Morse theory on spaces of braids and Lagrangian dynamics

Scalar parabolic PDE's and braids

The comparison principle for scalar second order parabolic PDEs on functions $u(t,x)$ admits a topological interpretation: pairs of solutions, $u^1(t,\cdot)$ and $u^2(t,\cdot)$, evolve so as to not increase the intersection number of their graphs. We generalize to the case of multiple solutions $\{u^α(t,\cdot)\}_{α=1}^n$. By lifting the graphs to Legendrian braids, we give a global version of the comparison principle: the curves $u^α(t,\cdot)$ evolve so as to (weakly) decrease the algebraic length of the braid. We define a Morse-type theory on Legendrian braids which we demonstrate is useful f…

A smooth pseudo-gradient for the Lagrangian action functional

We study the action functional associated to a smooth Lagrangian function on the cotangent bundle of a manifold, having quadratic growth in the velocities. We show that, although the action functional is in general not twice differentiable on the Hilbert manifold consisting of H^1 curves, it is a Lyapunov function for some smooth Morse-Smale vector field, under the generic assumption that all the critical points are non-degenerate. This fact is sufficient to associate a Morse complex to the Lagrangian action functional.

拉格朗日系统与测地线的分支现象 I

Bifurcations for Lagrangian systems and geodesics I

本文研究了具有不同边界条件和约束的拉格朗日系统中的分支现象，通过三种主要配置探讨了莫尔斯理论与多重解存在的关系，提出了分支条件的必要充分条件。

流环与量子群

Flow loops and quantum groups

本文通过研究结的量子群不变量与莫尔斯流的动力学，定义了一个双变量级数不变量，用于描述结补集中的莫尔斯流环，推测该动力学级数与结补集的BPS q级数一致，后者源于量子群的Verma模，编码所有着色Jones多项式。

可逆拉格朗日系统中周期轨道的莫尔斯指标、稳定性与谱界

Morse Index, Stability and Spectral Bounds for Periodic Orbits in Reversible Lagrangian Systems

本文针对可逆拉格朗日系统，利用莫尔斯指标与狄利克雷指标给出周期轨道弗洛凯乘子的定量谱界，证明指标一刹车轨道的横向双曲性，并展示精细界的尖锐性。

Quantum corrections to the holomorphic structure of the mirror bundle along the caustic and the bifurcation locus

Given, in the Lagrangian torus fibration $R^4\to R^2$, a Lagrangian submanifold $L$, endowed with a trivial flat connection, the corresponding mirror object is constructed on the dual fibration by means of a family of Morse homologies associated to the generating function of $L$, and it is provided with a holomorphic structure. Morse homology, however, is not defined along the caustic $C$ of $L$ or along the bifurcation locus $B$, where the family does not satisfy the Morse-Smale condition. The holomorphic structure is extended to the subset $C\cup B$, except cusps, yielding the so called quan…

Periodic billiard trajectories and Morse theory on loop spaces

We study periodic billiard trajectories on a compact Riemannian manifold with boundary, by applying Morse theory to Lagrangian action functionals on the loop space of the manifold. Based on the approximation method due to Benci-Giannoni, we prove that nonvanishing of relative homology of a certain pair of loop spaces implies the existence of a periodic billiard trajectory. We also prove a parallel result for path spaces. We apply those results to show the existence of short billiard trajectories and short geodesic loops. We also recover two known results on the length of a shortest periodic bi…

奇异拉格朗日系统的指标理论与Bessel型微分算子

Index theory for singular Lagrangian systems and Bessel-type differential operators

本文针对奇异拉格朗日系统发展指标理论，建立奇异Sturm Liouville算子的谱流公式与Morse指标定理，构建Bessel型微分算子的Morse指标理论并应用于引力n体问题，同时为Rellich观察到的变定义域谱现象提供新解释。

Tonelli Lagrangian systems on the 2-torus and topological entropy

We study Tonelli Lagrangian systems on the 2-torus in energy levels above Mañé's strict critical value and analyize the structure of global minimizers in the spirit of Morse, Hedlund and Bangert. In the case where the topological entropy of the Euler-Lagrange flow on the fixed energy level vanishes, we show that there are invariant tori for all rotation vectors indicating integrable-like behavior on a large scale. On the other hand, using a construction of Katok, we give examples of reversible Finsler geodesic flows with vanishing topological entropy, but having ergodic components of positive …

半流的代数：两个拓扑的故事

The algebra of semi-flows: a tale of two topologies

本文提出通过构造流拓扑将半流动力学重新表述为双拓扑空间，并引入 Morse 前序与同调信息实现其代数化，从而为离散抛物流和正辫子提供了新的不变量。

On the Floer homology of cotangent bundles

This paper concerns Floer homology for periodic orbits and for a Lagrangian intersection problem on the cotangent bundle of a compact orientable manifold M. The first result is a new uniform estimate for the solutions of the Floer equation, which allows to deal with a larger - and more natural - class of Hamiltonians. The second and main result is a new construction of the isomorphism between the Floer homology and the singular homology of the free loop space of M, in the periodic case, or of the based loop space of M, in the Lagrangian intersection problem. The idea for the construction of su…

非哈密顿系统的广义指标理论

A generalized index theory for non-Hamiltonian system

本文针对非哈密顿耗散系统，引入基于布劳威尔度的新拓扑不变量“度指标”作为马斯洛夫指标的替代，并通过抽象迹公式证明其与莫尔斯指标的等式，最后应用于一维反应扩散系统。

A dihedral Bott-type iteration formula and stability of symmetric periodic orbits

In 1956, Bott in his celebrated paper on closed geodesics and Sturm intersection theory, proved an Index Iteration Formula for closed geodesics on Riemannian manifolds. Some years later, Ekeland improved this formula in the case of convex Hamiltonians and, in 1999, Long generalized the Bott iteration formula by putting in its natural symplectic context and constructing a very effective Index Theory. The literature about this formula is quite broad and the dynamical implications in the Hamiltonian world (e.g. existence, multiplicity, linear stability etc.) are enormous. Motivated by the recent …