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Schubert Calculus of Flag Manifolds

Student-Run Applied & Math Seminar

Speaker: Nate Gallup, UC Davis
Location: 2112 MSB
Start time: Wed, Oct 23 2013, 12:10PM

In the field of algebraic topology, cohomology is an algebraic topological invariant that, because of its ring structure, is especially useful for providing information about topological spaces. Quantum cohomology, used mainly in algebraic geometry, is in general a slightly more refined version of cohomology in the sense that it often tells us more information about a topological space than its classical counterpart. In this talk we will study flag manifolds, in particular the Grassmannian and the full flag manifold, using classical and quantum Borel-Moore cohomology. The classical and quantum cohomology rings of flag manifolds each have a finite basis of Schubert classes and there exist formulas to express the classical product of simple Schubert classes in terms of these bases. In their 1999 paper Quantum Multiplication of Schur Polynomials, A. Bertram, I. Ciocan-Fontanine, and Fulton provide an algorithm to express the quantum products of two Schubert classes in the quantum cohomology ring of the Grassmannian in terms of this basis. They do so by first computing the classical product in a higher dimensional Grassmannian, and then removing rim hooks from the Young diagrams that index the classes while adding in powers of the quantum variable based on properties of the rim hooks removed. This talk describes work towards finding an analog of the rim hook rule for general flag manifolds.