Author Archives: zirui

A stationary set method for estimating oscillatory integrals

The HADES seminar on Tuesday, October 5th, will be given by Ruixiang Zhang at 5 pm in 740 Evans.

Speaker: Ruixiang Zhang (UC Berkeley)

Abstract: 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.

The Benjamin-Ono approximation for low frequency gravity water waves with constant vorticity

The HADES seminar on Tuesday, September 21st, will be given by James Rowan from at 5 pm in 740 Evans.

Speaker: James Rowan (University of California, Berkeley)

Abstract: It is well-known that the cubic nonlinear schrodinger equation gives a good approximation for frequency-localized solutions to the irrotational 2D gravity water waves equations, at least on a cubic timescale.  Replacing the assumption of irrotationality with one of constant vorticity allows the model to apply to waves in settings with countercurrents, but the new terms introduced by the vorticity break the scaling symmetry, and in the low-frequency regime, they should have a large effect.  We show that, for low-frequency solutions, the Benjamin-Ono equation gives a good approximation to the 2D gravity water waves equations with constant vorticity.  This work is joint with Mihaela Ifrim, Daniel Tataru, and Lizhe Wan.  Along the way to this result, I will give a brief introduction to some topics in nonlinear dispersive PDE and fluid dynamics.

A Nonnegative Version of Whitney’s Extension Problem

The HADES seminar on Tuesday, November 17th will be given by Kevin O’Neill via Zoom from 3:40 to 5 pm.

Speaker: Kevin O’Neill (UC Davis)

Abstract: Whitney’s Extension Problem asks the following: Given a compact set $E\subset\mathbb{R}^n$ and a function $E\to\mathbb{R}$, how can we tell if there exists $F\in C^m(\mathbb{R}^n)$ such that $F|_E=f$? The classical Whitney Extension theorem tells us that, given potential Taylor polynomials $P^x$ at each $x\in E$, there is such an extension $F$ if and only if the $P^x$’s are compatible under Taylor’s theorem. However, this leaves open the question of how to tell solely from $f$. A 2006 paper of Charles Fefferman answers this question. We explain some of the concepts of that paper, as well as recent work of the speaker, joint with Fushuai Jiang and Garving K. Luli, which establishes the analogous result when $f\geq 0$ and we require $F\geq 0$.