Reporter: Xin Fang (University of Cologne)
Time: 17:00-18:30, March 7, 14, 21, 28, 16:00-17:30, April 4, 2023
Place: Tencent Meeting ID：942 663 0176, no password
Abstract：Schubert varieties first appeared in the work of Hermann Schubert in the study of the following question: how many geometric shapes with definite definitions fulfill given conditions? Schubert gave a non-rigorous approach to the general question by transforming geometric conditions to symbolic calculus; to put it on a rigorous foundation is Hilbert’s 15th problem. A typical example of such a question is: given four lines in a three-dimensional complex space, how many lines intersect them all?
The first goal of this lecture is to establish an appropriate algebro-geometric setup (Schubert calculus on Grassmann varieties), in order to provide a solid treatment to this typical example. On the way we will encounter Grassmann varieties and their Schubert varieties, Young tableaux and standard monomials, symmetric functions and Littlewood-Richardson coefficients, etc… In the second part of the lecture we will move to flag varieties. Concrete description of their Schubert calculus is still open; if time permits, I plan to introduce certain recent work (Kirichenko-Smirnov-Timorin, Fujita) from the perspective of polyhedral geometry (a.k.a. toric geometry) using Gelfand-Tsetlin polytopes.
Prerequisite: Basics on affine and projective varieties (Hartshorne Chapter 1, Section 1 and 2). Knowledge on basic algebraic topology (Cohomology ring, Poincaré duality) would be helpful, but not necessary.
Video recording and notes can be downloaded here: https://rec.ustc.edu.cn/share/1535bbc0-bcef-11ed-b3c2-537cf43fb255, password: 2023igp
