Algebraic geometry of toric varieties and its applications to combinatorics.

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We study problems originating in the geometry of toric varieties. Toric varieties are geometric objects that are defined combinatorially by a convex polytope or a fan. The geometric properties of a toric variety are therefore reflected in the combinatorics of the polytope or fan. The theory of toric varieties has applications in both algebraic geometry and combinatorics. In algebraic geometry toric varieties are considered as being much simpler than general varieties, so hard conjectures can first be tested and proved on toric varieties. Examples of such results first proved for toric varieties include the minimal model program, mirror symmetry, and others. Toric varieties also have applications in combinatorics. These typically involve interpreting some combinatorial property of a polytope or a fan geometrically. The most well-known example of this is the description of how many faces of each dimension a simple polytope can have. These face numbers are described in terms of cohomology of the toric variety, and the relations coming from geometry give all the necessary conditions on these numbers. The first part of this proposal relates to the combinatorics of lattice points in a polytope. The geometric counterpart of this is the study of Cox rings of toric varieties and their blowups. The remaining parts of the proposal study the number of faces or chains of faces of a polytope. This in algebraic geometry corresponds to studying various cohomology theories, such as intersection cohomology, cd-index, algebraic cobordism theory and others.