Rabinowitz Floer Homology

Rabinowitz作用泛函是一种拉格朗日乘子泛函，其临界点是固定能量的周期轨道。这一行动泛函的Floer同调是由这一提议的作者首先构造的。同时，Rabinowitz Floer同源已经发现了许多应用，例如，参见本提案作者之一的ICM 2022谈话。临界点的拉格朗日乘数对应于周期，其中负值意味着周期轨道在时间上反向遍历。这一特征将Rabinowitz Floer同调与辛同调和辛场理论区分开来，在辛场理论中，周期轨道只能在正演时间内穿越。它导致了Rabinowitz-Floer同调与Tate同调和Poincare对偶之间的深刻联系，也是Rabinowitz-Floer同调具有分次拓扑量子场论结构的原因。这个提议的目的是更深入地理解这种结构，研究它如何推广到Hamil-Tung时滞方程，并探索这种结构在半经典极限下在量子力学中的应用。

The Rabinowitz action functional is a Lagrange multiplier functional whose critical points are periodic orbits of fixed energy. A Floer homology for this action functional was first constructed by the au-thors of this proposal. Meanwhile Rabinowitz Floer homology has found numerous applications, see e.g. the ICM 2022 talk of one of the authors of this proposal. The Lagrange multiplier at a critical point corresponds to the period, where a negative value means that the periodic orbit is traversed back-wards in time. This feature distinguishes Rabinowitz Floer homology from symplectic homology and symplectic field theory where periodic orbits can only be traversed in forward time. It leads to deep relations of Rabinowitz Floer homology with Tate homology and Poincare duality, and it is the reason that Rabinowitz Floer homology has the structure of a graded Topological Quantum Field Theory. The goal of this proposal is to understand this structure in more depth, to study how it extends to Hamil-tonian delay equations, and to explore applications of this structure to quantum mechanics in the semiclassical limit.