The tropical and algebraic geometry of character varieties

Thursday, March 14, 2019 - 1:00pm
110 Math Science Building
Christopher Manon
(U. Kentucky)

Abstract:  Free group character varieties appear as moduli spaces in higher Teichmuller theory, the theory of geometric structures, and the study of Higgs bundles.  The coordinate rings of these spaces are classical rings from invariant theory which encode the necessary relations that have to hold among the traces of the products of matrices (hence the name "character variety").  I'll describe some older work relating the tropical geometry of these spaces to a simplicial complex studied by Culler and Vogtmann called outer space.  This construction involves finding a special generating set for the coordinate ring called a Khovanskii basis which has useful combinatorial and computational properties.  Then I'll describe how this construction naturally leads to a class of Fano compactifications of the character variety.