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Abstract: Standard Groebner basis theory tells us how to start with a polynomial ideal I and pass to its initial ideal (the monomial ideal of initial terms of polynomials in I) with respect to a fixed term order. Although there are infinitely many term orders, there are finitely many initial ideals of a fixed ideal I. What happens if we intersect all these initial ideals? This joint work with Diane Maclagan (UC Berkeley) investigates what we call the "vertex ideal" of I when I is toric or in general a lattice ideal. I will talk about minimal generators, standard monomials and standard pair decomposition, the associated primes, the Hilbert function etc. of the vertex ideal of a lattice ideal. I will also present a counterexample to a conjecture of Bernd Sturmfels and Rekha Thomas regarding the complexity of codimension three toric Groebner fans.