Quantum Harmonic Analysis and Applications

The project aims to conduct a systematic study of quantum harmonic analysis, an emerging new area that extends the core ideas of classical harmonic analysis into the quantum setting. While classical harmonic analysis is concerned with representing functions or signals as combinations of basic waves (like sine and cosine functions), quantum harmonic analysis applies these methods to quantum states and observables within the framework of quantum mechanics. This field plays a foundational role in quantum information science, providing the mathematical tools needed to describe, analyze, and manipulate quantum systems. By developing a deeper theoretical understanding, the project will help lay the groundwork for future breakthroughs in quantum science and technology, including quantum computing, quantum cryptography, and quantum communication. The project also provides research training opportunities for graduate students. The project proposes new methods from topological dynamics and harmonic analysis to address numerous open problems in quantum harmonic analysis. In addition to tackling foundational questions, the project introduces several new research directions aimed at advancing this young and rapidly evolving area. The scope of the project includes a broad spectrum of problems with varying levels of difficulty, some specifically designed to be accessible to graduate students, thereby fostering involvement from early-career researchers. Moreover, the project provides a comprehensive framework to help organize, systematize, and clarify the recent surge of activity in quantum harmonic analysis. 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.