Coloring graphs, counting solutions, and attaching cells

The chromatic polynomial of a graph counts the number of its colorings. We give an affirmative answer to the conjecture of Read and Rota that the absolute values of the coefficients of the chromatic polynomial form a log-concave sequence. In this talk we introduce a sequence of numerical invariants of projective hypersurfaces analogous to the Milnor number of local analytic hypersurfaces and discuss its basic properties. This talk is based upon arXiv:1008.4749.