Zoom link: https://illinois.zoom.us/j/84600044860?pwd=M0tkN0htTlFhNGNKeGVxR1pYY3Y2dz09
Meeting ID: 846 0004 4860
Passcode: 441458
Abstract: Cluster algebras are certain commutative algebras whose generators are defined recursively. Fomin and Zelevinsky invented these algebras in 2002 in order to establish a combinatorial framework to understand the theory of total positivity due to Lusztig and the dual canonical bases of quantum groups due to Lusztig/Kashiwara. In particular, Fomin-Zelevinsky conjectured that the cluster monomials (certain monomials in the generators) belong to the dual canonical basis.
In this talk, we briefly introduce cluster algebras by using concrete examples. We introduce the (common) triangular bases, which are Kazhdan-Lusztig type bases naturally parametrized by the tropical points of the corresponding varieties. They allow us to verify the above motivational conjecture in full generality. We also discuss the relationship between such bases and monoidal categorification.
Fan Qin is an algebraist, working in representation theory and cluster algebras, with interests in quantum groups, geometric representation theory, categorification, algebraic geometry, tropical geometry and higher Teichmuller theory. He is a 2012 graduate of Université Paris VII (Advisor: Bernhard Keller) and is currently at Shanghai Jiao Tong University. Faculty have access to a complete background via mathjobs.org.
