Possible Project Topics
- Cluster algebras related to surfaces
- Cluster algebras associated to surfaces with punctures,
(including, in particular, cluster algebras of type D), see
Fomin-Shapiro-Thurston.
- Cluster algebras associated to a surface and their significance
in Teichmüller theory, see Williams.
- Tropical cluster algebras from surfaces, see
Fock-Goncharov.
- Particular examples of cluster algebras
- Coxeter-Conway frieze patterns, see
Coxeter-Conway 1,
Coxeter-Conway 2 (from 1973),
and Propp.
- Other examples of the Laurent phenomenon, some of them not quite
cluster algebras, see
Fomin-Zelevinsky.
- Combinatorial interpretations of rank 2 affine cluster algebras, see
Musiker-Propp.
- Cluster algebra structures on homogeneous co-ordinate rings of
Grassmannians, see
Scott.
- (Pre)-history
- Structural properties of cluster algebras
- Generalizations of cluster algebras
- Structure of finite type cluster algebras
- Nathan Reading and David Speyer have a multi-paper project that understands the structure of (mainly finite type) cluster algebras using Weyl groups.
One place to begin is here.
Somewhat later, but with a good review of what's going on,
here.
- Generalized associahedra
- Representation-theoretical approaches
- Other things I might add
- Dual semi-canonical basis
- Carroll-Price formula for cluster variables in terms of perfect matchings
- g-vectors
- More general coefficients on surfaces: laminations.
If a topic occurs to you which is not on this list, please ask — I am
pretty flexible.