The quantitative nature of reduced Floer theory

On the Colored Jones Polynomial, Sutured Floer homology, and Knot Floer homology

Let K in S^3 be a knot, and let \widetilde{K} denote the preimage of K inside its double branched cover, Σ(K). We prove, for each integer n > 1, the existence of a spectral sequence from Khovanov's categorification of the reduced n-colored Jones polynomial of the mirror of K to the knot Floer homology of (Σ(K),\widetilde{K}) (when n odd) and to (S^3, K # K) (when n even). A corollary of our result is that Khovanov's categorification of the reduced n-colored Jones polynomial detects the unknot whenever n>1.

Floer homology for negative line bundles and Reeb chords in pre-quantization spaces

In this article we prove existence of Reeb orbits for Bohr-Sommerfeld Legendrians in certain pre-quantization spaces. We give a quantitative estimate from below. These estimates are obtained by studying Floer homology for fibre-wise quadratic Hamiltonian functions on negative line bundles.

相对 Rabinowitz-Floer 复形的持续性

The persistence of a relative Rabinowitz-Floer complex

本文通过 Lagrangian 余配边迹构造 Legendrian 的 DGA 稳定 tame 同构，建立满足定量不变性的相对 Rabinowitz-Floer 复形，并利用其持续同调条形码证明位移能量界、非退化猜想及实射影空间的不可位移性。

Floer 理论中的标架 $E_2$ 结构

Framed $E_2$ structures in Floer theory

本文在 Hamilton Floer 理论中构造标架 $E_2$ operad 作用，建立带支撑辛上同调链模型及其函子性，并给出同伦 $BV$ 代数运算与几何有界情形推广。

定量 Heegaard Floer 上同调与 Calabi 不变量

Quantitative Heegaard Floer cohomology and the Calabi invariant

本文在任意亏格紧致曲面上定义与 Lagrangian links 相关的新谱不变量，证明其极限恢复 Calabi 不变量，并借此解决 Hamiltonian 同胚群非单性、Calabi 同态延拓及二维球面 quasimorphisms 构造等问题。

接触哈密顿动力学中的定量刻画——I

Quantitative characterization in contact Hamiltonian dynamics -- I

本文基于接触哈密顿Floer理论，为容许接触哈密顿系统构造间隙模块，定义接触谱不变量，并证明其稳定性与三角不等式，为接触动力学提供定量刻画。

BV双代数结构在Floer理论与弦拓扑中的应用

BV bialgebra structures in Floer theory and string topology

本文从模空间紧化拓扑导出BV双代数结构，并构造于约化辛同调、Rabinowitz Floer同调及非等变弦拓扑，同时展示等变情形下Lie双代数结构的形式推论。

定量托马斯-丘唯一性

Quantitative Thomas-Yau uniqueness

该研究在弗洛尔理论条件下，通过拉格朗日角小尺度性得到两个拉格朗日量接近程度的定量估计，应用于特殊拉格朗日量的强弱唯一性定理及变分收敛性刻画。

定量切触哈密顿动力学

Quantitative contact Hamiltonian dynamics

本文基于Merry-Uljarević的切触哈密顿弗洛尔理论，对弱⁺-单调辛流形边界的任意可接受切触哈密顿函数开展定量研究，发展切触谱不变量理论，建立多项基本结果并发现弗洛尔群的非传统滤过结构。

Reduction of symplectic homeomorphisms

In a previous article, we proved that symplectic homeomorphisms preserving a coisotropic submanifold C, preserve its characteristic foliation as well. As a consequence, such symplectic homeomorphisms descend to the reduction of the coisotropic C. In this article we show that these reduced homeomorphisms continue to exhibit certain symplectic properties. In particular, in the specific setting where the symplectic manifold is a torus and the coisotropic is a standard subtorus, we prove that the reduced homeomorphism preserves spectral invariants and hence the spectral capacity. To prove our main…

Traceless Character Varieties, the Link Surgeries Spectral Sequence, and Khovanov Homology

In arXiv:1611.09927, we constructed a well-defined Lagrangian Floer invariant for any closed, oriented $3$-manifold $Y$ via the symplectic geometry of so-called traceless $\mathrm{SU}(2)$-character varieties. This invariant, $\mathrm{SI}(Y)$, which we refer to as the symplectic instanton homology of $Y$, was also shown to satisfy an exact triangle for Dehn surgeries on knots which is typical of Floer-theoretic invariants of $3$-manifolds. In this article, we demonstrate further structural properties of this symplectic instanton homology. For example, Floer theories are expected to roughly sati…

条形码熵与相对辛上同调

Barcode entropy and relative symplectic cohomology

本文研究了Liouville域K嵌入到辛流形M中的相对辛上同调的条形码熵，并建立了其与边界上Reeb流拓扑熵的定量联系。

整数域上的谱不变量

Spectral invariants over the integers

本研究在哈密顿Floer同调框架下探究谱不变量对系数的依赖性，发现$\boldsymbol{\boldsymbol{\boldsymbol{Z}}}$系数下谱范数在复射影空间无界、域系数下一致有界的独有现象，解答了Hingston问题的辛版本，并给出相关动力学与几何应用。

边界深度与辛上同调的形变

Boundary Depth and Deformations of Symplectic Cohomology

本文研究 Liouville 域的环境与内蕴辛上同调关系，通过滤链与定量控制证明边界深度足够小时前者是后者的形变，并用于构造 SYZ 镜像局部片段。

Eigenvalue Distributions of Reduced Density Matrices

Given a random quantum state of multiple distinguishable or indistinguishable particles, we provide an effective method, rooted in symplectic geometry, to compute the joint probability distribution of the eigenvalues of its one-body reduced density matrices. As a corollary, by taking the distribution's support, which is a convex moment polytope, we recover a complete solution to the one-body quantum marginal problem. We obtain the probability distribution by reducing to the corresponding distribution of diagonal entries (i.e., to the quantitative version of a classical marginal problem), which…