Log geometry and multiplier ideals

I work in combinatorics, algebraic geometry, convex geometry and commutative algebra while staying informed on certain topics in category theory and ring theory. In particular, I focus on toric varieties and singularity theory. The study of toric varieties lies at the intersection of combinatorics, algebraic geometry, convex geometry and integer programming. There is a correspondence between certain combinatorial objects in convex geometry, cones and fans, and the geometry of toric varieties. For example, there are techniques based on toric geometry for counting the lattice points in a lattice polytope. Because of this correspondence between combinatorial objects in convex geometry and toric geometry, objects in toric geometry tend to be more concrete and computable than they are in algebraic geometry in general. At the same time, toric varieties provide a large enough class of geometric objects to test many conjectures. For example, the class of toric varieties includes products of projective spaces, many spaces with mild singularities, and some compact nonprojective varieties. The jumping numbers of a singular variety Y embedded in a smooth complex variety X form an interesting invariant of the pair (X,Y ). The jumping numbers are a sequence of positive rational numbers computed from— and reflecting subtle information about— an embedded resolution of singularities of the pair. For example, in the simplest case where Y is a smooth hypersurface in X, the jumping numbers are simply the positive integers. But the sequence of jumping numbers becomes increasingly complicated as a resolution of singularities requires more blowings up or as functions vanishing on Y vanish to higher orders along the resulting exceptional divisors. Jumping numbers, also known as jumping coefficients, were first explicitly defined in [4] as those numbers λ for which the multiplier ideal of the pair (X,λY ) makes a discrete “jump”, though these natural invariants arose earlier in several contexts; see [9], [10], and [19]. This research statement starts with my study of toric geometry. The first four sections start by talking about my ideas about the classic theory and move toward logarithmic geometry of schemes that behave like toric varieties, sections 5 and 6, explain my work on multiplier ideals, and the last section lists some my future plans. In Section 1, I describe an invariant of a (classical) toric variety that measures its failure to be smooth and generalizes the notion of the