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Summer Graduate School

Representation stability June 24, 2019 - July 05, 2019
Parent Program: --
Location: MSRI: Simons Auditorium, Atrium
Organizers Thomas Church (Stanford University), LEAD Andrew Snowden (University of Michigan), Jenny Wilson (University of Michigan)
Lecturer(s)

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Speaker(s)

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Description
Image
An illustration of an adaptation of Quillen's classical homological stability spectral sequence argument

This summer school will give an introduction to representation stability, the study of algebraic structural properties and stability phenomena exhibited by sequences of representations of finite or classical groups -- including sequences arising in connection to hyperplane arrangements, configuration spaces, mapping class groups, arithmetic groups, classical representation theory, Deligne categories, and twisted commutative algebras.  Representation stability incorporates tools from commutative algebra, category theory, representation theory, algebraic combinatorics, algebraic geometry, and algebraic topology. This workshop will assume minimal prerequisites, and students in varied disciplines are encouraged to apply. 

Suggested prerequisites:
Representations of finite groups over C and the classification of Sn-irreps
Representation Theory of Finite Groups:
- Fulton--Harris, "Representation Theory, A first course", Chapters 1-3
- Serre, "Linear representations of finite groups", Parts I and II
Representations of S_n:
- Fulton--Harris, "Representation Theory, A first course", Chapter 4
- James, "The representation theory of symmetric groups"

Commutative algebra (Noetherian rings, tensor product, free resolutions)
Tensor products:
- Dummit--Foote, "Abstract Algebra", Chapter 10.4
- Atiyah--MacDonald, "Introduction to Commutative Algebra", Chapter 2
Noetherian rings:
- Dummit--Foote, "Abstract Algebra", Chapter 15.1
- Atiyah--MacDonald, "Introduction to Commutative Algebra", Chapter 6-7

Gröbner bases
- Dummit--Foote, "Abstract Algebra", Chapter 9.5-9.6
- Cox--Little--O'Shea, "Ideals, Varieties, and Algorithms", Chapters 2.1-2.6
- Eisenbud, "Commutative algebra" Chapter 15

Representation theory of GLnC
- Henderson, "Representations of Lie Algebras", whole book
- Fulton--Harris, "Representation Theory, A first course", Chapter 15

Homological algebra (Tor, Ext, derived functors)
- Dummit--Foote, "Abstract Algebra", Chapter 10.5, 17.1
- Rotman, "An Introduction to Homological Algebra", Chapter 6.1-6.2 & 7.1-7.2
- Weibel, "An introduction to homological algebra", Chapters 2, 3

Topology (homology and cohomology of spaces and/or groups)
Spaces:
- Hatcher,  "Algebraic Topology", Chapters 2-3
Groups:
- Brown, "Cohomology of Groups", Chapters I-III
- Dummit--Foote, "Abstract Algebra", Chapter 17.2
- Weibel, "An introduction to homological algebra", Chapters 6, 7

Symmetric functions
- MacDonald, "Symmetric Functions and Hall Polynomials", Chapter I
- Stanley, "Enumerative Combinatorics, Vol 2", Chapter 7

Category theory
- Mac Lane, "Categories for the Working Mathematician", Chapter
I.1-I.5, II.1-II.4, & IV.1-IV.4.

Backgroud articles:
- Church--Ellenberg--Farb, "FI-modules and stability for
representations of symmetric groups"
- Sam--Snowden, "Introduction to twisted commutative algebras"
- Draisma, "Noetherian up to symmetry"

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

Keywords and Mathematics Subject Classification (MSC)
Tags/Keywords
  • Representation stability

