A note on generating functions

A note on C^0 rigidity of Hamiltonian isotopies

We show that a symplectic isotopy that is a $C^0$ limit of Hamiltonian isotopies is itself Hamiltonian, if the corresponding sequence of generating Hamiltonians converge in $L^{(1, \infty)}$ topology.

On Legendrian cobordisms and generating functions

This note concerns Legendrian cobordisms in one-jet spaces of functions, in the sense of Arnol'd \cite{Arnold} -- consisting of big Legendrian submanifolds between two smaller ones. We are interested in such cobordisms which fit with generating functions, and wonder which structures and obstructions come with this notion. As a central result, we show that the classes of Legendrian concordances with respect to the generating function equipment can be given a group structure. To this construction we add one of a homotopy with respect to generating functions.

The generating function of the embedding capacity for 4-dimensional symplectic ellipsoids

Quite recently, McDuff showed that the existence of a symplectic embedding of one four-dimensional ellipsoid into another can be established by comparing their corresponding sequences of ECH capacities. In this note we show that these sequences can be encoded in a generating function, which gives several new equivalent formulations of McDuff's theorem.

Contact spectral invariants and persistence

This sketch shows that the usual generating function based capacities have an interpretation in the language of persistent homology as persistences of certain homology classes in the persistence module formed by the corresponding generating function homology groups. This viewpoint suggests various new invariants, in particular a $\mathbb{Z}_k$-equivariant capacity which can be used to prove orderability of lens spaces, proved by Milin (2008) using contact homology and by Sandon (2010) using equivariant generating function homology. These are informal notes originally circulated in January 2014…

通过生成函数的拉格朗日同伦球的框架配边

Framed bordism of Lagrangian homotopy spheres via generating functions

利用Bökstedt-Waldhausen关于管道的导数映射结果和Abouzaid-Courte-Guillermou-Kragh的管道型生成函数存在定理，证明了邻近拉格朗日同伦球的光滑结构限制，即若同伦n-球L拉格朗日嵌入到另一同伦n-球M的余切丛中，则θ_n/bP_{n+1}中的差[L]-[M]是Hopf元素η∈π^1_s的倍数，从而为2-挠元，且当n为偶数时L#L微分同胚于M#M；特别地，若同伦8-球L拉格朗日嵌入T^*S^8，则L微分同胚于S^8。

Quantitative Tamarkin category

This is a lecture note from a seminar course given at Tel Aviv University in Spring 2018. Part of the preliminary section is built from Kazhdan's seminar organized in the Hebrew University of Jerusalem in Fall 2017. The main topic of this note is a detailed introduction of Tamarkin category theory from a quantitative perspective, followed by a demonstration of various applications in symplectic topology. Many examples are provided in order to obtain certain geometric intuitions of those abstract algebraic constructions in Tamarkin category. In this note, we try to explain how standard symplect…