Tropical Geometry: Foundations and Combinatorial Applications

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Tropical geometry is an area of mathematics that translates problems in algebraic geometry to problems in combinatorics. Algebraic geometry is the study of algebraic varieties which are geometric objects that are described by polynomial equations. Algebraic varieties include familiar figures like circles, lines, and parabolas but also more complicated geometric objects in high dimensional spaces. Algebraic varieties are special in that they have a lot of structure. Combinatorics is the theory of counting but includes the study of discrete objects. One class of discrete objects is that of polyhedral complexes which are shapes made up of points, line segments, polygons, and their higher-dimensional analogues glued together. These polyhedral complexes have a rich theory and can be quite involved. The main technique of tropical geometry is tropicalization which takes as input an algebraic variety and outputs a polyhedral complex which is a combinatorial shadow of the variety. These polyhedral complexes, called tropical varieties capture a surprising amount of data about the original algebraic varieties. In particular, the tropical variety reflect the intersection theory, Hodge theory, and enumerative geometry of the original algebraic variety. Tropical geometry has applications to enumerative geometry, the study of moduli spaces, mirror symmetry in string theory, the theory of numerical algebraic computations, low dimensional topology, number theory, and algebraic dynamics. The research program has applications to more classical areas of combinatorics, enumerative geometry, and Hodge theory. I suggest four research projects that follow a general theme of relating combinatorics and algebraic geometry. Two projects involve discovering which properties of an algebraic variety are combinatorial in nature. Another involves finding a combinatorial analog of the theory of algebraic surfaces. The fourth involves refining the procedure that translates algebraic geometry into combinatorics. These projects address open problems and open up new areas of study. This proposal involves the training of undergraduate, master's, and doctoral students.