CAREER: New Techniques for High Dimensional Systems

随着“大数据”的兴起，大数据在工业和学术领域的价值日益得到重视。通常，此类数据集的分析从将数据建模为高维系统开始，因此了解此类系统的行为(以及处理它们的技术)变得同样重要。该奖项支持使用研究人员在之前与合作者的工作中介绍的技术对此类系统的行为进行数学调查。这些技术已经解决了许多不同数学领域的公开问题，包括Kadison-Singer问题和各种大小和度数的Ramanujan图的存在。这个项目旨在进一步发展与许多科学领域的研究的实质性联系，包括凸优化、实代数几何、泛函分析、组合学和概率论。在技术层面上，该项目侧重于两个领域的研究：(A)发展一种“有限自由概率”理论，这是一种集合了随机矩阵理论、凸优化、实代数几何和多项式几何的思想的集合；(B)推广上述技术，以扩大其潜在的应用。这包括将概念扩展到二元多项式以及非厄米特环境。在(B)的方向上，该项目旨在通过研究更一般情况下的相似行为来解决阻碍当前技术应用的一些障碍(例如，需要相关多项式的实根性)。例如，这将是将(A)扩展到非埃尔米特设置的必要部分，因为这样的对象不再满足应用当前技术所必需的条件。在这个方向上的扩展也将打开应用于新领域的可能性，例如定量几何。

The onset of "big data" has led to an increased appreciation of the value of large data sets in both industrial and academic settings. Typically, the analysis of such data sets begins by modeling the data as a high dimensional system, and so an understanding of the behavior of such systems (and techniques for dealing with them) has become equally important. This award supports a mathematical investigation of the behavior of such systems using techniques introduced by the researcher in previous work with collaborators. These techniques have led to the resolution of a number of open problems across various mathematical fields, including the Kadison-Singer problem and the existence of Ramanujan graphs of all sizes and degrees. This project aims to further develop substantial connections with research across numerous fields of science, including convex optimization, real algebraic geometry, functional analysis, combinatorics, and probability. The award also supports plans to focus interdisciplinary activities of particular benefit to students associated with the project.On a technical level, the project focuses on two areas of research: (A) to develop a theory of ''finite free probability,'' a collection of ideas lying in the intersection of random matrix theory, convex optimization, real algebraic geometry, and polynomial geometry, and (B) to generalize the previously mentioned techniques so as to widen their potential application. This includes extending the ideas to bivariate polynomials as well as a non-Hermitian setting. In the direction of (B), the project aims to address some of the obstacles that hinder the application of the current techniques (for example, the need for real rootedness of associated polynomials) by investigating analogous behaviors in more general settings. This would be a necessary part of extending (A) to non-Hermitian settings, for example, as such objects no longer satisfy the conditions necessary to apply the current techniques. Extensions in this direction would also open the possibility of application to new areas such as quantitative geometry.