Math 847 - Representation Stability (Fall 2017)

Lecture times and location: TR 9:30AM - 10:45AM, Van Vleck B235
Course website: http://math.wisc.edu/~svs/847/
Instructor: Steven Sam email
Instructor office hours: 321 Van Vleck, Tuesday 12:30-1:30, or by appointment

Notes for course (last updated 12/12/17)
Homework 1 (due Nov. 21)

There is no textbook for the course. Instead, I will maintain typed lecture notes, which can be found above. Some references you may find useful:

Representation theory and homological stability - paper by Church and Farb which first attempts to formalize representation stability. Contains many examples.
Benson Farb's ICM lecture on representation stability - overview of subject and has some examples
Lecture notes from symmetric functions, Spring 2017 - useful for more detailed representation theory calculations
Noetherian properties in representation theory - conference talk I gave illustrating the use of noetherianity in representation stability
Introduction to twisted commutative algebras - Twisted commutative algebras are some structures we will discuss. Also contains summary of representation theory of symmetric and general linear groups.

Schedule


Sep 7	Section 1: Introduction and motivation
Sep 12	Section 2.1: Schur-Weyl duality
Sep 14	Section 2.2: Symmetric functions Section 2.3: Polynomial representations Section 2.4: Partitions

Sep 19	Section 2.5: Bases for symmetric functions Section 2.6: Schur functors
Sep 21	Section 2.7: Pieri's rule

Sep 26	Section 2.8: Tensor categories
Sep 28	no class

Oct 3	Section 2.8 continued Section 2.9: Categorical Schur-Weyl duality
Oct 5	Section 2.10: Infinite general linear group Section 2.11: Littlewood-Richardson coefficients Section 2.12: A few more formulas

Oct 10	Sections 3.1, 3.2: Twisted commutative algebras
Oct 12	Guest lecture: John Wiltshire-Gordon

Oct 17	Section 3.3: Noetherianity of bounded tca's Section 3.4: Noetherianity of functor categories
Oct 19	Section 3.5: Noetherian posets Section 3.6: Monomial representations

Oct 24	Section 3.7: Gröbner categories Section 3.8: Noetherianity of FI_d-modules Section A.1-A.3: Definitions of group (co)homology
Oct 26	Section 4.1: The complex of injective words

Oct 31	Section A.4: Spectral sequences Section 4.2: Nakaoka stability
Nov 2	Section 4.3: Homological stability of FI-modules Section 5.1: FI-structure on cohomology of configuration spaces

Nov 7	Section 5.2: Representation stability for configuration spaces Section 6.1: Review of Zariski topology
Nov 9	Section 6.2: Tensor rank

Nov 14	Section 6.4.1: Flattenings Section 6.4.2: Spaces of infinite tensors
Nov 16	Section 6.4 continued

Nov 21	Section 6.4.3: Proofs Section 7.1: Asymptotic combinatorial properties of FI-modules
Nov 23	no class - holiday

Nov 28	Section 7.1: Asymptotic combinatorial properties of FI-modules Stability of Kronecker coefficients
Nov 30	Section 7.2: Serre quotient categories Section 7.3: Generic FI-modules

Dec 5	Section 7.4: Semi-induced FI-modules Section 7.5: Cohomology of FI-modules
Dec 7	Section 7.5: Cohomology of FI-modules

Dec 12	Section 8: Δ-modules