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Calogero-Moser correspondence: at the crossroads of representation theory, geometry and integrable systems

卡洛杰罗-莫泽对应：处于表示论、几何和可积系统的十字路口

基本信息

批准号：
EP/K004999/1
负责人：
Oleg Chalykh
金额：
$ 30.22万
依托单位：
University of Leeds
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2013
资助国家：
英国
起止时间：
2013 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FK004999%2F1
关键词：
Calogero Moser correspondence crossroads representation

项目摘要

When we want to organise things, we tend to make lists. This is also true for mathematicians, although in mathematics the lists are usually infinite. When dealing with an infinite list or family of objects, you first need to come up with a way to label them: for example, do you label the objects by natural numbers 1,2,3,...(indices) or by real numbers (parameters)? Labelling by real numbers (or by k-tuples of real numbers) means that you parameterise the objects in the family by points on the real line or in k-dimensional space. Obviously, as with any list, you need to make sure that you haven't missed anything or didn't count something twice. For that reason, quite often the space of parameters is more complicated than just a real line or space, and it is not always easy to get it right. For example, look at all circles of radius 1 in the plane. Each circle is uniquely described by giving its centre, i.e. the whole family is parameterised by points of the plane. If instead we look at the family of all lines in the plane, then a parameterisation is less obvious; in this case, the space of parameters has a shape of the Mobius band. In both cases the parameterisation on a small scale looks similar, but globally there is a significant difference: the plane surface has two sides while the Mobius band has only one.It is usually difficult to parameterise an infinite family of objects in an orderly fashion by the points of some nice, smooth space, so whenever that happens it makes mathematicians very happy. This project will look into particular instances of a recently emerged pattern when the objects which look completely unrelated at first, turn out to admit nice parameterisation by the same space. For instance, solutions of some complicated nonlinear differential equations governing water waves turn out to correspond to solutions of some algebraic equations, which in their turn correspond to the motion of certain interacting particles. Uncovering deep underlying reasons and mechanisms for such correspondences is very important, because we can then hope to use them in other situations in order to solve otherwise inaccessible problems. And that's exactly what the proposed project will try to achieve.

当我们想要组织事物时，我们倾向于列出清单。数学家也是如此，尽管在数学中列表通常是无限的。当处理一个无限的对象列表或对象族时，你首先需要想出一种方法来标记它们：例如，你是否用自然数1，2，3，.（指数）还是真实的数字（参数）？使用真实的数字（或使用真实的数字的k元组）进行标记意味着使用真实的直线上或k维空间中的点来参数化族中的对象。显然，与任何列表一样，您需要确保没有遗漏任何内容或没有重复计算某些内容。因此，参数空间通常比真实的线或空间更复杂，而且要正确处理并不总是那么容易。例如，观察平面上半径为1的所有圆。每个圆通过给出其中心来唯一描述，即整个族由平面上的点参数化。相反，如果我们看平面上所有直线的族，那么参数化就不那么明显了;在这种情况下，参数空间具有莫比乌斯带的形状。这两种情况下的参数化在小尺度上看起来很相似，但在全局上有一个显著的区别：平面有两个面，而莫比乌斯带只有一个面。通常很难通过一些漂亮的光滑空间的点来有序地参数化一个无限大的物体族，所以每当发生这种情况时，数学家们都会非常高兴。这个项目将研究最近出现的模式的特定实例，当最初看起来完全不相关的对象被同一空间很好地参数化时。例如，某些复杂的控制水波的非线性微分方程的解与某些代数方程的解相对应，而这些代数方程又与某些相互作用的粒子的运动相对应。揭示这种对应的深层原因和机制非常重要，因为我们可以希望在其他情况下使用它们来解决其他无法解决的问题。而这正是拟议中的项目将试图实现。