Excellence in Research: Numerical Analysis of Quasiperiodic Topology

哈密顿动力系统在数学和物理学中都是极其重要的。这些系统由一个称为哈密顿量的能量函数定义，通常允许第一积分，即其他函数，独立于哈密顿量，但也像哈密顿量一样，不改变它们在运动方程的解上的值。自从W.R.汉密尔顿爵士在近两个世纪前提出这种形式主义以来，它逐渐涵盖了大多数非耗散现象的物理学，以及由此自然产生的几个重要的几何分支，并成为量子力学的主要组成部分之一。然而，直到最近，一类重要的哈密顿系统被忽略了，即当某个第一积分是多值的时候(多值函数的一个例子是圆上的角坐标)。这样的系统自然产生于量子力学(在所谓的半经典近似下)，特别是在强磁场下金属的低温导电性理论中。从七十年前开始，许多金属都收集了关于这一现象的实验数据，但由于缺乏具有多值哈密顿量的哈密顿系统理论，这些数据无法与理论精确地进行核对。自八十年代以来，菲尔兹奖牌获得者S.P.诺维科夫和他的学校开始填补这一空白，大约十年前，实验数据在最简单的情况下成功验证，即金和银。同时，随着这个哈密顿系统的推广和形式化，一个新的拓扑学领域--拟周期拓扑学应运而生，并在对最简单的情形进行数值研究后发现了一些重要的理论结果。本项目的目的一方面是为了完成对尚未验证的几种金属的这些基本实验数据的验证，另一方面是为了扩展和深化准周期拓扑的数值研究。本项目的主要目的是继续和深化拟周期拓扑学的数值研究，特别是n维环面上多值映射水平集的拓扑。它的主要子目标是对数值实验所建议的性质进行上下文分析研究。这将通过以下具体目标和方法来实现：(1)在Perl/PYTHON/C++中开发和实施用于n-环面上多值映射的数值和分析研究的新旧算法；(2)这些地图的数值探索和分类；(3)此类地图的次要和主要性质的分析研究。作为这些结果的应用，该项目的另一个重要的子目标是应用(1)中生成的代码来第一次从第一性原理验证一些关于金属导电性的重要实验数据，这些实验数据在50年代和60年代测量，其数学描述涉及准周期函数。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

Hamiltonian dynamical systems are of paramount importance in both Mathematics and Physics. These systems are defined by an energy function, called Hamiltonian, and often admit first integrals, namely other functions, independent on the Hamiltonian but that also, just like the Hamiltonian, do not change their value over the solutions of the equations of motion. Since this formalism was introduced by Sir W.R. Hamilton almost two centuries ago, it grew to embrace the physics of most non-dissipative phenomena, as well as several important branches of geometry that arose naturally out of it, and it became one of the main building block of quantum mechanics. Nevertheless, until relatively recently, an important class of Hamiltonian systems was overlooked, namely the case when some first integral is multivalued (an example of multivalued function is the angle coordinate on a circle). Such systems arise naturally from quantum mechanics (in the so-called semiclassical approximation), in particular in the theory of conductivity of metals at low temperature under a strong magnetic field. Experimental data on this phenomenon were collected for many metals starting from seventy years ago but they could not be checked against the theory exactly for the lack of a theory of Hamiltonian systems with multivalued Hamiltonians. Since the Eighties, Fields medalist S.P. Novikov and his school started filling this gap and about ten years ago the experimental data were successfully verified in the simplest cases, namely for Au and Ag. At the same time, a new field of topology, called Quasiperiodic topology, arose as the generalization and formalization of this Hamiltonian system and important theoretical results have been found after the numerical study of the simplest cases. This project aims, on the one hand, at completing the verification of these fundamental experimental data for the several metals still unchecked and, on the other hand, at extending and deepening the numerical study of quasiperiodic topology. The main goal of this project is to continue and deepen the numerical study of Quasiperiodic Topology focusing, in particular, on the topology of level sets of multivalued maps on the n-dimensional tori. Its main subgoal is the contextual analytical study of the properties suggested by the numerical experiments. This will be achieved through the following specific objectives and methods: (1) development and implementation in Perl/Python/C++ of old and new algorithms for the numerical and analytical study of multivalued maps on n-tori; (2) numerical exploration and classification of these maps; (3) analytical study of minor and major properties of such maps. As an application of these results, a further important subgoal of the project is the application of the code generated in (1) to verify from first principles for the first time some important experimental data about conductivity in metals measured in Fifties and Sixties and whose mathematical description involves quasiperiodic functions.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.