Geometric Aspects of Complex Differential Equations

微分方程在描述从机械系统、流体动力学、光学现象到量子力学等各种动力学过程中起着关键作用，在这些过程中，它们将物理系统的状态与系统瞬间经历的变化率联系在一起。在应用程序中，重要的是知道系统何时具有在某种数学意义上行为良好的解，或者这些解是否表现出混沌行为。虽然在应用中人们通常对系统的时间演化感兴趣，时间是一个真实的一维参数，但人们很早就明白，当时间被视为扩展的数系的成员时，方程的解的性质被最好地解释，在数学上被实现为二维平面上的点，称为复数。在这个二维复平面中考虑的微分方程解可以有奇点，即解变得无限或以其他方式不明确的点。这些奇点的性质反过来又影响到解在一维时间参数空间中的行为。它允许人们确定一个方程是否可积，即在某种数学意义上是精确可解的。这个项目是关于调查某些广泛的方程类的奇点的性质，这些方程是由物理应用引起的。为了研究奇点，我们将考虑因变量(那些描述系统状态及其变化率的变量)空间中方程的解，其中添加了某些点，以包括这些点变为无穷大的点。通过这种方式，人们可以研究当解发展成奇点时会发生什么。然而，事实证明，即使在这些扩大的空间中，也有一些点系统的变化率是不确定的，当表示变化率的分数的分子和分母同时接近于零时，就会发生这种情况。这样的“基点”可以通过一种被称为爆炸的定义明确的数学过程来去除。一个点的爆破是一个几何概念，它通过引入一条‘例外线’将更多的点添加到解空间中，每个点对应于从该点出现的一个方向。通过这种方式，方程的解在例外线上分离出来，使进一步研究它们成为可能。当被认为是复杂几何(其中坐标以复数为值)中的对象时，爆炸引入的“异常线”被证明等同于一个球体。这样，原来溶液变化率不明确的点就膨胀成了球体，因此这个过程被称为爆破。结果表明，对于大多数方程，方程在某一点的一次爆破不足以使方程摆脱不确定性，也就是说，即使在进行了一次爆破之后，也可能需要进一步的爆破。对于数学物理中的一类重要方程--Painlevé方程，总共需要9次爆破才能使这些方程变成一种不存在未知点的形式。这样构造的解空间由日本数学家K.Okamoto最先获得，称为方程的初值空间。对于Painlevé方程，使用这个空间可以解释奇点的性质，在这种情况下奇点是极点：解在受控方式下趋于无穷的点。在拟议的项目中，我们将把构造初值空间的方法应用到更广泛的方程类中，以研究如何利用这种方法来确定某些微分方程可能表现出的更复杂奇点的性质。

Differential equations play a key role in describing a wide range of dynamical processes, from mechanical systems, fluid dynamics, optical phenomena and quantum mechanics amongst many others, where they relate the state of a physical system to the rate of change that the system is momentarily undergoing. It is important in applications to know when a system has solutions that are, in a certain mathematical sense, well-behaved, or whether the solutions exhibit chaotic behaviour. Although in applications one is usually interested in the time-evolution of a system, time being a real, 1-dimensional parameter, it has long been understood that the nature of the solutions of an equation is best explained when time is considered as member of an extended number system, mathematically realised as points in a 2-dimenional plane, known as the complex numbers.The solutions of a differential equation, considered in this 2-dimensional complex plane, can have singularities, i.e. points where a solution becomes infinite or is otherwise ill-defined. The nature of these singularities, in turn, has an impact on the behaviour of the solutions in the 1-dimensional time parameter space. It allows one to determine whether an equation is integrable, i.e. exactly solvable in a certain mathematical sense. This project is about investigating the nature of the singularities of certain wide classes of equations that are motivated from physical applications. To investigate the singularities, we will consider the solutions of the equations in spaces of the dependent variables (those variables describing the state of a system and its rate of change) to which certain points have been added to include points where these become infinite. In this way, one can investigate what happens when the solutions develop a singularity. It turns out, however, that even in these augmented spaces there are points at which the rate of change of a system is indeterminate, which happens when both the numerator and the denominator of the fractions expressing the rate of change approach zero at the same time. Such 'base points' can be removed by a well-defined mathematical procedure known as a blow-up. The blow-up of a point is a geometrical notion which adds further points to the solution space by introducing an 'exceptional line', each point of which corresponds to a direction emerging from the point in question. In this way, the solutions of an equation are separated out over the exceptional line, making it possible to investigate them further. The 'exceptional line' introduced by the blow-up, when considered as an object in complex geometry (where the coordinates take complex numbers as values) turns out to be equivalent to a sphere. In this way, the original point at which the rate of change of the solution was ill-defined, becomes inflated to a sphere, hence the name blow-up for this process. It turns out that for most equations, a single blow-up of the equation at one point is not sufficient to render the equation free of indeterminacies, i.e., even after one blow-up has been performed, further ones may be necessary. For the Painlevé equations, an important class of equations in mathematical physics, it turns out that a total of 9 blow-ups is required to bring these equations into a form where no points with indeterminate behaviour remain. The solution space thus constructed, first obtained by the Japanese mathematician K. Okamoto, is called the 'space of initial values' of the equation. For the Painlevé equations, using this space one can explain the nature of the singularities, which in this case are poles: points at which the solutions tend to infinity in a controlled way.In the proposed project, we will apply the method of constructing the space of initial values to much wider classes of equations to investigate how this method can be utilised to determine the nature of more complicated singularities that certain differential equations can exhibit.