Random plane geometry

这个提议的目标是创建和研究一个涉及不确定性的平面几何的现代版本。谷歌地图为我们的提示选择的路线不是直线或曲线。我们还能从数学上理解它吗？什么是正确的语言？概率学家以前就研究过这类问题。随机函数是什么样的？巴切利耶、爱因斯坦和维纳在各自的著作中回答了这个问题：这是布朗运动，我们看到的曲线，例如股票价格波动。布朗运动概念的力量来自于它的普遍性：从本质上讲，大多数随机曲线看起来都像布朗运动：随机漫步、股票奖励、粒子扩散，甚至二维增长的表面。与J. Ortmann和D. Dauvergne一起，我们构建了定向景观，一个在平面几何世界中扮演布朗运动角色的物体。它被认为是一个普适的物体，就像布朗运动一样：每一个随机的平面几何应该看起来像定向景观。我们2019年的论文（约70次引用）被邀请到Acta Mathematica。这开辟了新的研究方向。1. 统计和实际应用。在使用液晶的实验中观察到了随机平面几何中出现的分布；它们也准确地描述了咖啡渍。但任何二维数据，如地图、脑表面测量、城市距离、湍流环境中飞行的最短路径，都应该具有随机的几何结构。注意：重尾随机性的存在会改变指数。我将收集平面数据集，以了解自然界中出现的指数，并为所有指数类开发数学。该理论告诉我们，采取何种测量方法，以一种新的方式对地理数据进行分类。2. 普遍性。我们知道只有少数模型可以证明定向景观是极限。我一直在和学生们一起寻找更多的例子，证明一些极限情况下的普适性。在简单的情况下，我们甚至没有一个好的指数界。3. 数学的应用。就像普通几何一样，随机平面几何常常隐藏在表面后面。其中一个问题是在一个时空随机环境中的一维热流。相应的卡尔达-帕里西-张方程为物理学的这整个研究领域赋予了它的名字。最近，我证明了这个方程收敛于定向景观。与一名学生一起，我们正在努力将这些结果扩展到一般定向聚合物。在另一个项目中，我们致力于理解二维随机热流（抛物线安德森模型）以及它如何与定向景观相连接。4粒子系统。被称为tasep的简单粒子系统与随机几何非常相似。生物学家用它来了解细胞内蛋白质合成的交通堵塞。我们的工作表明，第二类粒子在tasep中的行为与定向景观中的测地线非常相似。

The goal of this proposal is to create and study a modern version of plane geometry that involves uncertainty. The route google maps chooses for our tip is not a straight line or a curve. Can we still understand it mathematically? What is the right language to use? Probabilists have worked on such questions before. What is a random function like? This was answered by Bachelier, Einstein, and Wiener in separate works: it is Brownain motion, curve we see, e.g. in stock price fluctuations. The strength of the concept of Brownian motion comes from its universality: deep down, most random curves look like Brownian motion: random walks, stock prizes, diffusion of particles, even surfaces of two-dimensional grwoth. With J. Ortmann and D. Dauvergne we construrced the directed landscape, an object that plays the role of Brownian motion in the world of plane geometry. It is expected to be universal object just like Brownian motion: every random plane geometry should look like the directed landscape. Our 2019 paper (about 70 citations) was invited to Acta Mathematica. This opens up new lines of research. 1. Statistical and real-world applications. The distributions that arise in random plane geometry have been observed in experiments using liquid crystals; they also accurately describe coffee stains. But any two-dimensional data, e.g. maps, brain surface measurements, distances in cities, shortest paths for flights in turbulent environment should have a random geometric structure. Caveat: the presence of heavy-tailed randomness will change exponents. I will collect planar data sets to understand which exponents arise in nature, and develop mathematics for all exponent classes. The theory tells us what measurements to take to classify geographic data in a new way. 2. Universality. We know only a few models in which the directed landscape can be proven to be the limit. I have been working with students to find more examples and prove universality in some limiting cases. We don't even have any good bounds on the exponents in simple cases. 3. Mathematical applications. Just like ordinary geometry, random plane geometry is often hiding behind the surface. One of these problems is one-dimensional heat flow in a spacetime random environment. The corresponding Kardar-Parisi-Zhang equation lent its name to this entire research area in physics. Recently, I showed that this equation converges to the directed landscape. With a student, we are working to extend such results to general directed polymers. In another project, we work to understand two-dimensional random heat flow (the parabolic Anderson model) and how it is connected to the directed landscape. 4 Particle systems. The simple particle system called tasep is deeply equivalent to random geometry. It is used by biologists to understand traffic jams in protein synthesis inside cells. We work to show that second-class particles in tasep behave very similarly to geodesics in the directed landscape.