Toward Modern Mathematics Seminar No. 958: Prof. Miao Changxing on 'Function Spectral Geometry in Harmonic Analysis and PDEs'
According to the Institute of Mathematics, the 958th “Toward Modern Mathematics” seminar will be held on July 6, 2026, featuring Prof. Miao Changxing from the Beijing Institute of Applied Physics and Computational Mathematics.
The talk is titled “Function Spectral Geometry in Harmonic Analysis and PDEs,” invited by Lou Zengjian and Yu Haixia.
| Details | Information |
|---|---|
| Title | Function Spectral Geometry in Harmonic Analysis and PDEs |
| Speaker | Prof. Miao Changxing (Beijing Institute of Applied Physics and Computational Mathematics) |
| Time | July 6, 2026, 10:00 |
| Tencent Meeting ID | 212-691-984 |
Abstract: The Fourier transforms of solutions to free dispersive equations are supported on smooth hypersurfaces with non-zero Gauss curvature. How does the geometric curvature of these hypersurfaces affect the structural interference of solutions in physical space? This leads to Fourier restriction norm methods for studying nonlinear dispersive and wave equations. The dual forms of restriction theorems — Strichartz estimates, wave packet decompositions arising from frequency support on smooth hypersurfaces, and corresponding square function and decoupling estimates — provide frameworks and methods for studying nonlinear dispersive equations. For elliptic and parabolic equations, however, the Fourier spectrum of solutions no longer lies on hypersurfaces with non-trivial geometric curvature, but rather exhibits nearly spectrally compactified characteristics on hyperplanes. Solutions then display averaging properties in physical space and satisfy Harnack inequalities, providing a framework — variational principles or regular approximations — for studying weak solution existence, with regularity analyzed via De Giorgi iteration, Nash-Moser iteration, and other classical methods. This talk will examine from a function-theoretic perspective how different types of PDEs dictate distinct evolution laws of Fourier spectra, thereby revealing the dynamical behavior and singularity propagation of PDE solutions.
Prof. Miao Changxing is a recipient of the National Science Fund for Distinguished Young Scholars, the Yu Min Mathematical Physics Science Award, and the Distinguished Expert Award of the China Academy of Engineering Physics. He is a renowned mathematician trained in China with international influence in PDEs and harmonic analysis. He has published over 100 papers in top-tier journals including CPAM, JEMS, Ann. PDE, PLMS, CMP, Adv. Math, ARMA, JMPA, TAMS, IMRN, AIHP, CPDE, and JFA. His work on scattering theory for nonlinear dispersive equations and mathematical theory of fluid dynamics equations has resolved several internationally significant problems, earning high praise from colleagues including Kenig and Constantin. He has authored six monographs, playing an important role in advancing this core area of mathematics in China.
Source: Institute of Mathematics OA Notice