The main result of this paper is the construction a family superintegrable Hamiltonian systems on moduli spaces flat connections principle $G$-bundle surface. space Poisson variety with Atiyah-Bott structure. Among particular cases such are spin generalizations Ruijsenaars-Schneider models.

We describe a natural action on the Quot scheme, Rd compactifying the space of degree d maps from P to the Grassmannian of lines. We identify the fixed points components for this action and the weights of the normal bundle of these components. We compute the degree of this Quot scheme under the generalized Plücker embedding by applying Atiyah-Bott localization formula.

We give a detailed proof of the well-known classical fact that every symplectic circle action on a compact manifold admits a circle valued momentum map relative to some symplectic form. This momentum map is Morse-Bott-Novikov and each connected component of the fixed point set has even index. These proofs do not appear to be written elsewhere.

We discuss unifying features of topological field theories in 2, 3 and 4 dimensions. This includes relations among enumerative geometry (2d topological field theory) link invariants (3d Chern-Simons theory) and Donaldson invariants (4d topological theory). (Talk presented in conference on Geometry and Topology in honor of M. Atiyah, R. Bott, F. Hirzebruch and I. Singer, Harvard University, May ...

For Hamiltonian flows we establish the existence of periodic orbits on a sequence of level sets approaching a Bott-nondegenerate symplectic extremum of the Hamiltonian. As a consequence, we show that a charge on a compact manifold with a nondegenerate (i.e. symplectic) magnetic field has periodic orbits on a sequence of energy levels converging to zero.

We review the localization formula due to Berline-Vergne and Atiyah-Bott, with applications to the exact stationary phase phenomenon discovered by Duistermaat-Heckman. We explain the Weil model of equivariant cohomology and recall its relation to BRST. We show how to quantize the Weil model, and obtain new localization formulas which, in particular, apply to Hamiltonian spaces with group valued...

We study the topological K-theory spectrum of the dg singularity category associated to a weighted projective complete intersection. We calculate the topological K-theory of the dg singularity category of a weighted projective hypersurface in terms of its Milnor fiber and monodromy operator, and, as an application, we obtain a lift of the Atiyah-Bott-Shapiro construction to the level of spectra.