Anti-Concentration, Random Structures, and Sumsets

在这个项目中，目标是研究从实际问题中产生的随机变量的分布，如求解线性方程组或寻找高次多项式的根。这些问题在计算机科学和工程的许多发展中至关重要。求多项式的根是一个经典的基本问题，有着广泛的应用。 该项目的主要重点是证明随机变量的许多自然分布在单个点上没有大质量。 有了这个新工具，研究人员希望为长期存在的问题提供答案，例如：在典型情况下，n次多项式的根有多少是真实的？主要研究者的目标是发展一个理论的反集中现象。反集中不等式在许多涉及概率的领域中起着重要的作用。 最近在该领域的新见解，导致了显着的改进，具有广泛的应用范围的几个经典的结果。研究人员计划将新结果扩展到更一般的（非线性）设置。 这方面的成就将在不同领域产生直接影响。 该项目的另一部分继续研究随机离散结构，特别是随机矩阵和随机多项式。PI将研究各种基本问题，如随机行列式的性质和分布，多重特征值的不存在性及其与QR算法的关系，以及经典问题：一个随机多项式有多少根是真实的？ 他还将研究加法组合学中的和集问题（例如，Erdos和Moser关于一组中的和自由集的老问题）。

In this project, the goal is to investigate the distribution of random variables arising from practical problems such as solving linear systems of equations or finding the roots of high degree polynomials. These problems are critical in many developments in computer science and engineering. Finding roots of a polynomial is, in particular, a classical and fundamental problem with widespread applications. The main focus of the project is to prove that many natural distributions of random variables do not have large mass on a single point. With this new tool, the investigator hopes to provide answers to long-standing questions such as: In a typical case, how many roots of a polynomial of degree n are real? The principal investigator aims to develop a theory for the anti-concentration phenomenon. Anti-concentration inequalities have been playing an important role in many areas where probability is involved. Recent new insights in the field have led to a significant refinement of several classical results with a broad range of applications. The investigator plans to extend the new results to a more general (nonlinear) setting. Achievements in this direction will have an immediate impact in different fields. Another part of the project continues the study of random discrete structures, in particular, random matrices and random polynomials. The PI will investigate various basic problems, such as properties and distribution of random determinants, the non-existence of multiple eigenvalues and its relation to the QR algorithm, and the classical question: How many roots of a random polynomial are real? He will also investigate sum-set problems in additive combinatorics (for example, an old question of Erdos and Moser on sum-free sets in a group).