Function Theory on Symplectic Manifolds

摘要奖：DMS-1006610首席研究员：Leonid Polterovich建议的研究属于辛几何和拓扑学，这是一个发展迅速的数学领域，最初是作为经典力学问题的几何工具出现的。20世纪80年代的“辛革命”导致了涉及辛流形、辛子集和微分同胚的令人惊讶的刚性现象的发现。最近的一些进展表明，还有另一种辛刚性的表现，发生在与辛流形相关的函数空间中。这些空间展示了意想不到的性质和有趣的结构，在辛拓扑中产生了另一种直觉和新的工具，并提供了研究辛流形上的函数理论的动机。这一新理论及其应用的发展是拟议研究的主要目标。我们重点研究了以下主题。首先，研究了泊松括号的稳健性。泊松括号是一种基本运算，它涉及一对函数，并由它们的导数定义。泊松括号的某些特征对于一致范数中的小扰动表现出惊人的稳健性，即使这种扰动可以显著地改变导数。这一现象似乎与辛微分同态群上的Hofer几何密切相关。其次，我们讨论了辛准态理论的各个方面。考虑辛流形上的函数空间。辛拟态是这个空间上的单调泛函，它在每个泊松交换子代数上是线性的，但不一定在整个空间上是线性的。这个概念的起源可以追溯到量子力学的基础。高维流形上的非线性拟态是由现代辛拓扑的基石Floer理论提供的。准态是解决辛拓扑中许多问题的有用工具，例如辛交和拉格朗日纽结。最后，我们统一了这两个主题，并探索了辛准状态和泊松括号之间的相互关系。辛拓扑与几个科学领域进行了卓有成效的相互作用，这些领域通过对技术的应用而对社会产生了重大影响。其中一个领域是哈密顿动力学，这是一门数学学科，为模拟各种基本的物理和技术过程提供了有效的工具，如卫星的轨道运动、光纤中的光传播和带电粒子通过加速器的运动。另一种是量子理论，这是物理学的一个分支，研究物质在微观尺度上的行为，其潜在应用范围甚至延伸到密码学和计算机技术。本文中提出的辛流形上函数理论的发展使人们对哈密顿动力学中的稳健度量有了新的认识，并揭示了量子-经典对应的一个新方面，这是量子理论的基本原理。

AbstractAward: DMS-1006610Principal Investigator: Leonid PolterovichThe proposed research belongs to symplectic geometry and topology, a rapidly developing field of mathematics which originally appeared as a geometric tool for problems of classical mechanics. The "symplectic revolution" of the 1980s gave rise to the discovery of surprising rigidity phenomena involving symplectic manifolds, their subsets and diffeomorphisms. A number of recent advances show that there is yet another manifestation of symplectic rigidity, taking place in function spaces associated to a symplectic manifold. These spaces exhibit unexpected properties and interesting structures, giving rise to an alternative intuition and new tools in symplectic topology, and providing a motivation to study the function theory on symplectic manifolds. Development of this new theory and its applications is the main objective of the proposed research. We focus on the following topics. First, we study robustness of the Poisson bracket. The Poisson bracket is a basic operation which involves a pair of functions and is defined by their derivatives. Certain characteristics of the Poisson bracket exhibit surprising robustness properties with respect to small perturbations in the uniform norm, even though such perturbations can dramatically change the derivatives. This phenomenon appears to be closely related to Hofer's geometry on the group of symplectic diffeomorphisms. Second, we deal with various aspects of the theory of symplectic quasi-states. Consider the space of functions on a symplectic manifold. A symplectic quasi-state is a monotone functional on this space which is linear on every Poisson-commutative subalgebra, but not necessarily on the whole space. The origins of this notion go back to foundations of quantum mechanics. Non-linear quasi-states on higher-dimensional manifolds are provided by Floer theory, the cornerstone of modern symplectic topology. Quasi-states serve as a useful tool for a number of problems in symplectic topology such as symplectic intersections and Lagrangian knots. Finally, we unify both topics and explore interrelations between symplectic quasi-states and Poisson brackets.Symplectic topology fruitfully interacts with several areas of science, which have a significant impact on society through their applications to technology. One of these areas is Hamiltonian dynamics, a mathematical discipline providing efficient tools for modeling a variety of fundamental physical and technological processes such as orbital motion of satellites, propagation of light in optical fibers and motion of charged particles through accelerators. Another one is quantum theory, a branch of physics which studies behavior of matter on microscopic scales, and whose potential applications reach as far as cryptography and computer technology. Development of function theory on symplectic manifolds that is put forward in the present proposal leads to a new insight on robust measurements in Hamiltonian dynamics and reveals a new facet of the quantum-classical correspondence, a fundamental principle of quantum theory.