Quantitative Estimates In Spectral Theory and Their Complexity

在一个我们越来越依赖计算机来做任何事情的世界里，从订购食品杂货到设计航天飞机，知道它们工作的速度有多快，以及它们的计算是否保证会导致(或“收敛”)正确的答案是很重要的。这个项目旨在解决这两个问题。处理收敛速度的最重要的领域之一是“遍历理论”。这个领域主要研究物理系统的长期平均行为。通常认为，这种行为应该收敛到某个平均量(例如，一罐水在放入冰箱后温度会慢慢放松)。这种收敛的速度在应用中非常重要。例如，知道一罐水需要多长时间才能冷却到某个预定的阈值，这将是非常有用的。在这个项目的第一部分，我提出了一种新的方法来获得这样的比率，使用了纯数学领域中的方法，即所谓的“光谱分析”。简而言之，光谱分析员研究与手头的特定问题相关的频谱，这类似于问题的DNA：它是一个对象，编码了物理系统的所有重要属性。作为一个应用，我打算将这个理论用于研究等离子体和流体等物理现象。许多支配他们行为的方程式都适用于上述分析，使用这些新工具，我打算了解一些基本属性，如长期行为和稳定性。例如，等离子体是一种带电物质，工程师们希望能够利用这种物质在聚变反应堆中生产清洁能源。这方面的主要障碍是等离子体的不稳定性质。然而，我已经分别证明了，并不总是保证近似收敛到正确的结果。我和我的合作者一起提供了一些基本的计算例子(例如，计算光谱)，其中近似(如计算机所做的)注定会失败，并通过引入一种新的复杂性理论来解决这个问题，该理论允许比较两个“无限”复杂的问题的复杂性。拟议项目的第二部分集中在更好地理解这一新理论，并研究给定问题高度(或无限)复杂的“可能性”有多大。应用程序在这里也是至关重要的。我将把这个理论应用于一些具体的物理问题，这些问题是用计算机解决的，看看这些解决方案有时是否会出错。我预计情况确实会是这样，并计划制定警告机制。

In a world where we increasingly rely on computers for anything from ordering groceries to designing space shuttles, it is important to know how fast they work, and whether it's guaranteed that their computations lead (or "converge") to the correct answer. This project aims to address both questions.One of the most important fields that deals with rates of convergence is "ergodic theory". This field primarily deals with long-time averaged behaviour of physical systems. It is typically expected that this behaviour should converge to some averaged quantity (for example, the temperature of a jug of water slowly relaxes after it's placed in a refrigerator). The rate of this convergence is highly important in applications. For instance, it would be very useful to know how long it would take the jug of water to cool down to a certain predetermined threshold. In the first part of this project I propose a new method for obtaining such rates, using methods from a field in pure mathematics known as "spectral analysis". In a nutshell, spectral analysts study the spectrum associated to the particular problem at hand, which is akin to the DNA of the problem: it is an object that encodes all the significant properties of the physical system.As an application, I intend to use this theory for studying physical phenomena such as plasmas and fluids. Many of the equations that govern their behaviour are amenable to the aforementioned analysis, and using these new tools I intend to understand some basic properties, such as long-time behaviour and stability. Plasma, for instance, is a form of charged matter which engineers hope to be able to harness to produce clean energy in fusion reactors. The main obstacle to this is the unstable nature of plasma.However separately I have shown that it is not always guaranteed that approximations converge to the correct result. With my collaborators I provide some basic computational examples (for example, calculating spectra) where approximations (such as those a computer does) are doomed to fail and address this problem by introducing a new complexity theory that allows to compare the complexity of two problems that are "infinitely" complex. The second part of the proposed project is centered around understanding this new theory better and studying how "likely" it is for a given problem to be highly (or "infinitely") complex. The applications are crucial here too. I will apply the theory to some concrete physical problems that are solved using computers to see if these solutions might sometimes be wrong. I anticipate this to indeed be the case, and plan to develop warning mechanisms.