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.

LetG be a compact Lie group. LetM be a smoothG-manifold and V → M be an oriented G-equivariant vector bundle. One defines the spaces of equivariant forms with generalized coefficients on V and M . An equivariant Thom form θ on V is a compactly supported closed equivariant form such that its integral along the fibres is the constant function 1 on M . Such a Thom form was constructed by Mathai an...

The aim of this article is to introduce a new class SO(2)equivariant transversal maps T R(cl(Ω), ∂Ω) and to define degree theory for such maps. We define degree for SO(2)-equivariant transversal maps and prove some properties of this invariant. Moreover, we characterize SO(2)-equivariant transversal isomorphisms and derive formula for degree of such isomorphisms.

We introduce a Bredon motivic cohomology theory for smooth schemes defined over a field and equipped with an action by a finite group. These cohomology groups are defined for finite dimensional representations as the hypercohomology of complexes of equivariant correspondences in the equivariant Nisnevich topology. We generalize the theory of presheaves with transfers to the equivariant setting ...

Consider a Hamiltonian action of a compact Lie Group on a symplectic manifold which has the strong Lefschetz property. We establish an equivariant version of the Merkulov-Guillemin dδ-lemma for equivariant differential forms with smooth or distributional coefficient. As a corollary we also obtain a version of equivariant formality theorem in this case.

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 find presentations by generators and relations for the equivariant quantum cohomology of the Grassmannian. For these presentations, we also find determinantal formulae for the equivariant quantum Schubert classes. To prove this, we use the theory of factorial Schur functions and a characterization of the equivariant quantum cohomology ring.

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 present the review of noncommutative symmetries applied to Connes’ formulation of spectral triples. We introduce the notion of equivariant spectral triples with Hopf algebras as isometries of noncommutative manifolds, relate it to other elements of theory (equivariant K-theory, homology, equivariant differential algebras) and provide several examples of spectral triples with their isometries...

We explore the C2-equivariant spectra Tmf1(3) and TMF1(3). In particular, we compute their C2-equivariant Picard groups and the C2-equivariant Anderson dual of Tmf1(3). This implies corresponding results for the fixed point spectra TMF0(3) and Tmf0(3). Furthermore, we prove a Real Landweber exact functor theorem.