PDEA Seminar: Applying group duality in the analysis of operators

Abstract: Analysis of operators is of fundamental importance to science and engineering, and especially to quantum physics. There are types of operators—e.g., the Toeplitz type—whose properties are now well-understood theoretically, and which, in addition, are amenable to numerical pseudo-spectral methods of analysis. Regrettably, the commonly known techniques do not extend to other types of operators that are also of fundamental importance. In this talk, I will highlight examples of operators where the traditional techniques break down completely. These examples are rooted in the theory of interacting bosons and in the analytic number theory. At the same time, the task of analyzing such operators is not hopeless at all. I will demonstrate the effectiveness toward this end of the generalized Fourier transform specialized to the group of positive rationals Q+. This is a theoretical development, but it is also enabling for high accuracy numerical analysis. Another famous challenge in the analysis of operators is the so-called curse of complexity, which emerges in quantum engineering. In essence, it results from the exponential dependence of the size of relevant matrices on the number of qubits in the system that one wishes to model. However, an analysis based on an innovative application of multi-scale methods—tied to the infinite group Z2 ×Z2 ×Z2 ×. . .—enables one to fully solve certain types of quantum dynamics, even if they involve an infinite number of qubits. In fact, it furnishes an alternative framework for discussion of quantum information processing. While this framework is entirely equivalent to the canonical one, it is advantageous in the analysis of certain types of problems. Many of the results that I will discuss are an outcome of collaborations between mathematicians and physicists. The conceptual underpinnings of this work are rooted in harmonic analysis, multi-scale methods, and analytic number theory. Applications branch out from quantum theory to post-quantum cryptography and information science.

Participate on site or via Zoom link, meeting ID: 67226102216

This is a seminar in the PDEs and Applications seminar series