Homotopical Methods in Higher Representation Theory

对称性的研究是理解我们周围世界的基本工具，也是我们用来描述它的数学框架。随着我们不断突破数学理解的界限，探索理论物理的前沿，越来越清楚的是，下一代理论需要更精确的对称性概念。 这些类型的“更高”对称性的数学探索和发展揭示了它们存在的任何系统中的结构和形式，减少了这些系统中的不确定性程度，并使进步触手可及。 虽然在发展这些更高的对称性方面已经取得了很大的进展，但目前理解的剃刀边缘已经揭示，最令人兴奋的应用将需要比以前开发的更灵活的理论。 该项目将填补这一知识空白，提供一个强大的方法来更高的对称性，能够描述更复杂的现象，并提供必要的框架，以促进最紧迫的应用。该奖项还将支持研究生和博士后的培训和专业发展。该项目利用同伦方法扩展高级表示理论领域的现有技术。 主要目的是将分类量子群的概念从其加性框架扩展到更强大和同伦风味的理论，更适合解决该领域最紧迫的开放问题。 特别是，这样一个框架将揭示更高的表示的张量积，导致更丰富的更高的表示理论，允许无限维的表示，并提供直接的应用程序，以稳定的改进链接同源。 该奖项还将支持研究生和博士后的培训和专业发展。该奖项反映了NSF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 study of symmetry is a fundamental tool for understanding the world around us, as well as the mathematical frameworks we use to describe it. As we push the boundaries of our understanding of mathematics and explore the frontiers of theoretical physics, it is becoming increasingly clear that a more refined notion of symmetry is required for the next generation of theories. The mathematical exploration and development of these types of "higher" symmetries reveals structure and form in any system where they are present, reducing degrees of uncertainty in these system and bringing progress within reach. While a great deal of progress has been made developing these higher symmetries, the razor's edge of current understanding has revealed that the most exciting applications will require a more flexible theory than has been previously developed. This project will fill this knowledge gap by providing a robust approach to higher symmetries, capable of describing more complex phenomena and supplying the requisite framework to facilitate the most pressing applications. This award will also support the training and professional development of graduate students and postdocs. This project utilizes homotopical methods to expand existing technology in the field of higher representation theory. The primary aim is to expand the notion of categorified quantum group from its additive framework to a more robust and homotopically flavored theory that is better adapted for solving the most pressing open problems in the area. In particular, such a framework will shed light on tensor products of higher representations, lead to a richer higher representation theory that allows for infinite dimensional representations, and provide direct applications to stable refinements in link homology. This award will also support the training and professional development of graduate students and postdocs.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.