We analyze S equivariant cohomology from the supergeometrical point of view. For this purpose we equip the external algebra of given manifold with equivariant even super(pre)symplectic structure, and show, that its Poincare-Cartan invariant defines equivariant Euler classes of surfaces. This allows to derive localization formulae by use of superanalog of Stockes theorem.

We construct a Thom class in complex equivariant elliptic cohomology extending the equivariant Witten genus. This gives a new proof of the rigidity of theWitten genus, which exhibits a close relationship to recent work on non-equivariant orientations of elliptic spectra.

We give positive formulas for the restriction of a Schubert Class to a T -fixed point in the equivariant K-theory and equivariant cohomology of the Grassmannian. Our formulas rely on a result of Kodiyalam-Raghavan and Kreiman-Lakshmibai, which gives an equivariant Gröbner degeneration of a Schubert variety in the neighborhood of a T -fixed point of the Grassmannian.

We give positive formulas for the restriction of a Schubert Class to a T -fixed point in the equivariant K-theory and equivariant cohomology of the Lagrangian Grassmannian. Our formulas rely on a result of GhorpadeRaghavan, which gives an equivariant Gröbner degeneration of a Schubert variety in the neighborhood of a T -fixed point of the Lagrangian Grassmannian.

We establish an equivariant quantum Giambelli formula for partial flag varieties. The answer is given in terms of a specialization of universal double Schubert polynomials. Along the way, we give new proofs of the presentation of the equivariant quantum cohomology ring, as well as Graham-positivity of the structure constants in equivariant quantum Schubert calculus.

Let LX be the space of free loops on a simply connected manifoldX. When the real cohomology of X is a tensor product of algebras generated by a single element, we determine the algebra structure of the real cohomology of LX by using the cyclic bar complex of the de Rham complex Ω(X) of X. In consequence, the algebra generators of the real cohomology of LX can be represented by differential form...