July 30,-, 1309 application of Polynomial Method in Harmonic Analysis

Question: Application of Polynomial methods in Harmonic Analysis
Speaker: Zhang Ruixiang
Time: January 1, July 30, 2014
Location: Room 1309, building 1, science

Recently, in terms of the problems and limitations of kakeya in harmonic analysis, the so-called "polynomial method" has developed very important and produced a series of important results. We will describe these results and try to explain the role and power of auxiliary polynomials introduced in these problems.

Lecture 1 (-) discrete model of the kakeya type Problem

Abstract: The problem of kakeya has a long history and is very difficult. Bourgain, Wolff, and others studied it using some geometric methods and contributed to the proposal of a series of related discrete geometric problems. Since Dvir proved the kakeya conjecture in the finite field, he has been paying attention to the new method called the polynomial method. This article describes how to simulate the discretization of the kakeya conjecture and the proof of the polynomial method, including the proof of the Dvir on the kakeya conjecture in the finite field, guth-Katz's proof of a conjecture on joint and bourgain, as well as some recent work on the discrete models of the kakeya conjecture and furstberg problems in r ^ n.

Lecture 2 (-) kakeya: from discrete to continuous

Abstract: The help of the discrete model on the kakeya problem is limited. When we change points into small balls, we will lose a lot of algebra due to the loss of the exact dot-line relationship. However, we can still use polynomial methods and some analytical and geometric techniques to draw some interesting and useful conclusions. This article discusses Guth's proof of the Multi-linear kakeya conjecture. Guth proves the local "flatness" and "texture" of the three-dimensional kakeya set ". If there is time, we will also introduce some recent ideas about using multiple cross-cut Polynomials to deal with more complex multi-linear problems.

Lecture 3 (-) polynomial methods and restricted Conjecture

Abstract: so far, the polynomial method has not provided better indicators on the kakeya conjecture. However, some applications, such as the multi-linear kakeya conjecture, have introduced the progress (bourgain-Guth) in the more important restrictive Conjecture of analysis ). Recently, Guth has made progress in using polynomial methods in 3D for the first time in restricted conjecture. This article will focus on the results and the functions of Polynomial methods.

Speaker profile:
Zhang Ruixiang graduated from the School of Mathematics, Peking University in 2012 and is now a Ph.D. student at Princeton University. He is mainly engaged in research in analytical number theory, harmonic analysis, and other related fields. He has made some interesting results in his research on the very difficult problem of kakeya in harmonic analysis.‍