Random walks and super-approximation

PI将研究由对称性给出的随机行走，并探索它们在数学各个分支及其他领域的应用。考虑一个具有很多对称性的空间，例如一个球体。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，每次根据我们从所选集合中随机选择的对称来移动它。这被称为随机漫步。 当经过几步后，观察者不再拥有有关粒子位置的太多信息时，就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游动有许多重要的问题。例如，一个主要的问题是，在合理的假设下，谱隙是否是空间的一个属性，而不是随机游走。另一个问题是，是否可能有两个随机行走，其中观察一个可以给我们关于另一个的信息。PI将解决从数论问题中出现的空间的这类问题;这被称为超近似猜想。PI将继续在此奖项下指导研究生和博士后。在无限空间上具有谱隙性质的单次随机游动是许多应用的来源。例如，通过离散化空间，构建了高度连通的稀疏图，称为扩展器。在球面上进行带谱隙的随机游走可以得到球面上点的有效分布，最近被应用于量子计算中。在过去的15年里，已经表明，随机游动的选择的灵活性，它与超近似，使其成为一个强大的工具，为广泛的应用。以下是一些例子：仿射筛法、群论筛法、伽罗瓦表示的变化、扎伦巴猜想、几何有限双曲流形中的混合和轨道等价刚性。该奖项反映了NSF的法定使命，并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，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.