Pseudodifferential operators were first explicitly defined by Kohn-Nirenberg and Hörmander to connect singular integrals and differential operators. The theory of pseudodifferential operators serves as a unifying framework in modern harmonic analysis, which has substantial impact on linear and non-linear PDEs and differential geometry. In this talk, we report our recent work on the spectral asymptotics of pseudodifferential operators, and explain how the spectral asymptotics serve as a key ingredient in quantum calculus in the setting of noncommutative geometry introduced by Alain Connes. We will also mention an application to the semiclassical Weyl law.

Speaker Profile: Xiong Xiao, Professor and Executive Vice Dean of the Institute of Mathematics, Harbin Institute of Technology. The research fields include harmonic analysis, non-commutative analysis and their applications, etc. The research interests mainly focus on non-commutative analysis, which is a branch of functional analysis and mainly involves harmonic analysis and operator algebra. It is one of the most active and fruitful frontier interdisciplinary research fields in the discipline of mathematics in recent years. The research work mainly focuses on the harmonic analysis of operator values, non-commutative geometry, and harmonic analysis on groups. The main results were published in Mem. Amer. Math. Soc., Comm. Math. Phys., Adv. Math., J. Math. Pures Appl., J. Funct. Anal. Such as international authoritative journals. Received the "Frontier Science Award" at the 2024 International Conference on Basic Science.