Bourgain-Demeter的分离性方法及其应用

The decoupling method originated from Wolff's study on the regularity of solutions to wave equations and Bourgain's research on the Strichartz estimate of solution to the Schrödinger equation. Bourgain and Demeter have laid the foundation for the decoupling method in the study of the "restriction theory", which provides a powerful approach for solving important conjectures in the fields of harmonic analysis, partial differential equations, number theory, incidence geometry and geometric measure theory. For example, Bourgain-Guth-Demeter solved the Vinogradov conjecture that has been sleeping for almost a century; up to the loss of $N^\epsilon$, Bourgain-Demeter solved the Strichartz estimates of the Schrödinger equation for both rational and irrational torus. Furthermore, a breakthrough was made on such issues as Riemann-Zeta function conjecture, discrete restriction conjecture, exponential summation estimate, the estimate of counting integer solutions of Diophantine inequality and unit distance conjecture. The core of decoupling method reveals the inherent vanishing structure of $L^p$ space, and in a sense, it is a compensation for lack of orthogonality to non-Hilbert spaces. Hope that through the implementation of the Tianyuan Advanced Mathematics Seminar Project, young mathematicians will grasp the relevant theories and methods as soon as possible and make breakthroughs in the fields of modern harmonic analysis, PDE, number theory and geometric measure theory to solve the open problems in relevant research fields or making substantive progress on these open issues, making an important contribution to the progress of Chinese mathematics.

分离性方法源于Wolff关于波动方程解的正则性及Bourgain关于薛定谔方程解的Strichartz估计.Bourgain等在研究限制性理论过程中奠定分离性方法的基础,为解决调和分析、偏微分方程、数论、关联几何与几何测度论、堆垒组合等领域中的重要猜想提供强有力方法.例如：Bourgain等解决近百年的Vinogradov猜想;解决有理与无理环上薛定谔方程解的Strichartz估计,进而在Riemann-Zeta函数猜想、离散限制性猜想、指数求和估计、丢番图不等式整数解估计、单位距离猜想等一系列问题上取得突破.分离性方法的核心揭示了空间内在消失结构,从某种意义上来讲,是非Hilbert空间缺少正交性的补偿.希望通过该天元数学高级研讨班项目的实施，让年轻数学家尽快掌握相关的理论与方法，在现代调和分析、PDE、数论、几何测度论等研究领域取得突破,为中国数学的进步做出重要贡献。