03:00 PM
  Tea Break
03:00 PM - 04:00 PM
  Triangulations, Rigid Motions and Applications to Representation Theory
Asilata Bapat (University of Chicago)
04:00 PM - 05:00 PM
  Problem Session 1
05:00 PM - 06:00 PM
  Office Hour
06:00 PM - 07:00 PM
  Dinner
07:00 PM - 07:30 PM
  Tea Break
07:30 PM - 09:00 PM
  Problem Session 2
09:30 PM - 11:30 PM
  Public Lecture
Jun 30, 2022
Thursday
01:30 PM - 02:30 PM
  Kazhdan-Lusztig Polynomials: Representation, Geometry and Combinatorics
Geordie Williamson (University of Sydney)
02:30 PM - 03:00 PM
  Tea Break
03:00 PM - 04:00 PM
  Triangulations, Rigid Motions and Applications to Representation Theory
Asilata Bapat (University of Chicago)
04:00 PM - 05:00 PM
  Problem Session 1
05:00 PM - 06:00 PM
  Office Hour
06:00 PM - 07:00 PM
  Dinner
07:00 PM - 07:30 PM
  Tea Break
07:30 PM - 09:00 PM
  Problem Session 2