Quantitative Estimates In Spectral Theory and Their Complexity

在一个我们越来越依赖计算机的世界里，从订购杂货到设计航天飞机，重要的是要知道它们的工作速度，以及是否能保证它们的计算导致（或“收敛”）正确的答案。这个项目旨在解决这两个问题。处理收敛速度的最重要领域之一是“遍历理论”。该领域主要研究物理系统的长期平均行为。人们通常认为，这种行为应该收敛到某个平均值（例如，一壶水在放入冰箱后温度会慢慢降低）。这种收敛的速度在应用中非常重要。例如，知道水罐冷却到某个预定阈值需要多长时间将是非常有用的。在这个项目的第一部分，我提出了一个新的方法来获得这样的利率，使用方法从一个领域在纯数学称为“频谱分析”。简而言之，光谱分析师研究与特定问题相关的光谱，这类似于问题的DNA：它是一个编码物理系统所有重要属性的对象。作为应用，我打算使用这个理论来研究物理现象，如等离子体和流体。许多控制其行为的方程都可以通过上述分析来解决，我打算使用这些新工具来了解一些基本性质，例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它在聚变反应堆中产生清洁能源。主要的障碍是等离子体的不稳定性，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。我与我的合作者提供了一些基本的计算示例（例如，计算光谱），其中近似（例如计算机所做的近似）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”复杂问题的复杂性。该项目的第二部分是围绕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”。应用程序在这里也很重要。我将把这个理论应用到一些用计算机解决的具体物理问题上，看看这些解决方案有时是否是错误的。我预计情况确实如此，并计划制定预警机制。

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.