  • Representation theory

  • homological stability

  • functor categories

  • Gröbner methods

  • Noetherian

  • group cohomology

  • Schur-Weyl duality

  • twisted commutative algebra

  • FI-module

  • VI-module

  • VIC-module

  • pure braid groups

  • hyperplane arrangements

  • mapping class groups

  • Moduli space

  • Torelli groups

  • configuration spaces

  • congruence subgroups

  • Deligne categories

Primary Mathematics Subject Classification
Secondary Mathematics Subject Classification
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Schedule, Notes/Handouts & Videos
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Jun 24, 2019
Monday
08:40 AM - 09:00 AM
  Introduction to MSRI
09:00 AM - 09:45 AM
  Putman Lectures Series (Talk #1)
Andrew Putman (University of Notre Dame)
09:45 AM - 10:30 AM
  Putman Problem Session
Andrew Putman (University of Notre Dame)
10:30 AM - 10:45 AM
  Short Break
10:45 AM - 11:30 AM
  Snowden Lecture Series: Overview
Andrew Snowden (University of Michigan)
11:30 AM - 12:15 PM
  Snowden Problem Session
Andrew Snowden (University of Michigan)
12:15 PM - 01:45 PM
  Lunch
01:45 PM - 02:45 PM
  Background module: Category theory
02:45 PM - 03:15 PM
  Problem Session
03:15 PM - 03:45 PM
  Tea
03:45 PM - 05:00 PM
  Problem Session
Jun 25, 2019
Tuesday
09:00 AM - 09:45 AM
  Snowden Lectures Series: The ABC's of FI-modules
Andrew Snowden (University of Michigan)
09:45 AM - 10:30 AM
  Snowden Problem Session
Andrew Snowden (University of Michigan)
10:30 AM - 10:45 AM
  Short Break
10:45 AM - 11:30 AM
  Patzt Lecture Series: Free groups and their automorphisms
Peter Patzt (Purdue University)
11:30 AM - 12:15 PM
  Patzt Problem Session
Peter Patzt (Purdue University)
12:15 PM - 01:45 PM
  Lunch
01:45 PM - 02:30 PM
  Putman Lecture Series (Talk #2)
Andrew Putman (University of Notre Dame)
02:30 PM - 03:15 PM
  Putman Problem Session
Andrew Putman (University of Notre Dame)
03:15 PM - 03:45 PM
  Tea
03:45 PM - 05:00 PM
  Problem Session
Jun 26, 2019
Wednesday
09:00 AM - 09:45 AM
  Patzt Lectures Series: VIC-modules
Peter Patzt (Purdue University)
09:45 AM - 10:30 AM
  Putman Problem Session
Andrew Putman (University of Notre Dame)
10:30 AM - 11:30 AM
  Background module: Homological algebra
11:30 AM - 02:00 PM
  BBQ Lunch at Tilden Park
02:00 PM - 02:45 PM
  Putman Lecture Series (Talk #3)
Andrew Putman (University of Notre Dame)
02:45 PM - 03:15 PM
  Putman Problem Session
Andrew Putman (University of Notre Dame)
03:15 PM - 03:45 PM
  Tea
03:45 PM - 05:00 PM
  Problem Session
Jun 27, 2019
Thursday
09:00 AM - 09:45 AM
  Putman Lectures Series (Talk #4)
Andrew Putman (University of Notre Dame)
09:45 AM - 10:30 AM
  Putman Problem Session
Andrew Putman (University of Notre Dame)
10:30 AM - 10:45 AM
  Short Break
10:45 AM - 11:30 AM
  Snowden Lecture Series: The noetherian property
Andrew Snowden (University of Michigan)
11:30 AM - 12:15 PM
  Snowden Problem Session
Andrew Snowden (University of Michigan)
12:15 PM - 01:45 PM
  Lunch
01:45 PM - 02:30 PM
  Patzt Lecture Series: The homology of IA_n
Peter Patzt (Purdue University)
02:30 PM - 03:15 PM
  Patzt Problem Session
Peter Patzt (Purdue University)
03:15 PM - 03:45 PM
  Tea
03:45 PM - 04:45 PM
