Double Affine Hecke Algebras

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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.