Lagrangian Multiforms for Symmetries and Integrability: Classification, Geometry, and Applications

无论什么东西移动或变化，都可以用微分方程进行数学建模。解微分方程意味着确定系统（运动的东西）在未来任何时候的状态，从它的当前状态。对于许多微分方程来说，这是不可能的：它们所描述的系统表现出复杂的行为（想象风暴海洋上的波浪——它们的长期运动很难预测），而且不可能写出一个描述未来状态的公式。可积系统是例外：它们是可以求解的微分方程，并代表了看起来有序的动力学（想象一下，一艘船突然停下来，在运河上产生了一个波浪——它以一种可预测的方式继续前进）。可积系统的有序行为是由微分方程的某些隐藏的数学结构引起的。这种隐藏的结构可以有许多不同的形式。某些可积系统具有多种形式的隐结构，但没有一种形式适用于所有可积系统。关于可积系统的数学没有普遍的理论。本奖学金探讨了可积系统的最新发展，其中心思想来自物理学。几乎每一个物理理论都可以用某物被最小化这一事实来描述。这样的描述称为变分原理。例如，在光学中，一束光总是走最快的可能路径。在其他情况下，最小化的数量可能不那么直观，但变分原理总是提供强大的数学工具。“拉格朗日多重形式”理论使用变分原理来捕捉可积系统的隐藏结构。这是最近的一项发展，其优点才刚刚开始显现。一个优点是拉格朗日多重形式适用于离散系统的方式与适用于连续系统的方式相同。这提供了洞察两种类型的可积系统之间的关系。在这里，“离散”意味着空间和时间不是连续的，而是以固定的步骤工作（就像数字视频由有限数量的像素和每秒有限数量的帧组成）。本奖学金研究拉格朗日多形式在三个主要领域的好处：1。不同类型可积系统之间的关系及其分类。这种关系的一个例子可以在半离散系统（包括离散变量和连续变量）的拉格朗日多形式理论中找到。一些半离散拉格朗日多形表现出与完全连续可积系统的惊人联系。这只是拉格朗日多元形式提供不同类型方程之间关系的几种情况之一。该奖学金将对这一现象进行广泛的研究，利用它在不同类型的方程之间传递见解，并对感兴趣的方程进行分类。几何学。在拉格朗日多形理论中，描述系统对称性的参数与时间变量的处理方式相同。它们一起构成了“多时间”。本奖学金将研究拉格朗日多形式理论的几何方面。特别感兴趣的是多时间内的几何结构，与可积系统的特殊解有关，以及多时间本身的几何。这将使我们能够获得更大的一类微分方程，并将拉格朗日多元形式的见解转移到可积系统的领域之外。应用程序。本奖学金将研究拉格朗日多形在计算科学和基础物理中的应用。变分原理在这两个领域都有很多应用，但从严格的数学角度来看，并不是所有的原理都能完全理解。本研究将采用拉格朗日多形以及其他最新发展来确保这些应用的数学基础

Whenever something moves or changes, it can be modelled mathematically using a differential equation. Solving a differential equation means determining the state of the system (the thing which is moving) at any time in the future from its current state. For many differential equations this is impossible: the systems they describe exhibit complicated behaviour (picture waves on a stormy ocean - their long-term movements are very hard to predict) and it is impossible to write a formula describing future states. Integrable systems are the exceptions: they are differential equations that can be solved and represent dynamics that look orderly (picture a wave on a canal produced by a boat that has suddenly stopped - it keeps traveling forward in a predictable way). The orderly behaviour of an integrable system is caused by some hidden mathematical structure of the differential equation. This hidden structure can take many different forms. Some integrable systems possess several forms of hidden structures, but no form applies to all integrable systems. There is no universal theory of the mathematics of integrable systems.This Fellowship explores a recent development in integrable systems, the central idea of which comes from physics. Almost every physical theory can be described by the fact that something is minimised. Such a description is called a variational principle. In optics, for example, a ray of light will always take the fastest possible path. In other cases, the quantity that is minimised may be less intuitive, but a variational principle always provides powerful mathematical tools.The theory of "Lagrangian multiforms" uses variational principles to capture the hidden structures of integrable systems. It is a recent development, the advantages of which are only starting to come to light. One advantage is that Lagrangian multiforms apply to discrete systems in the same way as to continuous systems. This provides insight into relations between integrable systems of both types. Here, "discrete" means that space and time do not form a continuum, but work in fixed steps (like digital video consists of a finite number of pixels and a finite number of frames per second).This Fellowship investigates the benefits of Lagrangian multiforms in three main areas:1. Relations between integrable systems of different types and their classification. One example of such relations is found in the Lagrangian multiform theory of semi-discrete systems (which involve both discrete and continuous variables). Some semi-discrete Lagrangian multiforms exhibit surprising connections to fully continuous integrable systems. This is only one of several contexts in which Lagrangian multiforms provide relations between equations of different types. This Fellowship will deliver a broad investigation of this phenomenon, employ it to transfer insights between equations of different types, and classify equations of interest.2. Geometry. In the theory of Lagrangian multiforms, parameters describing symmetries of the system are treated in the same way as the time-variable. Together they form "multi-time". This Fellowship will study geometric aspects of Lagrangian multiform theory. Of particular interest are geometric structures within multi-time, related to special solutions of the integrable system, as well as the geometry of multi-time itself. This will allow us to capture a larger class of differential equations and transfer the insights of Lagrangian multiforms beyond the realm of integrable systems.3. Applications. This Fellowship will investigate applications of Lagrangian multiforms in computational science and in fundamental physics. Variational principles have many applications in both these areas, but not all are fully understood from a rigorous mathematical perspective. This research will employ Lagrangian multiforms as well as other recent developments to secure the mathematical foundation of these applications