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（DAHAS）及其在几何，数字理论，拓扑，谐波，谐波分析和组合中的应用，以及在现代数学物理学中的令人兴奋的应用，例如在字符串理论中。这包括通过DAHAS物理学和数字理论之间的新联系，该连接可以将物理中的相关函数与几何学有限领域的点计数相关联。 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主张Hecke代数与PI的差异高几何函数理论有关，以及（3）数字理论中的应用，包括Q-Zeta函数和Rogers-Ramanujan身份。关键是DAHA超级分析的最新联系（与降低的Khovanov-Rozansky多项式有关）与平面曲线奇异性的Zeta功能之间的联系。这种连接猜想在物理学中识别了所谓的超二倍度（例如，M理论中的S二重性），并且在DAHA理论中具有数字理论中的功能方程式。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子和广泛影响的评估来通过评估来支持的。