Topology of Algebraic Varieties and Enumerative Combinatorial Geometry

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Enumerative combinatorial geometry studies fundamental counting questions for geometric combinatorial objects. One example of such a question is to determine the possible number of intersection points among a given number of lines in the plane. This project investigates such combinatorial questions using methods from algebraic geometry. Some combinatorial invariants have been shown to have algebraic geometric nature, that is, there are algebraic varieties associated to special combinatorial objects, and it is possible to express combinatorial invariants using the corresponding algebraic varieties. Once such algebraic geometric nature is realized, one can translate combinatorial questions into questions about algebraic varieties, with the potential to reduce them to known results in algebraic geometry. The underlying algebraic geometry structures beneath combinatorial objects can also lead to deeper understanding of the combinatorial objects. Matroid theory is a fundamental subject in combinatorial geometry. The first goal of this project is to explore the algebraic geometric nature of the numerical invariants of matroids that are realizable over some field. More precisely, the project aims to express the numerical invariants in terms of the (étale or singular) cohomology ring of some associated algebraic varieties, obtaining new properties of the matroids. The more ambitious goal is to develop combinatorial generalizations of theorems in algebraic geometry and to derive properties for non-realizable matroids.