Joint Mathematics Colloquium
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Abstract |
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The characteristic polynomial of a matroid is
an invariant that generalizes the chromatic polynomial of a graph. The theory of hyperplane arrangements provides it with numerous topological and algebraic interpretations. I will describe another such interpretation, a characteristic polynomial formula obtained by a fundamental geometric construction involving critical points of products of polynomial functions. In this story, the theory of Chern-Schwartz-MacPherson classes comes up as the perfect tool from the point of view of combinatorics. In joint work with June Huh, we observe the characteristic polynomial as the bidegree of a biprojective variety in a new way that provides, in particular, some new inequalities that the coefficients of the characteristic polynomial of a complex hyperplane arrangement must satisfy.
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