【1月9日-1月13日】 金龙 微分几何短期授课班

发布者:卢珊珊发布时间:2021-01-05浏览次数:969

课程题目:量子混沌与分形不确定性原理


授课人:金龙,清华大学


授课时间:1月9号, 14:00-16:00 (腾讯会议号:202801247;会议密码:662607)
                1月11号, 16:00-18:00(腾讯会议号:930556440;会议密码:662607)
                1月13号,19:00-21:00(腾讯会议号:760787847;会议密码:662607)
               
课程概述:
在这个短课程中,我们简要讨论量子混沌领域中的一些最新进展。特别地,我 们引入一个新的工具称为分形不确定性原理(Fractal Uncertainty Principle,简 记为 FUP)。课程计划如下:
(1) 引论:量子混沌中的一些问题;
(2) 简单模型:开放量子映射与离散 FUP;
(3) 预备知识:双曲曲面的几何与动力系统性质;
(4) 双曲曲面上的半经典微局部分析;
(5) 紧致双曲曲面上的控制问题;
(6) 分形不确定性原理和未解问题。
Dyatlov–Zahl 在 2016 年的文章最早提出了 FUP 这一工具来研究凸余紧双曲曲 面的谱隙;Bourgain–Dyatlov 在一维情形给出了完整的证明,并证明了谱隙的存 在性;Dyatlov–Jin 以及 Dyatlov–Jin–Nonnenmacher 将其应用到紧致双曲曲面 的半径典测度的研究中。我们希望能够在这个短课程中讲述 Bourgain–Dyatlov 以及 Dyatlov–Jin 对于双曲曲面上 Laplace 特征函数的半经典测度的全支集性 质的证明的主要思想。下面是关于 FUP 的一些参考文献。


参考文献:
1. 概述性文章
• SemyonDyatlov,Controlofeigenfunctionsonhyperbolicsurfaces: an application of fractal uncertainty principle, Journées équations aux dérivées partielles, 2017. • Semyon Dyatlov, An introduction to fractal uncertainty principle, J. Math. Phys. 60 (2019), 081505. • Steve Zelditch, Mathematics of Quantum Chaos in 2019, Not. Am. Math. Soc. 66(2019). • LongJin,Quantumchaosandfractaluncertaintyprinciple,toappear in Proceedings of ICCM 2018.
2. 量子混沌的一些新进展
• Semyon Dyatlov and Joshua Zahl, Spectral gaps, additive energy, and a fractal uncertainty principle, Geom. Funct. Anal. 26 (2016), 1011–1094. • Semyon Dyatlov and Long Jin, Resonances for open quantum maps anda fractaluncertainty principle,Comm. Math. Phys. 354(2017), 269–316. • Semyon Dyatlov and Long Jin, Semiclassical measures on hyperbolic surfaces have full support, Acta Math. 220 (2018), 297–339. • Semyon Dyatlov, Improved fractal Weyl bounds for hyperbolic manifolds, with an appendix with David Borthwick and Tobias Weich, J. Eur. Math. Soc. 21 (2019), 1595–1639. • SemyonDyatlovandMaciejZworski, Fractal uncertainty for transfer operators, Int. Math. Res. Not. (2020), 781–812. • Semyon Dyatlov, Long Jin and Stéphane Nonnenmacher, Control of eigenfunctionsonsurfacesofvariablecurvature,preprint,arXiv:1906.08923.
3. 分形不确定性原理
• Jean Bourgain and Semyon Dyatlov, Fourier dimension and spectral gaps for hyperbolic surfaces, Geom. Funct. Anal. 27 (2017), 744– 771.
• Jean Bourgain and Semyon Dyatlov, Spectral gaps without the pressure condition, Ann. of Math. (2) 187 (2018), 825–867. • Semyon Dyatlov and Long Jin, Dolgopyat’s method and the fractal uncertainty principle, Anal. PDE, 11 (2018), 1457–1485. • Long Jin and Ruixiang Zhang, Fractal uncertainty principle with explicit exponent, Math. Ann. 376 (2020), 1031-1057. • RuiHanandWilhelmSchlag,AhigherdimensionalBourgain-Dyatlov fractal uncertainty principle, Anal. PDE, 13 (2020), 813–863. • Laura Cladek and Terence Tao, Additive energy of regular measures in one and higher dimensions, and the fractal uncertainty principle, preprint, arXiv:2012.02747
4. 在偏微分方程中的应用
• Long Jin, Control for Schrödinger equation on hyperbolic surfaces, Math. Res. Lett. 25 (2018), 1865–1877. • Jian Wang, Strichartz esitmates for convex co-compact hyperbolic surfaces, Proc. Amer. Math. Soc. 147 (2019), 873–883. • Long Jin, Damped wave equations on compact hyperbolic surfaces, Comm. Math. Phys. 373 No. 3 (2020), 771-794. • JeffereyGalkowskiandSteveZelditch,LowerboundsforCauchydata on curves in a negatively curved surface, preprint, arXiv:2002.09456.