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

该项目涉及随机行走和数论。随机漫步是以某种随机的方式选择在漫步中的特定点选择走哪条路的路径。这个项目的主要目的是调查固定紧凑组中随机行走的主要特征是否与我们选择不同的随机行走相同。紧群的一个例子是球面。主办方将开设与该项目相关的阅读课程。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。国际数学家协会将每周举办一次研讨会，邀请年轻和杰出的数学家。举办如此活跃的研讨会对于我们的研究生发挥他们的潜力是至关重要的。非政府组织继续促进代表人数不足的少数群体参加任何此类活动。在一个紧凑的空间中随机行走，如果它有所谓的光谱间隙，那么它被理解得最好。这一性质或多或少表明，到达(伪)随机点的速度相当快。出于这个原因，随机游走在计算机科学中被用来产生伪随机数。如果手中的紧致空间(连续)通过有限集分解，则得到高度连通(稀疏)图-称为扩展器，它在计算机科学和通信中有许多应用。即使是在球面上找到一个有光谱间隙的随机游动也是非常不平凡的；而且，这种随机游动是给出肯定答案的Banach-Ruziewicz问题的来源；最近，它们被用于量子计算。在过去的十年里，(局部)光谱间隙已被证明在数学的各个分支中起到了重要作用：仿射筛选、伽罗瓦表示的变化、群论、双曲几何、轨道等价刚性等。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.