R440, Astronomy-Mathematics Building, NTU

**Speaker(s):**

Zane Kun Li (University of Wisconsin–Madison)

**Organizer(s):**

Chun-Yen Shen (National Taiwan University)

**1. Outline**

Lectures 1-2:

In these two lectures, I will introduce what is decoupling and, in particular, concentrate on decoupling for the parabola. This was first proven by Bourgain and Demeter in 2014, but the proof we follow will be due to myself in 2018. I will mention how this proof of parabola decoupling was inspired by the number theoretic efficient congruencing methods. I will also compare this method with the original Bourgain and Demeter proof. Along the way, I will discuss (and make use of) the uncertainty principle and how to make it rigorous.

**Lecture 3:**

In this lecture, I plan to talk about Vinogradov's Mean Value Theorem and decoupling for the moment curve, first proved by Bourgain, Demeter, and Guth in 2015. Expanding upon ideas mentioned in Lecture 2, we will sketch the key ideas of the short proof of decoupling for the moment curve given by myself joint with Shaoming Guo, Po-Lam Yung, and Pavel Zorin-Kranich in 2019 that is based off Wooley's nested efficient congruencing proof.

**Lectures 4-5:**

In these final two lectures, I plan to discuss the proof of decoupling for the paraboloid in R^3 proved by Bourgain and Demeter in 2014 (though the proof I follow will be closer to the so called "study guide" proof). To set up the proof, I will first talk about the Bourgain-Guth reduction (often now more commonly called the broad-narrow argument) to trilinear decoupling. Assuming the multilinear Kakeya inequality, I will show how one proves decoupling for the paraboloid in R^3. I will end with a discussion of the short proof of multilinear Kakeya due to Larry Guth.

**2. References**

**Lectures 1-2:**

An l^2 decoupling interpretation of efficient congruencing: the parabola: https://arxiv.org/abs/1805.10551

Section 4 of The Vinogradov Mean Value Theorem [after Wooley and Bourgain, Demeter, and Guth]: https://arxiv.org/abs/1707.00119

Section 2 of these notes by Tao: https://terrytao.wordpress.com/2020/04/13/247b-notes-2-decoupling-theory/

**Lecture 3:**

A short proof of l^2 decoupling for the moment curve: https://arxiv.org/abs/1912.09798

Lemma 4.5 of A bilinear proof of decoupling for the cubic moment curve: https://arxiv.org/abs/1906.07989

**Lectures 4-5:**

n = 3 in A study guide for the l^2 decoupling theorem: https://arxiv.org/abs/1604.06032

A short proof of the multilinear Kakeya inequality: https://arxiv.org/abs/1409.4683

**5/25 Online Course: https://nationaltaiwanuniversity-ksz.my.webex.com/nationaltaiwanuniversity-ksz.my/j.phpMTID=m5030bd7d542f63167328d4015de6742b**

**3. Registration**

https://forms.gle/tawRAjpe3Y3A8PYQ7

**4. Course Video**

https://ncts.ntu.edu.tw/bib_detail.php?gid=192

**Contact:**
murphyyu@ncts.ntu.edu.tw