Schubert Calculus and the Cohomology of the Grassmannian

Abstract

This talk presents the defense of an undergraduate senior thesis conducted at Lahore University of Management Sciences (LUMS) and Seoul National University (SNU) as part of the BS Mathematics degree. I begin with motivating problems in enumerative geometry, then develop the necessary mathematical machinery: Young tableaux and their algebraic and combinatorial properties, cellular cohomology, symmetric polynomials, and some fundamental objects from algebraic geometry. Finally, I present an introduction to Schubert calculus and the theory of intersections of Schubert varieties, demonstrating how these tools solve problems in arithmetic and incidence geometry.

The proofs of two main theorems were also sketched during the talk: first, how the Schubert cell structure gives rise to the cohomology of the Grassmannian; and second, an isomorphism between said cohomology ring and the ring of symmetric polynomials modulo the ideal generated by Schur functions defined in the appropriate way via Young tableaux.