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

这个项目涉及随机游走和数论。随机行走是一种路径，其中在行走的特定点上选择的路径以某种随机方式选择。这个项目的主要目标是研究当我们选择不同的随机漫步时，固定紧群中的随机漫步的主要特征是否保持不变。紧群的一个例子是球面。PI将对与该项目相关的主题进行阅读课程。这个项目的一个重要特点是它与许多数学领域的联系。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，并邀请年轻和杰出的数学家。举办这样一个活跃的研讨会对我们的研究生发挥他们的潜力至关重要。PI继续促进代表性不足的少数群体参与任何此类活动。紧凑空间中的随机游走，如果有所谓的谱间隙，就能得到最好的理解。这个性质或多或少地表明，一个人很快就会到达一个（伪）随机点。因此，随机漫步在计算机科学中被用于生成伪随机数。如果紧化空间中的（连续的）因子通过有限的集合，那么就会得到高度连接的（稀疏的）图——称为展开图，它在计算机科学和通信中有很多应用。在一个球体中找到一个具有谱间隙的随机漫步是非常重要的；这种随机漫步是Banach-Ruziewicz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，（局部）谱隙已被证明在数学的各个分支中起着重要作用：仿射筛、伽罗瓦表示的变化、群论、双曲几何、轨道等效刚性等。该奖项反映了国家科学基金会的法定使命，并通过基金会的知识价值和更广泛的影响审查标准进行了评估，认为值得支持。

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.