Double Affine Hecke Algebras

该项目的目的是研究双仿射Hecke代数（DAHAs）及其在几何、数论、拓扑学、谐波分析和组合学中的应用，以及在现代数学物理中的令人兴奋的应用，例如弦理论。这包括通过DAHAs将物理学和数论联系起来，它可以将物理学中的相关函数与几何中有限域上的点计数联系起来。这项工作与美国国家科学基金会（NSF）十大创意之一的量子飞跃（Quantum Leap）紧密相连，通过开发软件来计算统一根的daha -傅里叶变换。本文的主要研究课题是：(1)DAHA与代数环和平面曲线奇点不变量的动机理论，包括相应的Riemann假设；(2)DAHA的调和分析，它推广了仿射Hecke代数理论，与PI的差分超几何函数理论有关；(3)DAHA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（推测与约化Khovanov-Rozansky多项式有关）与平面曲线奇点的ζ函数之间的联系。这种联系在理论上将物理学中所谓的超对偶性（比如m理论中的s -对偶性）和DAHA理论中的超对偶性与数论中的泛函方程联系起来。该奖项反映了美国国家科学基金会的法定使命，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 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.