Combinatorics Seminar

The Matrix-Tree Theorem is a mid-19th century result by G. Kirchhoff, representing the minors of the Laplace matrix as sums of positive monomials of matrix elements indexed by directed rooted forests. We will present higher-degree generalizations of both notions appearing above: a minor and a forest. The theorem involves an analog, due to O. Bernardi, of the classical Tutte polynomial for directed graphs. Applications include a formula, due to B. Epstein and M. Polyak, for the Casson-Walker invariant of a rational homology 3-sphere. Speaker(s): Yurii Burman (Higher School of Economics, Moscow)