  Background module: Spectral sequences
Jun 28, 2019
Friday
09:00 AM - 09:45 AM
  Patzt Lectures Series: Central stability homology
Peter Patzt (Purdue University)
09:45 AM - 10:30 AM
  Patzt Problem Session
Peter Patzt (Purdue University)
10:30 AM - 10:45 AM
  Short Break
10:45 AM - 11:30 AM
  Snowden Lecture Series: Nagpal's shift theorem
Andrew Snowden (University of Michigan)
11:30 AM - 12:15 PM
  Snowden Problem Session
Andrew Snowden (University of Michigan)
12:15 PM - 01:45 PM
  Lunch
01:45 PM - 02:30 PM
  Putman Lecture Series (Talk #5)
Andrew Putman (University of Notre Dame)
02:30 PM - 03:15 PM
  Putman Problem Session
Andrew Putman (University of Notre Dame)
03:15 PM - 03:45 PM
  Tea
03:45 PM - 05:00 PM
  Problem Session
Jul 01, 2019
Monday
09:00 AM - 09:45 AM
  Snowden Lectures Series: Generic FI-modules
Andrew Snowden (University of Michigan)
09:45 AM - 10:30 AM
  Snowden Problem Session
Andrew Snowden (University of Michigan)
10:30 AM - 10:45 AM
  Short Break
10:45 AM - 11:30 AM
  Patzt Lecture Series: Highly connected simplicial complexes
Peter Patzt (Purdue University)
11:30 AM - 12:15 PM
  Patzt Problem Session
12:15 PM - 01:45 PM
  Lunch
01:45 PM - 02:45 PM
  Lecture Series: Computing the tail of a rational FI-module
John Wiltshire-Gordon (University of Wisconsin-Madison)
02:45 PM - 03:30 PM
  Problem Session
03:30 PM - 04:00 PM
  Tea
04:00 PM - 05:00 PM
  Q&A with lecturers and TAs
Jul 02, 2019
Tuesday
09:00 AM - 09:45 AM
  Patzt Lectures Series: Quillen’s spectral sequence argument for homological stability
Peter Patzt (Purdue University)
09:45 AM - 12:15 PM
  Patzt Problem Session
Peter Patzt (Purdue University)
12:15 PM - 01:45 PM
  Lunch
01:45 PM - 03:15 PM
  Snowden Lecture Series: Schur--Weyl duality
Andrew Snowden (University of Michigan)
03:15 PM - 03:45 PM
  Tea
03:45 PM - 05:00 PM
  Snowden Problem Session
Andrew Snowden (University of Michigan)
Jul 03, 2019
Wednesday
09:00 AM - 09:45 AM
  Snowden Lectures Series: FI_d-modules
Andrew Snowden (University of Michigan)
09:45 AM - 10:30 AM
  Snowden Problem Session
Andrew Snowden (University of Michigan)
10:30 AM - 10:45 AM
  Short Break
10:45 AM - 11:30 AM
  Lecture Series
10:45 AM - 11:30 AM
  Wilson Lectures (Talk #1)
Jenny Wilson (University of Michigan)
11:30 AM - 12:15 PM
  Problem Session
12:15 PM - 01:45 PM
  Lunch
01:45 PM - 02:30 PM
  Patzt Lectures: Polynomial functors
Peter Patzt (Purdue University)
02:30 PM - 03:15 PM
  Patzt Problem Session
Peter Patzt (Purdue University)
03:15 PM - 03:45 PM
  Tea
03:45 PM - 05:00 PM
  Problem Session
Jul 05, 2019
Friday
09:00 AM - 09:45 AM
  Snowden Lectures Series: FIM-modules
Andrew Snowden (University of Michigan)
09:45 AM - 10:30 AM
  Snowden Problem Session
Andrew Snowden (University of Michigan)
10:30 AM - 10:45 AM
  Short Break
10:45 AM - 11:30 AM
  Patzt Lecture Series: Central stability homology for polynomial VIC(Z)–module
Peter Patzt (Purdue University)
11:30 AM - 12:15 PM
  Patzt Problem Session
Peter Patzt (Purdue University)
12:15 PM - 01:45 PM
  Lunch
01:45 PM - 02:30 PM
  Wilson Lectures (Talk #2)
Jenny Wilson (University of Michigan)
02:30 PM - 03:15 PM
  Wilson Problem Session
Jenny Wilson (University of Michigan)
03:15 PM - 03:45 PM
  Tea
03:45 PM - 05:00 PM
  Problem Session