Daoji Huang: The "lifting dream" of Schubert calculus

Daoji Huang (IAS)

In the 19th century, German Mathematician Hermann Schubert was interested in questions in enumerative geometry, like "given 4 generic lines in 3-space, how many lines intersect all four lines?" The modern treatment of such questions centers around the study of the cohomology ring of the complete flag variety, with a basis given by the classes of Schubert varieties. While this topic and its variations have been extensively studied, a central question, which seeks a combinatorial interpretation of the structure constants, remains open. However, the same question has been fully solved using many different methods in the special case when the classes are pullbacks from the Schubert classes in Grassmannians, in which case the Schubert structure constants are known as the Littlewood-Richardson coefficients. The "lifting dream" of Schubert calculus hopes to find ways to lift the Littlewood-Richardson rules to the general case. I will talk about the mathematical pipeline that turns the geometric questions into combinatorics, potential venues to lift the combinatorial and geometric techniques for the Grassmannian case using bumpless pipe dreams and positroid varieties, and some success stories so far.