Degeneracy loci, toric degenerations, and equivariant algebraic geometry

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Fundamentally, algebraic geometry is the study of solutions to systems of polynomial equations. The systems of interest typically involve many equations and many variables, and it is essential to look for ways to simplify them. One way to do this is by tuning certain parameters to zero; if done carefully, the resulting system retains many properties of the original one, and may be more amenable to analysis. Another way to simplify is to look for symmetry in the set of solutions, and take advantage of this to reduce the problem and extract information about the original solution set from a smaller set. These techniques, respectively known as "degeneration" and "equivariant localization", can be used in concert: for example, one may degenerate to a system possessing extra symmetry. This project will use these methods to study several problems of interest in enumerative geometry.The aim of this project is to study the geometry of algebraic varieties with group actions using equivariant and degeneration techniques. The PI will study flag varieties, Schubert varieties, and toric varieties using methods from equivariant cohomology, as well as the theory of Newton-Okounkov bodies. Motivating problems are to determine criteria for a variety to admit a flat degeneration to a toric variety, and to produce a wider class of examples where the Newton-Okounkov body can be computed explicitly. The PI will also continue his study of degeneracy loci, and investigate their finer geometric properties by developing the combinatorics of diagrams for signed permutations. Finally, the PI will explore applications of the recently-developed operational K-theory, focusing on the relationship between equivariant localization and Riemann-Roch theorems in this context.