Date: Monday, March 12, 2012
Location: 3088 East Hall (4:00 PM to 5:00 PM)
Title: Why do type A cluster algebras work?
Abstract: Ptolemy's theorem states that the four side lengths a,b,c,d and the two diagonal lengths e,f of a cyclic quadrilateral satisfy the relation ef=ac+bd. Suppose we have a triangulated polygon with positive real weights on each of its sides and diagonals. We can perform a "flip" by erasing a diagonal and drawing its complementary diagonal weighted in such a way that Ptolemy's identity holds in the surrounding quadrilateral. Amazingly, the weight of any diagonal reached after a sequence of such flips is independant of the sequence of flips chosen.
I will outline a proof of this fact that involves hyperbolic geometry. Both the result and the proof can be generalized to the setting of triangulations of oriented surfaces. Cluster algebras, introduced by S. Fomin and A. Zelevinsky, describe an even more general class of dynamical systems. Time allowing, I will give basic definitions and results pertaining to cluster algebras.
Speaker: Max Glick
Institution: U Michigan
Event Organizer:
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