Lectures: Mondays 2-4pm, Room C5H41
Lecturer: Jon Wilson (jonathan.wilson@mnf.uni-tuebingen.de)
Office Hours: by appointment (office P27 Floor 5)
Fomin and Zelevinsky's cluster algebras are a 21st centrury construct which propogate through vast areas of mathematics.
This course will tour combinatorial, geometric and algebraic concepts which perhaps unexepctedly share this newfound structure.
Lecture 1: Motivation for cluster algebras Part I — positivity tests of Gr(2,n).
Lecture 2: Motivation for cluster algebras Part II — positivity tests of flag minors.
Lecture 3: Motivation for cluster algebras Part III — total positivity tests of nxn matrices.
Lecture 4: Definition of a cluster algebra and examples.
Lecture 5: Diagram mutation and the link between diagrams and skew-symmetrizable matrices.
Lecture 6: Proof of the Laurent phenomenon Part I.
Lecture 7: Proof of the Laurent phenomenon Part II.
Lecture 8: Y-seed patterns, cross ratios, and the Pentegram Map.
Lecture 9: Moving towards finite type classification Part I — tropical semi-fields and Y-dynamics in replace of extended part of exchange matrices.
Lecture 10: Moving towards finite type classification Part II — coverings of exchange graphs via change of coefficients, and the classification of rank 2 cluster algebras.
Lecture 11: Classification of type An seed patterns.
Lecture 12: Classification of type Dn seed patterns.
Lecture 13: Folding quivers to get new seed patterns.
Lecture 14: Classification of type Bn, Cn, F4, G2 seed patterns via folding, and a remark on the finiteness of E6, E7, E8.
Lecture 15: Classification of finite type cluster algebras via 2-finiteness.
Lecture 16: Basics of bordered surfaces and their ideal triangulations.
Lecture 17: Connectivity of ideal triangulations via flips.
Lecture 18: Cluster algebras from surfaces Part I — tagged triangulations and their associated quivers.