Combinatorial algebraic geometry: Modern Schubert calculus and generalized splines

Smith College, Northampton MA

Investigators

Abstract

Whenever equations model the world around us---whether describing how the gross domestic product depends on different sectors of the economy or predicting the trajectory of a spacecraft or a hurricane---those equations also describe a geometric object. For a simple system of equations like the line x+y=1, geometry might not seem very useful. When we have complicated systems with dozens of equations in hundreds of variables, methods from geometry can be one of the few ways to analyze important quantitative and qualitative aspects of the system, like where it has a maximum or how many pieces it has. This research project develops important geometric tools for describing complicated systems of equations. It then applies these tools to specific systems of equations with important applications in mathematics, physics, computer science, and other fields. The PI proposes to solve three linked questions in modern Schubert calculus: the first studies the cohomology of the affine Grassmannian; the second studies the intersection homology of Schubert varieties; and the third studies the cohomology of a family of subvarieties of the flag variety called Hessenberg varieties, in order to answer major open questions about the flag variety. To do this, the PI will use GKM theory, a combinatorial and algebraic algorithm to describe equivariant cohomology, as well as a recent extension of GKM theory called generalized splines. The goals of the project will be 1) to develop theoretical tools and algorithms in generalized splines and 2) to apply these tools to perform specific computational and combinatorial calculations in the various cohomology rings we study.