Double Affine Hecke Algebras

双仿射赫克代数

• 批准号: 1901796
• 负责人: Ivan Cherednik
• 金额: $ 57.5万
• 依托单位: University of North Carolina at Chapel Hill
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-07-01 至 2025-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1901796&HistoricalAwards=false
• 关键词: Double; Affine; Hecke; Algebras

The aim of the project is the study of double affine Hecke algebras (DAHAs) and their applications in geometry, number theory, topology, harmonic analysis, and combinatorics, as well as exciting applications in modern mathematical physics, for instance in string theory. This includes new connections between physics and number theory via DAHAs, which can connect correlation functions in physics with count of points over finite fields in geometry. This work is firmly aligned with Quantum Leap, one of the NSF's 10 Big Ideas, through software to be developed that will compute the DAHA-Fourier transform at roots of unity.The major themes of the proposed research are: (1) DAHA and motivic theory of invariants of algebraic links and plane curve singularities, including the corresponding Riemann hypothesis, (2) harmonic analysis on DAHA, which generalizes theory of affine Hecke algebras and is related to the PI's theory of difference hypergeometric functions, and (3) applications in number theory, including q-zeta functions and Rogers-Ramanujan identities. The key is a recent connection between the DAHA superpolynomials (conjecturally related to the reduced Khovanov-Rozansky polynomials) and the zeta-functions of plane curve singularities. This connection conjecturally identifies the so-called super-duality in physics (say S-duality in M-theory) and that in DAHA theory with the functional equation in number theory.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.

该项目的目的是研究双仿射Hecke代数（DAHA）及其在几何，数论，拓扑学，调和分析和组合学中的应用，以及在现代数学物理中令人兴奋的应用，例如在弦理论中。这包括物理学和数论之间通过DAHA的新联系，它可以将物理学中的相关函数与几何学中有限域上的点的计数联系起来。这项工作与NSF的10大理念之一量子飞跃（Quantum Leap）紧密结合，通过将开发的软件来计算DAHA-傅立叶变换的统一根。拟议研究的主要主题是：（1）DAHA与代数杆系和平面曲线奇点不变量的motivic理论，包括相应的Riemann假设，（2）DAHA的调和分析，（3）在数论中的应用，包括q-zeta函数和Rogers-Ramanujan恒等式。关键是最近DAHA超多项式（与约化的Khovanov-Rozansky多项式相关）和平面曲线奇点的zeta函数之间的联系。这种联系将物理学中所谓的超对偶性（如M理论中的S对偶性）和DAHA理论中的超对偶性与数论中的函数方程联系起来。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。