Notes
Miscellaneous
Miscellaneous
Chaoli forum (超理论坛) is a Chinese forum for mathematics and natural science. I have some research notes written in Chinese posted there:
During 2021 spring, I stayed at Yau Mathematics Science Center (YMSC) of Tsinghua University as a visiting scholar, sponsored by Professor Pin Yu (于品). I gave a total of nine lectures on modern nonlinear analytic techniques with applications to the study of nonlinear dispersive differential equations, and the first eight of them were based on lecture notes composed by myself. A temporary collected version can be found here:
It is still subjected to modification.
Lecture 1: Littlewood-Paley Theory
Brief description of Littlewood-Paley decomposition of functions; how regularity properties are related to convergence properties of the Littlewood-Paley building blocks; primary exposition of para-product of J. Bony.
Lecture 2: Para-differential Calculus
The forbidden (1,1) symbol class and corresponding pseudo-differential operators; para-differential operators; symbolic calculus of para-differential operators.
Lecture 3: Some Applications
Brief description of J. Bony's microlocal regularity theorem; classical quasilinear hyperbolic differential equation via para-differential calculus.
Lecture 4: The Water Waves System
The Dirichlet-Neumann operator in Euclidean spaces; local well-posedness of the gravity-capillary water waves system via para-differential calculus.
Lecture 5: Frameworks for Nash-Moser
"Second order property" of Newton iteration; failure of the classical implicit function theorem; "loss of regularity"; the category of tame spaces in the sense of R. Hamilton.
Lecture 6: The Nash-Moser Scheme
"Simple" Nash-Moser type theorems in graded Fréchet space with smoothing operators; the Nash embedding theorem, and M. Günthier's alternative proof; regularity loss caused by "small divisor" and KAM structural stability theorem.
Lecture 7: How to Avoid Nash-Moser Method?
Hörmander's low regularity Nash-Moser type theorem; avoidance of Nash-Moser scheme via para-differential calculus; weakly parabolic linear system in the sense of R. Hamilton; avoidance of Nash-Moser scheme for geometric flows via the Deturck technique.
Lecture 8: Normal Form Method
What is a normal form; Shatah's normal form reduction for quasilinear Klein-Gordon equations, and global well-posedness.
Lecture 9: Normal Form Reduction in Compact Settings
A brief exposition of Delort, J. M., & Szeftel, J. (2004). Long-time existence for small data nonlinear Klein-Gordon equations on tori and spheres. International Mathematics Research Notices, 2004(37), 1897-1966.
I was a mentor for the reading track of MIT PRIMES of the year 2019 and 2020. I tried to teach analytic number theory to high school students who have little preliminary knowledge in calculus, and I composed about half of the ingredients of this teaching experiment into several notes. Although my teaching was completely in English, I wrote the notes in Chinese and posted them on Chaoli forum because I hope that these materials become accessible for Chinese high school students as well.
Hadamard's Factorization Theorem
The notion of entire functions; the Weierstrass factorization theorem; order of an entire function, distribution of zeros; the factorization theorem.
Prime Number Theorem with Remainder
Perron's formula without remainder; Riemann's zeta function and its integral representation; non-zero regions and remainders in the prime number theorem; integral representation from the idèle group point of view. Contains an outline of teaching within a year, for high school students with little preliminary knowledge in calculus.
The Letter from Iwasawa to Dieudonné, with Further Expositions
I used the letter from Iwasawa to Dieudonné as supplementary material. I translated it into Chinese. The expositions were based on my (hand-written) lecture notes.
These notes were composed for my preparation of qualification exam. All theorems mentioned were proved in detail. Although my qualification exam was in English, I still wrote these notes in Chinese, since materials on these topics were not of great abundance in Chinese-speaking world.
Notes on Singular Integral Operators
Notes on Littlewood-Paley Theory
Notes on Local Well-posedness of Nonlinear Schrödinger Equation