Virtual Morse theory on $ΩHam(M,ω)$

虚拟Morse-Bott指数、配对模空间及其在光滑四维流形拓扑中的应用

Virtual Morse-Bott index, moduli spaces of pairs, and applications to topology of smooth four-manifolds

本文研究了圈作用下哈密顿函数的虚拟Morse-Bott指数，证明其与经典Morse-Bott指数一致，并利用Hirzebruch-Riemann-Roch定理计算了所有临界层的虚拟指数，验证了四维光滑流形的Bogomolov-Miyaoka-Yau不等式猜想。

Morse theory for the Hofer length functional

Following \cite{citeSavelyevVirtualMorsetheoryon$Omega$Ham$(Momega)$.}, we develop here a connection between Morse theory for the (positive) Hofer length functional $L: Ω\text {Ham}(M, ω) \to \mathbb{R}$, with Gromov-Witten/Floer theory, for monotone symplectic manifolds $ (M, ω) $. This gives some immediate restrictions on the topology of the group of Hamiltonian symplectomorphisms (possibly relative to the Hofer length functional), and a criterion for non-existence of certain higher index geodesics for the Hofer length functional. The argument is based on a certain automatic transversality p…

余切丛中哈密顿作用的莫尔斯同调

Morse homology for the Hamiltonian action in cotangent bundles

本文利用梯度流方程在余切丛的混合正则性环路空间上为哈密顿作用构造了Morse复形，通过引入强可积(0)-本质子丛保证紧致性，并利用伪梯度向量场扰动实现横截性，最终证明所得Morse同调与Floer同调和自由环路空间的奇异同调同构。

通过Morse-Bott-Floer谱序列对量子上同调进行滤波

Filtrations on quantum cohomology via Morse-Bott-Floer Spectral Sequences

本文通过Morse-Bott-Floer谱序列研究辛流形上量子上同调的滤波，提出构建Floer复形滤波的方法，并计算了多个例子以展示其应用。

Proof of the index conjecture in Hofer geometry

Let $γ$ be a non-degenerate Ustilovsky geodesic in $Ham (M, ω)$ generated by $H$. We give a simple proof of a generalization of the conjecture stated in \cite{virtmorse}, relating the Morse index of $ γ$, as a critical point of the Hofer length functional, with the Conley Zehnder index of the extremizers of $H$, considered as periodic orbits.

Index computations in Rabinowitz Floer homology

In this note we study two index questions. In the first we establish the relationship between the Morse indices of the free time action functional and the fixed time action functional. The second is related to Rabinowitz Floer homology. Our index computations are based on a correction term which is defined as follows: around a non-degenerate Hamiltonian orbit lying in a fixed energy level a well-known theorem says that one can find a whole cylinder of orbits parametrized by the energy. The correction term is determined by whether the periods of the orbits are increasing or decreasing as one mo…