Academy of Mathematics and Systems Science, CAS Colloquia & Seminars
Speaker:
Zhang Ruixiang,University of Wisconsin-Madison
Inviter:
Title:
The polynomial partitioning approach to the Fourier restriction problem
Time & Venue:
2019.7.2/3 9:30-10:30 7.3 14:00-15:00 N202
Abstract:
The Fourier extension operator has been a central object to study in harmonic analysis. Stein conjectured that it is a bounded linear operator between certain $L^p$ spaces. Recently people have found that auxiliary real polynomials can help one study Stein's above Restriction Conjecture. I will talk about this approach known as "polynomial partitioning" and mention some recent progress on Stein's Restriction Conjecture.In the first lecture (the colloquium), I will state Stein's Fourier Restriction (Extension) Conjecture and talk on why and how auxiliary (real) polynomials might help. In the second lecture, I will explain the framework set up by Guth concerning this approach. In the third lecture, I will talk about three interesting facts about the zero sets of real polynomials (Wongkew's theorem, the Polynomial Wolff Axiom, and a lemma of Guth), and how they can help one get better results in the Fourier Restriction Conjecture. These are relevant to some work in progress joint with Jonathan Hickman, Keith Rogers and Hong Wang.