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Geodesic Currents and Counting Problems

测地线流和计数问题

基本信息

批准号：
EP/T015926/1
负责人：
Viveca Erlandsson
金额：
$ 34.31万
依托单位：
University of Bristol
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2020
资助国家：
英国
起止时间：
2020 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FT015926%2F1
关键词：
Geodesic Currents Counting Problems

项目摘要

Imagine drawing a circle in the plane, centered at the origin and of radius R, and you want to count the number of points with integer coefficients enclosed by the circle. For example, if R=3 there are 13 such points, if R=10 there are 253, and if R=20 there are 1129 points. Clearly, the larger R is the more points there are, but exactly how are these numbers related? One can prove, using elementary mathematics, that the number of integer points inside a circle of radius R grows like pi*R^2, i.e. the area of the region it encloses. More precisely, the number of such points is asymptotic to the area, meaning that the ratio of the two quantities tends to 1 as R goes to infinity. The simple problem described above is closely related to counting curves on surfaces. To a topologist, a surface is a 2-dimensional object which can be obtained by cutting out a polygon in the plane and then gluing sides together in pairs. For example, if the polygon is a square and we glue two opposite sides together we get a tube. If we glue the two boundary circles of the tube together, we get a donut, which we call a torus. The torus comes with a metric, a way to measure distances, given by its identification with a square in the plane which has the usual flat (Euclidean) metric. A curve on the torus is a closed loop (think of a string wrapped around the surface where you tie the two endpoints together) which we "pull tight" so it becomes as short as possible. As it turns out, the number of curves on the torus of length at most R is exactly the same as number of integer points in the plane inside a circle of radius R.If we use another polygon instead of a square in the construction above we get a more complicated surface. In fact, in general we get a surface that looks like several tori glued together. The number of tori is called the genus g of the surface. However, to get a nice (constant curvature) metric on the surface, we need to cut the polygon out of the hyperbolic plane (which is negatively curved, like the inside of a bowl) instead of the usual Euclidean plane (which is flat). This drastically changes the growth of the number of curves: it was shown in the 60s by Huber that the asymptotic growth is exponential in the length when g>1. However, if we look instead only at curves that do not self-intersect there are much fewer curves and we again get a polynomial growth rate (this was first observed by Birman-Series in the 80s and proved in more detail by Rivin in 2001). Finding the exact asymptotic growth of these curves is a hard problem and was solved by a deep theorem by Mirzakhani in 2008. She proved that the number of simple curves of length at most R on a surface of genus g>1 is asymptotic to a constant times R^{6g-6}. Mirzakhani's result became instantly famous since it was a part of her triad of results on curve counting, volume growth, and the Witten conjecture (an important problem in physics) breaking ground in both the world of geometry and dynamics and having important implications to physics. In this project we use new methods to approach the problem of counting curves which allows us to generalize her result. In fact, we also get a new, and very different, proof of Mirzakhani's result. The original proof requires expert understanding of several fields of mathematics and is hard to grasp in full detail even for experts in the fields; the new approach has potential to open up the field to researchers from a wider field of expertise. The new proof also gives a new way to compute important constants related to Mirzakhani's theorem. The novelty of these methods is the use of so called geodesic currents, a space that unifies the study of curves, measured laminations, and hyperbolic metrics, all integral notions to curve counting.

想象一下，在平面上画一个圆，以原点为中心，半径为R，你想计算圆内具有整数系数的点的数量。例如，如果R=3，则有13个这样的点，如果R=10，则有253个，如果R=20，则有1129个点。显然，R越大，点就越多，但这些数字到底是如何相关的呢？我们可以用初等数学证明，半径为R的圆内的整数点的数量以pi*R^2的形式增长，即它所包围的区域的面积。更确切地说，这样的点的数量是渐近的面积，这意味着两个量的比例趋于1，因为R走向无穷大。上面描述的简单问题与曲面上的曲线计数密切相关。对于拓扑学家来说，曲面是一个二维的物体，它可以通过在平面上切出一个多边形，然后把边成对地粘在一起而得到。例如，如果多边形是一个正方形，我们把两个相对的边粘在一起，我们得到一个管子。如果我们把管的两个边界圆粘在一起，我们得到一个圆环，我们称之为环面。环面带有一个度量，一种测量距离的方法，通过它与平面上具有通常平坦（欧几里得）度量的正方形的标识给出。环面上的曲线是一个闭合的环（想想一根绳子缠绕在曲面上，你把两个端点绑在一起），我们把它“拉紧”，使它变得尽可能短。事实证明，长度最多为R的环面上的曲线数量与半径为R的圆内平面上的整数点数量完全相同。如果我们在上面的构造中使用另一个多边形而不是正方形，我们会得到一个更复杂的曲面。实际上，一般来说，我们得到的曲面看起来像是几个粘在一起的圆环。环面的个数称为曲面的亏格g。然而，为了在曲面上得到一个好的（常曲率）度量，我们需要将多边形从双曲平面（它是负弯曲的，就像碗的内部）而不是通常的欧几里得平面（它是平的）中切割出来。这极大地改变了曲线数量的增长：在60年代，Huber证明了当g>1时，曲线长度的渐近增长是指数增长的。然而，如果我们只看不自相交的曲线，那么曲线就少得多，我们又得到一个多项式增长率（这是由伯曼级数在80年代首次观察到的，并由Rivin在2001年更详细地证明）。找到这些曲线的精确渐近增长是一个困难的问题，并在2008年由Mirzakhani的一个深层定理解决。她证明了在亏格g>1的曲面上长度至多为R的简单曲线的个数渐近于常数乘以R^{6 g-6}。米尔扎哈尼的结果立即成为著名的，因为这是她的一部分三重结果曲线计数，体积增长，和维滕猜想（一个重要的问题，物理学）破土动工，在世界上的几何和动力学，并具有重要意义的物理。在这个项目中，我们使用新的方法来处理计数曲线的问题，这使我们能够推广她的结果。事实上，我们也得到了一个新的，非常不同的，证明米尔扎哈尼的结果。原来的证明需要对数学的几个领域的专家理解，即使是这些领域的专家也很难掌握全部细节;新的方法有可能向来自更广泛专业领域的研究人员开放该领域。新的证明也提供了一种新的方法来计算与Mirzakhani定理相关的重要常数。这些方法的新奇是使用所谓的测地线电流，一个空间，统一的研究曲线，测量叠层，双曲度量，所有积分的概念曲线计数。