• 2:10-2:25  Isabelle Charton, University of Haifa

Symplectic Fano Manifolds and Fano Varieties

A symplectic Fano manifold is a compact symplectic manifold, whose first Chern class is equal to the class given by the symplectic form. The algebraic counter parts of these are Fano varieties. In this talk I would like discuss the different of these objects under the assumption of the existence of suitable symmetries.

• 2:30-2:45  Jeffrey Carlson, Imperial College London

The topology of Gelfand–Zeitlin fibers

It is known that fibers of unitary and orthogonal Gelfand–Zeitlin systems are iterated pullbacks of homogeneous spaces, but one could wish the resulting expressions were more explicit. We provide a new interpretation of GZ fibers as biquotients HG/K of Lie groups which both sheds light on previously known facts and makes much previously unknown structure visible.

In no particular order, we find (1) a direct product decomposition into biquotient factors indexed by eigenvalues, (2) a torus direct factor, which we show to be that acting via the Thimm trick, (3) a continuous equivariant map from a coadjoint orbit onto a toric manifold (plausibly that underlying the time-one gradient-Hamiltonian flow of a known toric degeneration in the unitary case), (4) a new local model for the part of a coadjoint orbit lying over a ray in the GZ polytope, and (5) the cohomology rings and (6) first three homotopy groups of GZ fibers.

All of these expressions can be read off of the combinatorics of a triangle of inequalities between prescribed eigenvalues. This represents joint work with Jeremy Lane.

• 2:50-3:05  Maarten Mol, Max Planck Institute for Mathematics, Bonn

Stratification of the transverse momentum map

Let $G$ be a compact Lie group. The momentum map $J:(S,\omega)$ to $\mathfrak{g}^*$ of a Hamiltonian $G$-action descends to a map from the orbit space $S/G$ into the orbit space $\mathfrak{g}^*/G$, that we call the transverse momentum map. Although both of these orbit spaces are naturally stratified by orbit types, the transverse momentum map need not be a map of stratified spaces. The aim of this talk is to describe a natural refinement of the orbit type stratification of $S/G$ with respect to which this map does become a map of stratified spaces. We will discuss various properties of this stratification, such as its compatibility with the Poisson geometry of $S/G$ and the symplectic reduced spaces.

• 3:10-3:25  Wilmer Smilde, University of Illinois at Urbana-Champaign

Lie groups of Poisson diffeomorphisms

Although the symplectomorphism group is a well-studied object, little is known about the symmetry groups of other Poisson manifolds. This is partially due to a lack of understanding of what kind of object the Poisson diffeomorphism group actually is. 

In this talk, I will present a strategy to obtain infinite-dimensional Lie group structures on groups of Poisson diffeomorphisms on a Poisson manifold. The idea is to recast the graph of a Poisson diffeomorphism as a bisection of a Poisson groupoid. Then, if one can linearize the Poisson structure on the Poisson groupoid, the group of coisotropic bisections can be endowed with a Lie group structure.

The results can be applied to many classes of well-studied Poisson structures, including "almost everywhere symplectic Poisson manifolds" (e.g. log-symplectic, elliptic symplectic), as well as regular Poisson manifolds with a tractable transverse geometry, such as cosymplectic manifolds and regular Poisson manifolds of proper type.