Projektdetails
Abstract
Wider research context:
Symmetry provides a powerful framework for analyzing functions. The most successful example of this is the decomposition of a repeating wave according to its harmonics: for an integer n, these are basic waves given by sin(nx) and cos(nx). Any repeating wave can be graphed over a circle, and here, the frequency n of a harmonic is simply the measure of its circular symmetry. Harmonic analysis pairs this conceptual picture with a precise and efficient calculus, yielding a study that has enriched fields ranging from quantum mechanics to medical imaging. This project is concerned with notions of harmonic analysis when the circle is replaced with a quantum group. By “harmonic analysis on quantum groups'', we do not mean here a specific formal discipline, but rather a heuristic for finding and studying new functions distinguished by quantum symmetry. We take as a model the rich theory and applications of Macdonald polynomials.
Objectives:
We will study special functions arising from quantum groups and the algebras of operators acting on them. There are four specific paths of inquiry. First, the braided monoidal structure of the category of finite-dimensional representations of quantum groups allows connections to skeins on Riemann surfaces; we plan to apply Macdonald polynomials and their generalizations to relate double affine Hecke algebras (DAHAs) to quantized character varieties. Second, we will develop the theory of wreath Macdonald polynomials, especially regarding combinatorial formulas and Selbergtype integrals. Third, we aim to find and study new generalizations of Macdonald polynomials coming from quantum toroidal algebras of types D and E. Finally, we would like to study generalizations of Noumi–Shiraishi’s explicit formulas for Macdonald eigenfunctions.
Methods:
Each path requires different tools. The first path will involve DAHAs, deformation quantization, and skein algebras. The second path will require us to adapt many techniques from the theory of Macdonald polynomials and basic hypergeometric functions. The third path involves a deep study of quantum toroidal algebras and their associated shuffle algebras. The fourth path involves the refined topological vertex used in curve counts on Calabi–Yau threefolds.
Innovation:
This project will attack unsolved problems, some of which are decades old. The first path will clarify the isomorphism type of quantum character varieties and open up a topological approach to the representation theory of DAHAs. The second will derive new combinatorial and integral formulas that may be of use in random matrix theory. The third will uncover a new class of orthogonal polynomials. The fourth will introduce new functions and attempt to settle difficult problems concerning elliptic integrable systems.
Primary researchers involved:
Researchers include J. Wen (PI) and collaboration partners S. Albion Ferlinc, H. Dinkins, D. Laurie, A. Mellit, M. Romero, and Y. Zenkevich.
Symmetry provides a powerful framework for analyzing functions. The most successful example of this is the decomposition of a repeating wave according to its harmonics: for an integer n, these are basic waves given by sin(nx) and cos(nx). Any repeating wave can be graphed over a circle, and here, the frequency n of a harmonic is simply the measure of its circular symmetry. Harmonic analysis pairs this conceptual picture with a precise and efficient calculus, yielding a study that has enriched fields ranging from quantum mechanics to medical imaging. This project is concerned with notions of harmonic analysis when the circle is replaced with a quantum group. By “harmonic analysis on quantum groups'', we do not mean here a specific formal discipline, but rather a heuristic for finding and studying new functions distinguished by quantum symmetry. We take as a model the rich theory and applications of Macdonald polynomials.
Objectives:
We will study special functions arising from quantum groups and the algebras of operators acting on them. There are four specific paths of inquiry. First, the braided monoidal structure of the category of finite-dimensional representations of quantum groups allows connections to skeins on Riemann surfaces; we plan to apply Macdonald polynomials and their generalizations to relate double affine Hecke algebras (DAHAs) to quantized character varieties. Second, we will develop the theory of wreath Macdonald polynomials, especially regarding combinatorial formulas and Selbergtype integrals. Third, we aim to find and study new generalizations of Macdonald polynomials coming from quantum toroidal algebras of types D and E. Finally, we would like to study generalizations of Noumi–Shiraishi’s explicit formulas for Macdonald eigenfunctions.
Methods:
Each path requires different tools. The first path will involve DAHAs, deformation quantization, and skein algebras. The second path will require us to adapt many techniques from the theory of Macdonald polynomials and basic hypergeometric functions. The third path involves a deep study of quantum toroidal algebras and their associated shuffle algebras. The fourth path involves the refined topological vertex used in curve counts on Calabi–Yau threefolds.
Innovation:
This project will attack unsolved problems, some of which are decades old. The first path will clarify the isomorphism type of quantum character varieties and open up a topological approach to the representation theory of DAHAs. The second will derive new combinatorial and integral formulas that may be of use in random matrix theory. The third will uncover a new class of orthogonal polynomials. The fourth will introduce new functions and attempt to settle difficult problems concerning elliptic integrable systems.
Primary researchers involved:
Researchers include J. Wen (PI) and collaboration partners S. Albion Ferlinc, H. Dinkins, D. Laurie, A. Mellit, M. Romero, and Y. Zenkevich.
| Kurztitel | Harmonic analysis on quantum groups |
|---|---|
| Status | Laufend |
| Tatsächlicher Beginn/ -es Ende | 1/06/26 → 31/05/30 |