Undergraduate Research & Scholarships

Junhao Fan 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

I am very grateful for my donor's generous support of our research project. It is such a unique experience for me. It not only enabled me to spend the summer delving deep into a specific question that I'm interested in, but also offered me the freedom to explore various relevant theories, which was intellectually pleasing even for its own sake. Going through this research experience offered me new perspectives of viewing, learning, and doing mathematics that is so different from reading textbooks and attending lectures. Like a lighthouse in the distance, this experience has definitely offered me guidance in my journey towards unknown territory.
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Major: Math
Mentor: David Nadler
Sponsor: McKinley Fund
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