Random walks and super-approximation

PI将研究由对称性给出的随机漫步，并探索其在各种数学分支及其他领域的应用。考虑一个具有许多对称性的空间，例如球体。在这个空间中选择一个有限的对称集合，把一个粒子放在一个点上，每次根据我们选择的集合中随机选择的对称移动它。这被称为随机漫步。当经过几步后，观察者不再有关于粒子位置的太多信息时，我们说随机漫步具有谱间隙特性。关于同一空间的不同随机漫步存在许多重要问题。例如，一个主要问题是，在合理的假设下，谱间隙是否是空间的属性，而不是随机漫步。另一个问题是是否有可能有两个随机漫步，观察其中一个会给我们关于另一个的信息。PI将解决这类由数论问题产生的空间问题；这就是所谓的超逼近猜想。PI将在该奖项下继续指导研究生和博士后。在具有谱间隙属性的无限空间上进行单个随机漫步是许多应用的来源。例如，通过离散空间，构建了高度连通的稀疏图，称为展开图。利用球上有谱隙的随机漫步方法获得球上点的有效分布，近年来在量子计算中得到了应用。在过去的15年里，它已经证明了随机漫步选择的灵活性，它伴随着超逼近，使它成为一个强大的工具，用于广泛的应用。这里有几个例子：仿射筛，群论中的筛，伽罗瓦表示的变化，Zaremba猜想，几何有限双曲流形中的混合，轨道等效刚性。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The PI will study random-walks given by symmetries and explore their applications to various branches of mathematics and beyond. Consider a space with lots of symmetries, for example a sphere. Choose a finite set of symmetries of this space, put a particle at a point, and each time move it according to a randomly picked symmetry from our chosen set. This is called a random-walk. When after a few steps, an observer no longer has much information about the position of the particle, it is said that the random-walk has the spectral gap property. There are many important questions about different random-walks on the same space. For instance a major problem is whether under reasonable assumptions spectral gap is a property of the space rather than a random-walk. Another question is whether it is possible to have two random-walks where observing one gives us information about the other. The PI will solve these type of questions for spaces that emerge from number theoretic problems; this is known as super-approximation conjecture. The PI will continue his mentoring of graduate students and postdocs under this award. Having a single random-walk on an infinite space with the spectral gap property was the source of many applications. For instance, by discretizing the space, highly connected sparse graphs, known as expanders, were constructed. Having a random-walk with spectral gap on spheres were used to obtain an effective distribution of points on spheres, and recently used in quantum computing. In the past 15 years, it has been shown that the flexibility of the choice of random-walk which comes with the super-approximation makes it a powerful tool for a wide range of applications. Here are a few examples: affine sieve, sieve in group theory, variation of Galois representations, Zaremba’s conjecture, mixing in geometrically finite hyperbolic manifolds, and orbit equivalence rigidity.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.