Math 274  Topics in Algebra: Cluster Algebras  Spring 2011
Lectures: Tuesday and Thursday 1112:30pm, 9 Evans.
Lecturer: Lauren Williams
(office Evans Hall 913, email 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
 Ysystems 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 CDM03 conferences,
International Press, 2004.
 Fomin and Zelevinsky, Cluster algebras I: Foundations,
J. Amer. Math. Soc. 15 (2002), 497529.
 Fomin and Zelevinsky, Ysystems and generalized associahedra,
Ann. Math. 158 (2003), 9771018.
 Fomin and Zelevinsky, Cluster algebras II: Finite type
classification, Invent. Math. 154 (2003), 63121.
 Berenstein, Fomin and Zelevinsky, Cluster algebras III:
Upper bounds and double Bruhat cells, Duke Math. J. 126 (2005), 152.
 Fomin and Zelevinsky, Cluster algebras IV: Coefficients,
Compos. Math. 143 (2007), 112164.

Gekhtman, Shapiro, and Vainshtein, Cluster algebras and Poisson geometry,
Amer. Math. Soc, 2010.

Cluster Algebras Portal
Possible topics for final projects:
List of topics
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MSRI Summer Graduate workshop on Cluster Algebras and Cluster Combinatorics,
August 112, 2011
MSRI Semesterlong 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): Fpolynomials and gvectors
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, 45:30pm 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: ``KacMoody 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