Fabian Ziltener

A quantum Kirwan map: Bubbling and Fredholm theory for symplectic vortices over the plane

Consider a Hamiltonian action of a compact connected Lie group on a symplectic manifold

Discontinuous symplectic capacities

(M,\omega)

Coisotropic submanifolds, leaf-wise fixed points, and presymplectic embeddings

. Conjecturally, under suitable assumptions there exists a morphism of cohomological field theories from the equivariant Gromov-Witten theory of

The invariant symplectic action and decay for vortices

(M,\omega)

Symplectic vortices on the complex plane and quantum cohomology

to the Gromov-Witten theory of the symplectic quotient. The morphism should be a deformation of the Kirwan map. The idea, due to D. A. Salamon, is to define such a deformation by counting gauge equivalence classes of symplectic vortices over the complex plane

Coisotropic displacement and small subsets of a symplectic manifold

C

A Maslov Map for Coisotropic Submanifolds, Leaf-wise Fixed Points and Presymplectic Non-Embeddings

. The present memoir is part of a project whose goal is to make this definition rigorous. Its main results deal with the symplectically aspherical case. The first one states that every sequence of equivalence classes of vortices over the plane has a subsequence that converges to a new type of genus zero stable map, provided that the energies of the vortices are uniformly bounded. Such a stable map consists of equivalence classes of vortices over the plane and holomorphic spheres in the symplectic quotient. The second main result is that the vertical differential of the vortex equations over the plane (at the level of gauge equivalence) is a Fredholm operator of a specified index. Potentially the quantum Kirwan map can be used to compute the quantum cohomology of symplectic quotients.

A symplectically non-squeezable small set and the regular coisotropic capacity

We show that the spherical capacity is discontinuous on a smooth family of ellipsoidal shells. Moreover, we prove that the shell capacity is discontinuous on a family of open sets with smooth connected boundaries.