Math 274 - Topics in algebra: cluster algebras

Math 274 - Topics in Algebra: Cluster Algebras - Spring 2011

Lectures: Tuesday and Thursday 11-12:30pm, 9 Evans.

Lecturer: Lauren Williams (office Evans Hall 913, e-mail williams@math.berkeley.edu)

Office Hours: by appointment

Course description

This course will survey one of the most exciting recent developments in algebraic combinatorics, namely, Fomin and Zelevinsky's theory of cluster algebras. Cluster algebras are a class of combinatorially defined commutative rings that provide a unifying structure for phenomena in a variety of algebraic and geometric contexts. Introduced just 10 years ago, cluster algebras have already been shown to be related to a host of other fields of math, such as quiver representations, Teichmuller theory, Poisson geometry, total positivity, and statistical physics.

The following is a partial list of topics which I plan to cover:

Definition of cluster algebras
Finite type classification of cluster algebras
Cluster algebra combinatorics, including generalized associahedra
Examples from geometry, eg Grassmannians, double Bruhat cells, triangulated surfaces
Y-systems and more generally the dynamics of coefficients in cluster algebras

More topics will be added, based on the interests of the participants.

I will assume that people have some familiarity with combinatorics. Familiarity with root systems would also be helpful. I will not assume prior knowledge of cluster algebras.

References for cluster algebras: (many of them available here)

Fomin and Zelevinsky, Cluster algebras: Notes for the CDM-03 conferences, International Press, 2004.
Fomin and Zelevinsky, Cluster algebras I: Foundations, J. Amer. Math. Soc. 15 (2002), 497--529.
Fomin and Zelevinsky, Y-systems and generalized associahedra, Ann. Math. 158 (2003), 977--1018.
Fomin and Zelevinsky, Cluster algebras II: Finite type classification, Invent. Math. 154 (2003), 63--121.
Berenstein, Fomin and Zelevinsky, Cluster algebras III: Upper bounds and double Bruhat cells, Duke Math. J. 126 (2005), 1--52.
Fomin and Zelevinsky, Cluster algebras IV: Coefficients, Compos. Math. 143 (2007), 112--164.
Gekhtman, Shapiro, and Vainshtein, Cluster algebras and Poisson geometry, Amer. Math. Soc, 2010.
Cluster Algebras Portal

Possible topics for final projects: List of topics

MSRI Summer Graduate workshop on Cluster Algebras and Cluster Combinatorics, August 1-12, 2011

MSRI Semester-long program on Cluster Algebras, Fall 2012

Lectures

Lecture 1 (Jan. 18): Introduction, definition, examples

Lecture 2 (Jan. 20): Cluster algebras with coefficients, cluster monomials, the Grassmannian of 2 planes in n space

Lecture 3 (Jan. 25): Cluster algebras of rank 2 and canonical bases

Lecture 4 (Jan. 27): Cluster algebras of rank 2 and canonical bases (cont.)

Lecture 5 (Feb. 1): The finite type classification of cluster algebras

Lecture 6 (Feb. 3): More criteria for determining finite type, plus (start) generalized associahedra

Lecture 7 (Feb. 8): The cluster complex in terms of the root system; generalized associahedra

Lecture 8 (Feb. 10): Sketch of proofs of the finite type classification

Lecture 9 (Feb. 15): Proof of the Laurent phenomenon

Lecture 10 (Feb. 17): Upper and lower cluster algebras; the importance of an acyclic seed

Lecture 11 (Feb. 22): From total positivity for double Bruhat cells to cluster algebras

Lecture 12 (Feb. 24): The cluster algebra structure on double Bruhat cells

Lecture 13 (March 1): The dynamics of coefficients

Lecture 14 (March 3): F-polynomials and g-vectors

Lecture 15 (March 8): Total positivity on the Grassmannian (guest lecture by Kelli Talaska)

Lecture 16 (March 10): Total positivity on the Grassmannian (continued)

Lecture 17 (March 15): Plabic graphs, parameterizations of positroid cells, and toric varieties

Lecture 18 (March 17): The cluster algebra structure on the Grassmannian

Lecture 19 (March 29): 40 minute talk by Adam Chavin, plus start on cluster algebras from surfaces

Lecture 20 (March 31): Cluster algebras from surfaces

Lecture 21 (April 5): Teichmuller theory (and the connection to cluster algebras)

Lecture 22 (April 7): Teichmuller theory (the space of rational unbounded measure laminations, and shear coordinates)

Lecture 23 (April 12): A (very) cursory introduction to categorifications of cluster algebras

No lecture on April 14

Lecture 24 (April 19): An introduction to the KP equation

Lecture 24.5 (April 20): KP solitons, total positivity, and cluster algebras (RTGC seminar, 4-5:30pm in 891 Evans)

Lecture 25 (April 21): 40 minute talks by Harold Williams and Vu Thanh

Lecture 26 (April 26): 40 minute talks by Melody Chan and Felipe Rincon

Lecture 27 (April 28): 40 minute talks by Pablo Solis and Adam Boocher

Lecture 28 (May 3): 40 minute talks by Charley Crissmann and David Berlekamp

Student talks:

David Berlekamp: ``Quivers with potentials and their representations I: Mutations" (by Derksen, Weyman, Zelevinsky)

Adam Boocher: ``Real Schubert calculus: polynomials systems and a conjecture of Shapiro and Shapiro" (by Frank Sottile)

Morgan Brown: ``Grassmannians and cluster algebras" (by Scott)

Melody Chan: ``The positive tropical Grassmannian" (by Speyer, Williams)

Adam Chavin: ``Perfect matchings and the octahedron recurrence" (by Speyer)

Charley Crissman: "Laurent Expansions in Cluster Algebras via Quiver Representations" (by Caldero, Zelevinsky)

Felipe Rincon: ``The tropical Grassmannian of 3 planes in n space, and its positive part"

Pablo Solis: ``Kac-Moody groups and cluster algebras" (by Geiss, Leclerc, Schroer)

Vu Thanh: ``Cluster algebras of finite type via coxeter elements and principal minors" (by Yang, Zelevinsky)

Harold Williams: work of Fock and Goncharov