偏微分方程与数论中的decoupling定理

Fields Prize winner Bourgain developed the number theory method(Weyl-sum approach, Littlewood-Paley circle argument, minor/major arc decomposition )to study Strichartzestimates of the periodic solution of free dispersion equation，and built a bridge between partial differential equations and number theory research.Bourgain reduced many key problems in number theory and PDEs into l^2-decoupling estimates (discrete cases are equivalent to the sum of Gauss exponents in different scales). Wolff gives the l^p-decoupling(cone surface) which was regarded as start-point to solve the local smoothness conjecture on the wave equation; the l^2-decoupling estimates established by Bourgain-Demeter (parabolic or sphere ) solves almost Sharp- Strichartz estimates of solution for rational and irrationalSchro ̈dingerequations.Bourgian opens up the natural connection between number theory and PDE, and The decoupling method has achieved great success in the study of number theory. The Tianyuan advanced seminar will invite a few of mathematicanssuch as L.Guth, C.Sogge to give a series of lectures or min-course on decoupling theorems appearing in both number theory and PDEs , and to establish a powerful academic exchanges platform for young scholars at home and abroad. We try to do some breakthroughs in the study of modern harmonic analysis, PDE, number theory etc.

菲尔兹奖得主Bourgain发展了周期色散方程Strichartz估计的数论方法(Weyl求和法，Littlewood-Paley圆法、优弧/劣弧分解技术等),搭建了PDE与数论研究的桥梁.Bourgain将数论、PDE中许多关键问题归结于slab型求和的l^2-分离性估计.Wolff通过建立锥面对应的l^p-decoupling定理，开启波动方程解的局部光滑性猜想的研究之门;Bourgain等通过建立抛物面对应的l^2-decoupling定理,解决了Schro ̈dinger方程周期解的Strichartz估计.Bourgian开辟了数论与PDE之间的联系, decoupling方法在数论研究中获得巨大成功.该天元高级研讨班拟邀请Guth、Sogge等讲解PDE与数论中的decoupling定理，为海内外青年学者搭建学术交流与合作的平台，在现代调和分析、PDE、数论等研究领域取得突破.

：菲尔兹奖得主Bourgain等通过建立抛物面上的l^2-decoupling定理,解决了Schro ̈dinger方程周期解的Strichartz估计.通过建立不同光滑曲面（曲线）上的decoupling不等式，解决了数论中若干重要的问题. 从而搭建了数论与偏微分方程研究的桥梁.该天元高级研讨班邀请Sogge、Duyckaerts等讲解PDE与数论中的decoupling定理.另一方面， 通过腾讯会议举方式举办一次国际研讨会，邀请了海内外从事该领域的8位著名青年才俊 (Y. Deng, S. Guo, Z. Guo, L.Jin, B. Liu, Y. Xi, H. Wang, R. Zhang)分别做了精彩的学术报告并进行深入的探讨，内容涉及分离性方法、局部光滑猜想、多项式分解技术、平方函数估计、波包分解等，线上先后参与的年轻学者、研究生超过200人。美国霍普金斯大学的 Sogge教授认为本次研讨会的水平非常高，特别是在疫情期间能够通过线上会议形式进行学术交流尤为可贵，为海内外青年学者搭建学术交流与合作的平台。