Random walks and super-approximation

PI将研究对称给出的随机游动，并探索它们在数学的各个分支和更远的领域中的应用。考虑一个具有许多对称性的空间，例如一个球体。选择这个空间的一个有限对称集合，将一个粒子放在一个点上，每次根据我们选择的集合中随机挑选的对称来移动它。这被称为随机漫步。当经过几个步骤后，观测者不再有关于粒子位置的太多信息时，可以说随机游走具有谱隙性质。关于同一空间上的不同随机行走，有许多重要的问题。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一种性质，而不是一种随机游走。另一个问题是，是否有可能有两个随机漫步，其中一个观察给我们关于另一个的信息。PI将为数论问题中出现的空间解决这些类型的问题；这被称为超逼近猜想。该奖项将继续指导研究生和博士后。在具有谱间隙性质的无限空间上进行一次随机游走是许多应用的来源。例如，通过对空间进行离散化，构造了高度连通的稀疏图，称为扩展器。在球面上进行带隙随机游走是为了获得球面上点的有效分布，最近在量子计算中得到了应用。在过去的15年中，研究表明，超逼近带来的随机游走的灵活性使其成为一种强大的工具，适用于广泛的应用。这里有几个例子：仿射筛子，群论中的筛子，伽罗瓦表示的变化，Zaremba猜想，几何有限双曲流形中的混合，以及轨道等价刚性。这个奖项反映了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 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.