The 40th Anniversary of Shenzhen University and the 40th Anniversary of Mathematics Department

Liyuan scholar Colloquium thirty-nine

Lecture Title: Famous Conjectures in Reconciliation Analysis and Related Fields

Speaker: Miao Changxing, Researcher (Beijing Institute of Applied Physics and Computational Mathematics)

Lecture time: 16:30-17:30, September 7, 2023

Lecture location: Classroom 3, 1st floor, Huixing Building,YuehaiCampus, Shenzhen University

Overview: Besicovitch constructed the Besicovitch set (the set of arbitrarily oriented unit segments in n-dimensional Euclidean space) in the course of solving Kakeya's "spinning needle" problem, Fefferman was the first to solve the famous "disc conjecture" using the Besicovitch set, Bourgain's genius elevated Kakeya's conjecture to Kakeya's great function conjecture and Nikodym's great function conjecture, and built the geometric measure theory. Fefferman was the first to use the Besicovitch set to solve the famous "Disc Conjecture", and Bourgain's genius in elevating the Kakeya Conjecture to the Kakeya and Nikodym Conjectures, bridged the gap between geometric measure theory and the study of harmonic analysis (by bringing the geometric Kakeya Conjecture into the realm of modern harmonic analysis). The study of this conjecture has gone through many vicissitudes, and it has gradually been found to be closely related to the Fourier restriction conjecture, the Bochner-Riesz conjecture, and the locally smooth conjecture, forming the four famous conjectures in the field of harmonic analysis. We believe that these famous mathematical conjectures may be the manifestations of the same core problem in different fields of mathematical research. What is even more amazing is that so far, the study of the four conjectures has involved many mathematical fields such as harmonic analysis, partial differential equations, stacked number theory, algebraic geometry, geometric measure theory and associative geometry, arithmetic combinatorics, and so on. This presentation will focus on the four conjectures and introduce the famous conjectures related to them, the progress of research, and the modern mathematical methods derived from the study of these conjectures, in particular, the Bougain-Guth method, which mediates between structural interferences and the vanishing of the square root, the wave-packet decomposition and the scale induction method, the decoupling theory, and the decomposition of algebraic polynomials and the associative geometry method, etc. As applications, the presentation will also introduce the algebraic polynomial decomposition and the associative geometry method. As applications, the important role of these methods in the research fields of PDEs, geometric measure theory, number theory, etc. will also be presented.

Speaker Introduction: Changxing Miao, researcher of Beijing Institute of Applied Physics and Computational Mathematics. Yu Min Prize in Mathematical Sciences, Distinguished Expert of Chinese Academy of Engineering Physics, First Prize of Scientific and Technological Innovation of Chinese Academy of Engineering Physics, he is an outstanding mathematician trained by our country and influential in the field of partial differential equations in the international arena. In recent years, he has published more than 100 papers in leading international academic journals, such as CPAM, CMP, ARMA, MZ, JFA, JMPA, SIAM, AIHP, CPDE, PLMS, etc. His main contributions are in the fields of modulation analysis, scattering theory of nonlinear dispersion equations and mathematical theory of fluid dynamics equations, etc., and he has solved a number of mathematical problems with international influence. He has been highly praised by the famous mathematicians Kenig and Constantin and other international colleagues. He has successively published five monographs, including "Harmonic Analysis and its Application in Partial Differential Equations", "Harmonic Analysis Methods for Partial Differential Equations" and "Modern Methods for Nonlinear Fluctuation Equations", which have played a fundamental role in the research and development of this core mathematical field in China. At the same time, among the young talents he has trained, several students have stood out and achieved outstanding research results in the frontier field of harmonic analysis, which has attracted the attention of their international counterparts.

Teachers and students are welcome to participate!

Schoolof Mathematical Sciences

September7,2023