Random Walks in a Compact Group and Super-Approximation in Number Theory

该项目涉及随机漫步和数论。随机漫步是一条路径，其中关于在漫步中的特定点走哪条路的决定是以某种随机方式选择的。这个项目的主要目标是研究在一个固定的紧群中，当我们选择不同的随机游动时，随机游动的主要特征是否保持不变。紧群的一个例子是球面。PI将针对与本项目相关的主题开展阅读课程。这个项目的一个重要特点是它与数学的许多领域的联系。这可以帮助学生看到什么样的数学让他们更兴奋。PI将每周举办一次研讨会，邀请年轻和杰出的数学家。有这样一个积极的研讨会是至关重要的，我们的研究生达到他们的潜力。PI继续促进代表性不足的少数群体参与任何此类活动。如果紧致空间中的随机游动具有所谓的谱隙，那么它被理解得最好。这个性质或多或少表明，一个人到达（伪）随机点的速度相当快。由于这个原因，随机游走在计算机科学中被用来生成伪随机数。如果手头的紧凑空间（连续）通过有限集因子，那么得到高度连接（稀疏）的图-称为扩展器，在计算机科学和通信中有许多应用。在一个球面上找到一个具有谱隙的随机游动是非常重要的;而且，这样的随机游动是对巴拿赫-鲁济维奇问题给出肯定答案的来源;最近它们被用于量子计算。在过去的十年中，（本地）光谱间隙已被证明是数学的各个分支工具：仿射筛，伽罗瓦表示的变化，群论，双曲几何，轨道等效刚度等。该奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

The project concerns random walks and number theory. A random walk is a path where decisions about which way to go at a particular point in the walk is chosen in some random fashion. The main goal of this project is to investigate whether main features of random-walks in a fixed compact group stay the same as we choose different random-walks. An example of a compact group would be the surface of a sphere. The PI will conduct reading courses on the subjects related to this project. One of the important features of this project is its connections with many areas of mathematics. And this can help students to see what kind of mathematics makes them more excited. The PI will run a weekly seminar, and invites both young and prominent mathematicians. Having such an active seminar is crucial for our graduate students to reach to their potential. The PI continues to promote participation of underrepresented minorities in any such activities. A random-walk in a compact space is understood the best if it has the so-called spectral gap. This property more or less says one reaches to a (pseudo)-random point quite fast. For that reason random-walks have been used in computer science for generating pseudo-random numbers. If the compact space in hand (continuously) factors through finite sets, then one gets highly connected (sparse) graphs - called expanders, which have many applications in computer science and communications. Finding even a single random-walk with spectral gap in a sphere is highly non-trivial; and, such a random-walk was the source of giving an affirmative answer to Banach-Ruziewicz problem; and more recently they are used in quantum computing. In the past decade, (local) spectral gap have been proved to be instrumental in various branches of mathematics: affine sieve, variation of Galois representations, group theory, hyperbolic geometry, orbit equivalence rigidity, etc.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.