Undergraduate thesis research in algebraic geometry on Schubert varieties, intersection theory, and the structure of the cohomology ring of Grassmannians, culminating in a defended thesis presentation.
Available materials: Thesis (PDF) Defense Slides (PDF)
While counting is the simplest mathematical exercise, it can become notoriously involved when we try to enumerate geometric objects and in particular their intersections; this is precisely the subject of the 15th Hilbert Problem (of the famous 23 published in 1900), namely the formalization of Schubert Calculus. We will develop remarkable connections between Algebraic Geometry, Topology and Combinatorics through the enumerative tools of this subject. There is a surprising translation between structures on topological invariants (Cohomology Groups) and structures of purely combinatorial objects (Young Tableaux and Symmetric Polynomials) of a very important class of geometric objects: a differentiable manifold known as the Grassmannian. These translations rely on very powerful result in Topology, namely Poincaré Duality. We will develop necessary general preliminaries (e.g. an Introduction to Cohomology, the Calculus of Tableaux, structures on the Grassmannian) and then consider these translations in particular. Applications of Schubert Calculus and its ideas range from Intersection theory, to even purely geometric questions, e.g. the Toeplitz Square Peg problem, but the interesting nature of these unlikely mathematical connections warrants attention in its own right.