Nikhil Sahoo L&S Sciences
Schubert Calculus Through Toric Geometry
In our research, we will use toric geometry to study the cohomological structure of complex Grassmannians. The cohomology ring of a Grassmannian varieties is described by the Littlewood-Richardson rule. One of the main open questions in Schubert calculus concerns the generalization of the Littlewood-Richardson rule to flag varieties. Such a generalization is highly desirable, because it is a manifestly positive formula that can be applied to other areas: in algebraic geometry, it helps describe complicated intersections; in representation theory, it helps to find irreducible, direct-sum decompositions of tensor products; in physics, it can be applied to calculate certain physical quantities. Our research aims to give a new, geometric proof of the Littlewood-Richardson rule,by applying toric degeneration to Weyl-group-translated Schubert varieties. More specifically, we will study the intersection behavior of Schubert varieties, in terms of face-intersections of Gelfand-Cetlin polytopes. A new geometric perspective would help give a deeper understanding of the Littlewood-Richardson rule, particularly in its relation to Schubert calculus. New methods could also suggest how to generalize the Littlewood-Richardson rule to arbitrary flag varieties.
Message To Sponsor
Having now completed my SURF project, I would like to extend my sincerest thanks to the McKinley fund for their support. This past summer afforded me chances to work at an advanced level, dive deep into the literature with the support of an expert in the field, and share my results with a vibrant group of academic peers. Such an opportunity is unprecedented in my career as a student and will be with me for a long time. Although the team planned for this project to be concluded at the end of the summer, I am now seeing new questions and directions of inquiry. I intend to keep thinking about these topics and working on these questions for a long time to come. Lastly, this program afforded me the chance to make important connections with people in my field. I am continuing to work with my advisor and some of my teammates, towards future goals that I would not have thought possible without this program.