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When

Start: 04/09/2008 - 3:15pm

End : 04/09/2008 - 4:15pm

End : 04/09/2008 - 4:15pm

Category

Colloquium

Speaker

Mathias Beck (San Francisco State University)

Abstract

A common theme of enumerative combinatorics are counting functions given

by polynomials that are evaluated at positive integers. For example, one

proves in any introductory graph theory course that the number of proper

k-colorings of a given graph G is a polynomial in k, the "chromatic

polynomial" of G. Combinatorial reciprocity theorems give interpretations

of these polynomials at negative integers. For example, when we evaluate

the chromatic polynomial of G at -1, we obtain (up to a sign) the number

of acyclic orientations of G, that is, those orientations of G that do not

contain a coherent cycle.

Combinatorics is abundant with polynomials that count something when

evaluated at positive integers, and many of these polynomials have a

completely different interpretation when evaluated at negative

integers. We follow a common thread of chromatic and flow polynomials of

graphs, the Euler characteristic of polyhedra, Ehrhart polynomials

enumerating integer points in polytopes, and characteristic polynomials of

hyperplane arrangements.

Where

Beckman Auditorium, B126, Harvey Mudd College