Joint IAS/Princeton/Montreal/Paris/Tel-Aviv Symplectic Geometry Zoominar

Yao Xiao : Equivariant Lagrangian Floer Theory on Compact Toric Manifolds

We introduce an equivariant Lagrangian Floer theory on compact symplectic toric manifolds. We define a spectral sequence to compute the equivariant Floer cohomology. We show that the set of pairs $(L,b)$, each consisting of a Lagrangian torus fiber and a weak bounding cochain, that have non-zero equivariant Lagrangian Floer cohomology forms a rigid analytic space (over the non-Archimedean Novikov field). We prove that the dimension of such a rigid analytic space is equal to that of the acting group in certain cases. We will discuss some examples.

Yoav Zimhony : Commutative Control Data for Smoothly Locally Trivial Stratified Spaces

For a compact Lie group G and a Hamiltonian G-space M, can we find a smooth weak deformation retraction from a neighbourhood of the zero level set of the momentum map onto it? If we do not require smoothness then this is already known, in fact one can obtain a strong deformation retraction. We will outline the construction of such smooth weak deformation retraction with the following steps.
First we show that the zero level set, stratified by orbit types, satisfies a condition stronger than Whitney (B) regularity - smooth local triviality with conical fibers. Using this local condition we construct control data in the sense of Mather with the additional properties that the fiber-wise multiplications by scalars, coming from the tubular neighbourhood structures, preserve strata and commute with each other. Finally we use this control data to obtain the neighbourhood smooth weak deformation retraction.
We will also discuss a key technical tool used in the construction of the control data - Euler-like vector fields.

Qi Feng : Symplectic Squeezing of Domains in $T^*T^n$

The symplectic squeezings in the cotangent bundle of a torus is distinct from those in $R^{2n}$, due to the nontrivial topology of the torus. In this talk, we will show that for $n\ge2$ any bounded domain of $T^*T^n$ can be symplectically embedded into a trivial subbundle with an irrational cylinder fiber. These symplectic embeddings are constructed based on Arnold's cat map, Dirichlet's approximation theorem, and Bézout’s identity. Our result resolves an open problem posted by Gong-Xue (stated in $n=2$) and also generalizes it to higher dimensional situations. This talk is based on joint work with Jun Zhang