Project description

Our goal is to lift classical Schubert calculus result from ordinary non-equivariant cohomology of Grassmannians and flag manifolds to the C_C-equivariant setting. To do this we will use recent developments in equivariant homotopy theory. In particular, the existence of pushforwards along bundle maps as developed by Costenoble and Waner. Specifically, we wish to do this for projective bundles associated to the universal bundles that such Grassmannians and flag manifolds come naturally equipped with. This will lead us to a notion of Segre classes, which will constitute the basic building block of a Giambelli formula expressing the fundamental classes of all Schubert varieties. On the other hand, in the special case of P^1-bundles, this will allow us to define an analogue of divided difference operators on the equivariant cohomology of flag manifolds. We would then like to compute the structure constants of equivariant real and complex Grassmannians. While many computations have been undertaken by Costenoble, Dugger, and Hogle, those computations do not include the data of these structure constants. Further, there is not yet an adequate interpretation of the relevant cohomology in terms of characteristic classes of vector bundles. Using techniques and insight gained from generalized Schubert Calculus, in the setting of algebraic cobordism, we hope to understand the appropriate C_2-equivariant generalization of the relevant combinatorial structure. We also hope to find the appropriate generalization of the ring of symmetric functions where the polynomials representing the Schubert classes live. Using these particular methods will ensure that our results will lend themselves to the analogous computations in the associated equivariant motivic cohomology theories.

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