Yael Karshon

Moment maps, cobordisms, and Hamiltonian group actions

Introduction Part 1. Cobordism: Hamiltonian cobordism Abstract moment maps The linearization theorem Reduction and applications Part 2. Quantization: Geometric quantization The quantum version of the linearization theorem Quantization commutes with reduction Part 3. Appendices: Signs and normalization conventions Proper actions of Lie groups Equivariant cohomology Stable complex and Spin

Euler-Maclaurin with remainder for a simple integral polytope

^{\mathrm{c}}

Centered complexity one Hamiltonian torus actions

structures Assignments and abstract moment maps Assignment cohomology Non-degenerate abstract moment maps Characteristic numbers, non-degenerate cobordisms, and non-virtual quantization The Kawasaki Riemann-Roch formula Cobordism invariance of the index of a transversally elliptic operator Bibliography Index.

The Gromov width of complex Grassmannians

We give an Euler Maclaurin formula with remainder for the sum of the values of a smooth function on the integral points in a simple integral polytope. This formula is proved by elementary methods.

Cobordism theory and localization formulas for Hamiltonian group actions

We consider symplectic manifolds with Hamiltonian torus actions which are “almost but not quite completely integrable”: the dimension of the torus is one less than half the dimension of the manifold. We provide a complete set of invariants for such spaces when they are “centered” and the moment map is proper. In particular, this classifies the preimages under the moment map of all sufficiently small open sets, which is an important step towards global classification. As an application, we construct a full packing of each of the Grassmannians Gr(2,R5) and Gr(2,R6) by two equal symplectic balls.

Orbifolds as diffeologies

We show that the Gromov width of the Grassmannian of com- plex k-planes in C n is equal to one when the symplectic form is normalized so that it generates the integral cohomology in degree 2. We deduce the lower bound from more general results. For example, if a compact manifold N with an integral symplectic form ω admits a Hamiltonian circle action with a fixed point p such that all the isotropy weights at p are equal to one, then the Gromov width of (N, ω) is at least one. We use holomorphic techniques to prove the upper bound. AMS Classification 53D20; 53D45

Classification of Hamiltonian torus actions with two-dimensional quotients

We announce the following result and give several applications: A Hamiltonian

Smooth Lie group actions are parametrized diffeological subgroups

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Basic Forms and Orbit Spaces: a Diffeological Approach

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Convexity package for momentum maps on contact manifolds

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