Geodesic Currents and Counting Problems

想象在平面上画一个圆，以原点为圆心，半径为R，你想计算圆内带整数系数的点的个数。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。显然，R越大点就越多，但这些数字究竟是如何联系起来的呢？我们可以用初等数学证明，半径为R的圆内的整数点的数目像pi*R^2一样增长，即它所包围的区域的面积。更准确地说，这样的点的数量是渐近于面积的，这意味着当R趋于无穷时，这两个量的比值趋于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，曲面是一个二维物体，它可以通过在平面上切出一个多边形，然后将两边成对地粘合在一起来获得。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。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。环面有一个度规，一种测量距离的方法，通过它与平面上的一个正方形的标识来给出，这个正方形具有通常的平度规（欧几里得）。环面上的曲线是一个闭环（想象一下缠绕在表面上的绳子，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。事实证明，长度不超过R的环面上曲线的数量与半径为R的圆内平面上整数点的数量完全相同。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表面。事实上，通常我们得到的表面看起来就像几个环面粘在一起。环面的数量称为曲面的g属。然而，为了在表面上得到一个漂亮的（恒定曲率）度规，我们需要从双曲平面（负弯曲，像碗的内部）而不是通常的欧几里得平面（平坦）上切下多边形。这极大地改变了曲线数量的增长：Huber在60年代表明，当g bbbb1时，曲线长度的渐近增长呈指数增长。然而，如果我们只看那些不自相交的曲线，曲线就会少得多，我们再次得到一个多项式增长率（这是在80年代由Birman-Series首次观察到的，并在2001年由Rivin更详细地证明了）。找到这些曲线的精确渐近增长是一个难题，米尔扎哈尼在2008年用一个深奥的定理解决了这个问题。她证明了在g>1属的曲面上长度不超过R的简单曲线的数目渐近于常数乘以R^{6g-6}。米尔扎哈尼的结果很快就出名了，因为它是她关于曲线计数、体积增长和威滕猜想（物理学中的一个重要问题）的三位一体结果的一部分，在几何和动力学领域都有突破性的进展，对物理学有重要的影响。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，米尔扎哈尼结果的证明。最初的证明需要对几个数学领域的专家理解，即使是该领域的专家也很难掌握全部细节；这种新方法有可能为来自更广泛专业领域的研究人员打开这一领域。新的证明也为计算与米尔扎哈尼定理有关的重要常数提供了一种新的方法。这些方法的新颖之处在于使用了所谓的测地线电流，这是一个将曲线研究、测量分层和双曲度量统一起来的空间，所有积分概念都用于曲线计数。

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.