MIT Combinatorics Seminar Combinatorial Secant Varieties Seth Sullivant, (Harvard University) Wednesday, December 7, 2005   4:30 pm    Room 2-142 ABSTRACT Given two projective varieties X and Y, their join X*Y is obtained by taking the Zariski closure of the union of all lines spanned by a point in X and a point in Y. The join of a variety X with itself is called the secant variety of X. In this talk, I will describe the construction of joins and secants in the combinatorial context of monomial ideals. For ideals generated by squarefree quadratic monomials, the generators of the secant ideals are obstructions to graph colorings and this leads to a commutative algebra version of the Strong Perfect Graph Theorem. Questions about secant varieties of combinatorially defined varieties (e.g. Grassmannians, determinantal varieties, toric varieties) can often be reduced to the monomial case. I will try to emphasize the combinatorial aspects of all of this, including the connections to graph theory, regular triangulations, and partially ordered sets. This is joint work with Bernd Sturmfels.