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

当我们想要组织事情时，我们倾向于列清单。这对数学家来说也是正确的，尽管在数学中列表通常是无限的。当处理一个无限列表或对象族时，你首先需要想出一种标记它们的方法：例如，你是否用自然数1、2、3、…（指标）还是实数（参数）？用实数（或实数的k元组）标记意味着用实数线上的点或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将某些内容计算两次。由于这个原因，参数的空间通常比实际的行或空间更复杂，并且并不总是容易得到正确的。例如，看平面上所有半径为1的圆。每个圆都是通过给出其中心来唯一描述的，也就是说，整个家族都是由平面上的点来参数化的。如果我们观察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。在这两种情况下，在小尺度上的参数化看起来很相似，但在全局上有显著的差异：平面表面有两面，而莫比乌斯带只有一面。通常很难用一些漂亮的光滑空间的点来有序地参数化无限的对象族，所以每当这种情况发生时，数学家们都会很高兴。该项目将研究最近出现的模式的特定实例，当对象最初看起来完全不相关时，结果承认相同空间的良好参数化。例如，一些控制水波的复杂非线性微分方程的解与一些代数方程的解相对应，而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，以解决其他难以解决的问题。而这正是拟议中的项目要努力实现的目标。

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.