Random plane geometry

随机平面几何形状

• 批准号: RGPIN-2022-04786
• 负责人: Virag, Balint
• 金额: $ 3.13万
• 依托单位: University of Toronto
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=749633
• 关键词: Random; plane; geometry

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.

这个计划的目标是创建和研究一个现代版本的平面几何，涉及不确定性。谷歌地图为我们的提示选择的路线不是一条直线或曲线。我们还能从数学上理解它吗？什么是正确的语言使用？概率论者以前也研究过这类问题。什么是随机函数？这是回答巴舍利耶，爱因斯坦和维纳在不同的作品：这是布朗运动，曲线，我们看到，例如在股票价格波动。布朗运动概念的力量来自于它的普遍性：在内心深处，大多数随机曲线看起来像布朗运动：随机行走，股票奖励，粒子扩散，甚至二维增长的表面。Ortmann和D.在Dauvergne，我们创造了有向景观，一个在平面几何世界中扮演布朗运动角色的物体。期望它像布朗运动一样是普适对象：每个随机平面几何都应该看起来像有向景观。我们2019年的论文（约70次引用）被邀请到Acta Mathematica。这开辟了新的研究领域。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污渍。但是任何二维数据，例如地图、大脑表面测量、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。警告：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，并开发所有指数类的数学。该理论告诉我们采取什么样的测量方法来以一种新的方式对地理数据进行分类。2.普遍性。我们只知道几个模型，其中有向景观可以被证明是极限。我一直在与学生合作，寻找更多的例子，并证明在一些限制情况下的普遍性。在简单的情况下，我们甚至没有指数的好界限。3.数学应用。就像普通几何一样，随机平面几何往往隐藏在曲面后面。这些问题之一是在时空随机环境中的一维热流。相应的Kardar-Parisi-Zhang方程为整个物理学研究领域提供了名称。最近，我证明了这个方程收敛于定向景观。我们正在与一个学生一起努力将这样的结果推广到一般的定向聚合物。在另一个项目中，我们致力于了解二维随机热流（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完全等价。它被生物学家用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的行为非常相似的测地线在有向景观。