Cotangent Schubert Calculus in Grassmannians

dc.contributor.authorOetjen, David Christopheren
dc.contributor.committeechairMihalcea, Constantin Leonardoen
dc.contributor.committeememberShimozono, Mark M.en
dc.contributor.committeememberHaskell, Peter E.en
dc.contributor.committeememberOrr, Daniel D.en
dc.description.abstractWe find formulas for the Segre-MacPherson classes of Schubert cells in T-equivariant cohomology and the motivic Segre classes of Schubert cells in T-equivariant K-theory. In doing so we look at the pushforward of the projection map from the Bott-Samelson (Kempf-Laksov) desingularization to the Grassmannian. We find that the Segre-MacPherson classes are stable under pullbacks of maps embedding a Grassmannian into a bigger Grassmannian. We also express these formulas using certain Demazure-Lusztig operators that have previously been used to study these classes.en
dc.description.abstractgeneralSchubert calculus was first introduced in the nineteenth century as a way to answer certain questions in enumerative geometry. These computations relied on the multiplication of Schubert classes in the cohomology ring of Grassmannians, which parameterize k-dimensional linear subspaces of a vector space. More recently Schubert calculus has been broadened to refer to computations in generalized cohomology theories, such as (equivariant) K-theory. In this dissertation, we study Segre-MacPherson classes and motivic Segre classes of Schubert cells in Grassmannians. Segre-MacPherson classes are related to Chern-Schwartz-MacPherson classes, which are a generalization to singular spaces of the total Chern class of the tangent bundle. Motivic Segre classes are similarly related to motivic Chern classes, which are a K-theory analogue of Chern-Schwartz-MacPherson classes. This dissertation also studies the relationship between Schubert varieties and their Bott-Samelson desingularizations, specifically their (T-equivariant) cohomology and K-theory rings. Since equivariant cohomology (or K-theory) classes can be represented by polynomials, we can represent the Segre-MacPherson (or motivic Segre) classes as rational functions. Furthermore, we use certain operators that act on such polynomials (or rational functions) to find formulas for the rational function representatives of the aforementioned classes.en
dc.description.degreeDoctor of Philosophyen
dc.publisherVirginia Techen
dc.rightsIn Copyrighten
dc.subjectEquivariant Cohomologyen
dc.subjectEquivariant K-Theoryen
dc.subjectBott-Samelson Varietyen
dc.subjectSegre-MacPherson Classen
dc.subjectMotivic Segre Classen
dc.subjectDemazure-Lusztig Operatoren
dc.titleCotangent Schubert Calculus in Grassmanniansen
dc.typeDissertationen Polytechnic Institute and State Universityen of Philosophyen


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