On a question of Dusa McDuff

Dusa McDuff and symplectic geometry

I describe some of McDuff's contributions to symplectic geometry, with a focus on symplectic embedding problems.

单调吹胀在辛托罗形几何中的结构及麦杜夫的问题

The structure of monotone blow-ups in symplectic toric geometry and a question of McDuff

研究单调多面体的吹胀结构，证明仅简单形和其二维吹胀体允许顶点吹胀，并回答麦杜夫2011年提出的问题。

Whitehead products in symplectomorphism groups and Gromov-Witten invariants

Consider any symplectic ruled surface $(M^g_λ,ω_λ)$ given by $(Σ_g \times S^2, λσ_{Σ_g} \oplus σ_{S^2})$. We compute all natural equivariant Gromov-Witten invariants $EGW_{g,0}(M^g_λ;H_k, A-kF)$ for all hamiltonian circle actions $H_k$ on $M^g_λ$, where $A=[Σ_g \times pt]$ and $F= [pt \times S^2]$. We use these invariants to show the nontriviality of certain higher order Whitehead products that live in the homotopy groups of the symplectomorphism groups $G_λ^g$, $g \geq 0$. Our results are sharper when $g=0,1$ and enable us to answer a question posed by D.McDuff in the case $g=1$ and provide a…

Symplectic embeddings of 4-dimensional ellipsoids

We show how to reduce the problem of symplectically embedding one 4-dimensional rational ellipsoid into another to a problem of embedding disjoint unions of balls into appropriate blow ups of \C P^2. For example, the problem of embedding the ellipsoid E(1,k) into a ball B is equivalent to that of embedding k disjoint equal balls into \C P^2, and so can be solved by the work of Gromov, McDuff--Polterovich and Biran. (Here k is the ratio of the area of the major axis to that of the minor axis.) As a consequence we show that the ball may be fully filled by the ellipsoid E(1,k) for k=1,4 and all k…

环面纤维的哈密顿分类与对称探针

Hamiltonian Classification of toric fibres and symmetric probes

本文研究环面辛流形中环面纤维在哈密顿微分同胚下的等价分类问题，提出对称探针构造方法并结合 Chekanov 分类给出阻碍，进而猜想并部分证明对称探针给出全部等价，同时确定若干例子的哈密顿单值群。