Geodesic Currents and Counting Problems

设想在平面上绘制一个以原点为中心、半径为R的圆，并计算该圆所包含的具有整数系数的点数。例如，如果R=3有13个这样的点，如果R=10有253个，如果R=20有1129个点。显然，R越大，点越多，但这些数字到底是如何关联的呢？人们可以用初等数学证明，半径为R的圆内的整点的数目按pi*R^2的方式增长，即它所包围的区域的面积。更准确地说，这样的点的数量与面积是渐近的，这意味着当R趋于无穷大时，两个量的比率趋于1。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，表面是一个二维物体，它可以通过在平面上切割一个多边形，然后将两面成对地粘合在一起而获得。例如，如果多边形是一个正方形，我们将两个相对的边粘在一起，我们就得到了一个管子。如果我们把管子的两个边界圆粘在一起，我们就得到了一个甜甜圈，我们称之为环面。环面有一个度量，这是一种测量距离的方法，通过它与平面中具有通常平坦(欧几里得)度量的正方形的标识而给出。圆环上的一条曲线是一个闭合的环(想像一条缠绕在曲面上的绳子，在那里你把两个端点绑在一起)，我们把它拉紧，这样它就会变得尽可能短。结果表明，长度环面上的曲线数目至多为R，与半径为R的圆内平面上的整点数目完全相同。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，我们将得到一个更复杂的曲面。事实上，一般来说，我们得到的表面看起来像几个圆环粘在一起。环面的个数称为曲面的亏格g。然而，为了在曲面上获得良好的(常曲率)度量，我们需要从双曲平面(它是负弯曲的，就像碗的内部一样)而不是通常的欧几里得平面(它是平的)中切割出多边形。这极大地改变了曲线数量的增长：Huber在20世纪60年代表明，当g&gt；1时，渐近增长是指数长度。然而，如果我们只看不自交的曲线，则曲线要少得多，我们再次获得多项式增长率(这是Bman-Series在80年代首次观察到的，并在2001年由Rivin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，米尔扎哈尼在2008年用一个深刻的定理解决了这个问题。她证明了亏格g&gt；1的曲面上长度至多为R的简单曲线的个数与R^{6g-6}的常数倍渐近。米尔扎哈尼的结果立即成名，因为它是她关于曲线计数、体积增长和威腾猜想(物理学中的一个重要问题)的三个结果的一部分，在几何学和动力学领域都具有开创性，并对物理学具有重要意义。在这个项目中，我们使用新的方法来处理曲线的计数问题，这使得我们可以推广她的结果。事实上，我们还得到了米尔扎哈尼结果的一个新的、非常不同的证据。最初的证明要求专家理解几个数学领域，即使是这些领域的专家也很难完全掌握；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。新的证明还给出了计算与Mirzakhani定理有关的重要常数的新方法。这些方法的新颖性在于使用了所谓的测地线电流，这是一个统一了对曲线、测量的分层和双曲度量的研究的空间，所有这些都是曲线计数的完整概念。

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.