Research Area(s):
Algebra/Algebraic Geometry
Combinatorics
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Algebraic geometry is the study of the geometry of the solutions to systems of polynomial equations. The simplest example of a theorem of algebraic geometry is that a polynomial of degree n has n solutions; this statement relates a feature of the equation (the number of monomials it contains) to a feature of its solution set (how many points there are). More generally, we may have many equations and want to answer questions like how manysolutions there are, where they are located and what sort of shape they form. I study areas of algebraic geometry which have a great deal of combinatorial structure, allowing connections to the tools of discrete math. In particular, I am interested in flag varieties and related configuration spaces, cluster algebras and toric varieties. On the combinatorial side side, I use ideas from Schubert calculus, matroids, lattice point enumeration and Coxeter groups.
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