数学学院基础数学学术报告

Analysis Seminar 2021-05-13

报告题目: A stationary set method for estimating oscillatory integrals

报告人：Ruixiang Zhang (IAS, Princeton)

时间： 2021-05-13 10:00-11:00

地点： Tencent meeting ID：819 207 732

摘要： Given a polynomial $P$ of constant degree in $d$ variables and consider the oscillatory integral $$I_P = \int_{[0,1]^d} e(P(\xi)) \mathrm{d}\xi.$$ Assuming $d$ is also fixed, what is a good upper bound of $|I_P|$? In this talk, I will introduce a ``stationary set'' method that gives an upper bound with simple geometric meaning. The proof of this bound mainly relies on the theory of o-minimal structures. As an application of our bound, we obtain the sharp convergence exponent in the two dimensional Tarry's problem for every degree via additional analysis on stationary sets. Consequently, we also prove the sharp $L^{\infty} \to L^p$ Fourier extension estimates for every two dimensional Parsell-Vinogradov surface whenever the endpoint of the exponent $p$ is even. This is joint work with Saugata Basu, Shaoming Guo and Pavel Zorin-Kranich.

报告人简介：Prof. Ruixiang Zhang is currently a member of IAS, Princeton. He obtained Ph.D. from Princeton University under the advisement of Peter Sarnak. After that, he became a Van Vleck Visiting Assistant Professor at University of Wisconsin-Madison, after one-year member at IAS. Prof. Ruixiang Zhang is a leading young mathematician, working in harmonic analysis, where he already made important contributions. He published more than 20 papers in leading math journals, including Ann. Math and Invent. Math.. He is also an IMO gold medalist and silver medalist of New World Mathematics Awards.

邀请人：张安 欢迎大家参加！