Yael Karshon
Professor University of Toronto, Department of Mathematics
University of Toronto, Department of Mathematics
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Basic forms and orbit spaces: a diffeological approach
SIGMA 122016
If a Lie group acts on a manifold freely and properly, pulling back by the quotient map gives an isomorphism between the differential forms on the quotient manifold and the basic differential forms upstairs. We show that this result remains true for actions that are not necessarily free nor proper, as long as the identity component acts properly, where on the quotient space we take differential forms in the diffeological sense.
Non-compact symplectic toric manifolds
SIGMA 112015
A key result in equivariant symplectic geometry is Delzant's classification of compact connected symplectic toric manifolds. The moment map induces an embedding of the quotient of the manifold by the torus action into the dual of the Lie algebra of the torus; its image is a unimodular ("Delzant") polytope; this gives a bijection between unimodular polytopes and isomorphism classes of compact connected symplectic toric manifolds.
Counting toric actions on symplectic four-manifolds
Mathematical Reports of the Academy of Science2015
Given a symplectic manifold, we ask in how many different ways can a torus act on it. Classification theorems in equivariant symplectic geometry can sometimes tell that two Hamiltonian torus actions are inequivalent, but often they do not tell whether the underlying symplectic manifolds are (non-equivariantly) symplectomorphic.
Distinguishing symplectic blowups of the complex projective plane
Journal of Symplectic Geometry2014
A symplectic manifold that is obtained from the complex projective plane by k blowups is encoded by k+1 parameters: the size of the initial complex projective plane, and the sizes of the blowups. We determine which values of these parameters yield symplectomorphic manifolds.
Classification of Hamiltonian torus actions with two dimensional quotients
Geometry and Topology2014
We construct all possible Hamiltonian torus actions for which all the non-empty reduced spaces are two dimensional (and not single points) and the manifold is
connected and compact, or, more generally, the moment map is proper as a map to a convex set.
