Yining Liu

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

I want to thank McKinley fund for giving us the opportunity to commit to research this summer. From this experience, I gained more confidence as both a researcher and a person. I was able to develop research skills and explore my interests. It was a wonderful experience working with my amazing teammates and our mentor. I learned a lot of mathematics and problem solving skills this summer, and the experience wouldn't have been possible without my teammates, our mentor, the SURF program and our donor. Thank you again for giving us this opportunity!
  • Major: Math
  • Sponsor: McKinley Fund
  • Mentor: David Nadler