拓扑Hamilton流生成Hamilton算子的唯一性

通过生成函数研究哈密顿辫

Hamiltonian Braids via Generating Functions

本文利用生成函数刻画平面紧支撑哈密顿微分同胚不动点的辫型与环绕数，并给出拓扑熵对 Hofer 范数下半连续性定理的有限维证明。

Uniqueness of real Lagrangians up to cobordism

We prove that a real Lagrangian submanifold in a closed symplectic manifold is unique up to cobordism. We then discuss the classification of real Lagrangians in $\mathbb{C} P^2$ and $S^2\times S^2$. In particular, we show that a real Lagrangian in $\mathbb{C} P^2$ is unique up to Hamiltonian isotopy and that a real Lagrangian in $S^2\times S^2$ is either Hamiltonian isotopic to the antidialgonal sphere or Lagrangian isotopic to the Clifford torus.

论Hüssen几何的唯一性

On the Uniqueness of Hofer's Geometry

本文研究了闭辛流形上光滑函数空间上的范数类，它们在Hamilton同态群的作用下是不变的。我们的主要结果表明，任何这样的范数是连续的关于$C^{\infty}$ -拓扑，是从上支配的$L_{\infty}$ -范数。作为推论，我们得到了由满足上述连续性假设的不变范数所生成的Hamilton同构群上的任何双不变Finsler伪度量，要么同为零，要么等价于Hessel度量.

哈密顿微分同胚的生成函数条形码的逼近

Approximation of Generating Function Barcode for Hamiltonian Diffeomorphisms

本文利用莫尔斯理论定义哈密顿微分同胚的生成函数条形码不变量，并给出逼近该条形码的有限步显式算法。

Generating families on arboreal graphs

The paper is devoted to the study of exact curves on Arborealized Liouville surfaces. We introduce the notion of a generating family for such curves. Our main statement is a hamiltonian lifting property: the set of curves admitting a generating family is closed with respect to Hamiltonian isotopes. This is part of the author's future thesis. It will be translated into English within a few weeks. In the future, we plan to generalize our results in several directions and find applications.

On periodic points of Hamiltonian diffeomorphisms of $\mathbb{C}\text{P}^d$ via generating functions

Inspired by the techniques of Givental and Théret, we provide a proof with generating functions of a recent result of Ginzburg-Gürel concerning the periodic points of Hamiltonian diffeomorphisms of $\mathbb{C}\text{P}^d$. For instance, we are able to prove that fixed points of pseudo-rotations are isolated as invariant sets or that a Hamiltonian diffeomorphism with a hyperbolic fixed point has infinitely many periodic points.

同调或上同调类中不变Lagrange图的唯一性

Uniqueness of Invariant Lagrangian Graphs in a Homology or a Cohomology Class

给定余切空间$T^*M$上的光滑紧致黎曼流形$M$和Hamilton流形$H$，证明了在给定的同调类或上同调类内，某些遍历不变Lagrange图的唯一性.特别地，在拟可积的哈密顿系统的情况下，我们的结果意味着具有旋转向量$ρ$的拉格朗日KAM环面的全局唯一性。这个结果一般地推广到KAM环面的$C^0$ -闭包。

The Morse index of Chaperon's generating families

This is an expository paper devoted to the Morse index of Chaperon's generating families of Hamiltonian diffeomorphisms. After reviewing the construction of such generating families, we present Bott's iteration theory in this setting: we study how the Morse index of a critical point corresponding to an iterated periodic orbit depends on the order of iteration of the orbit. We also investigate the precise dependence of the Morse index from the choice of the generating family associated to a given Hamiltonian diffeomorphism, which will allow to see the Morse index as a Maslov index for the linea…

Stein流形中的Lagrange合痕

Lagrangian isotopies in Stein manifolds

我们建立了某些四维Stein流形中Lagrange球面的唯一性直到Hamilton合痕。

Generating the Fukaya categories of Hamiltonian G-manifolds

Let $G$ be a compact Lie group and $\mathbf{k}$ be a field of characteristic $p \geq 0$ such that $H^* (G)$ does not have $p$-torsion. We show that a free Lagrangian orbit of a Hamiltonian $G$-action on a compact, monotone, symplectic manifold $X$ split-generates an idempotent summand of the monotone Fukaya category $\mathcal{F}(X; \mathbf{k})$ if and only if it represents a non-zero object of that summand (slightly more general results are also provided). Our result is based on: an explicit understanding of the wrapped Fukaya category $\mathcal{W}(T^*G; \mathbf{k})$ through Koszul twisted com…

Complex Germen on invariant isotropic tori under the Hamiltonian phases flow with in involution Hamilton functions

M. M. Nekhoroshev put forward the problem of to find the Complex Germ on a isotropic invariant torus with respect to Hamiltonian phases flows which come from k-functions in involution. This statement was partially solved in [9] establishing that if certain simplectic operator has a simple spectrum then the complex germ exist. In this work we solve this problem, providing a full solution, i.e. we present conditions for the existence and uniqueness of complex germ through the monodromy operator constructed in [9], but without the simple spectrum condition. We study also the Hamiltonian system wi…