Algebraic Geometry Seminar

The past decade has revealed a deep but hidden interplay between the
geometry of character varieties of surfaces and the representation theory
of quivers. One rough way of stating this is that the coordinate ring of a
(possibly wild) character variety can be viewed as a kind of Hall algebra
for a quiver with potential. Somewhat more precisely, there are geometric
and representation theoretic points of view on this ring with a priori
distinct prescriptions for "canonical" elements of it, which coincide
despite their definitions being fundamentally different in nature. We will
discuss recent progress developing this circle of ideas, in particular for
higher-rank gauge groups. A key organizing principle is the notion of
cluster algebras and their canonical bases, whose origins go back to the
work of Lusztig and Kashiwara in Lie theory.