James Leng

I'm James Leng, a math graduate student at UCLA. Before coming to UCLA, I was an undergraduate student at UC Berkeley from 2017-2020. 

Contact:

email: jamesleng + "at" + math + "dot" + ucla + "dot" + edu

Office: MS 5351-1.  You can often find me in MS 6620.

(replace "at", "dot" with appropriate symbols).

Teaching:

I have been a teaching assistant for the following classes:

• Summer 2023: Math 31A: Differential and Integral Calculus.

• Summer 2021: Math 170S, Introduction to Probability and Statistics

• Spring 2021:  Math 131A, Real Analysis

• Winter 2021, Math 32B, Calculus of Several Variables

• Fall 2020, Math 32B, Calculus of Several Variables

Research:

I'm working with Terence Tao.  I'm interested in arithmetic combinatorics, dynamical systems, and Fourier analysis, focusing mostly on the field Higher Order Fourier Analysis. My research is supported by an NSF Graduate Research Fellowship Grant No. DGE-2034835. My papers are:

• J. Leng, A. Sah, and M. Sawhney. Improved Bounds for Szemerédi's Theorem.

• J. Leng, A. Sah, and M. Sawhney. Quasipolynomial bounds on the inverse theorem for the Gowers U^{s+1}[N]-norm.

• J. Leng, A. Sah, and M. Sawhney. Improved bounds for five-term arithmetic progressions. 

• J. Leng. Efficient equidistribution of nilsequences

  • Here are slides.

• J. Leng. Efficient equidistribution of periodic nilsequences and applications

• A. Backus, J. Leng, and Z. Tao. The fractal uncertainty principle via Dolgopyat's method in higher dimensions. arxiv.org/abs/2302.11708

  • Aidan Backus has written nice notes on our paper here. 

• J. Leng. Improved Quadratic Gowers Uniformity for the Möbius Function. arxiv.org/abs/2212.09635 

  •  Here are slides to this paper.

• J. Leng. A Quantitative Bound for Szemeredi's Theorem for a Complexity One Polynomial Progressions over Z/NZ. https://arxiv.org/abs/2205.05540. Discrete Analysis, accepted (pending revisions). 

Some older papers are:

• B. Haddock, J. Leng, and C. E. Silva. Nonsingular Transformations that are Ergodic with Isometric Coefficients are not Weakly Doubly Ergodic. https://arxiv.org/abs/2011.14278. Indagationes Mathematicae. 8/23/2022.

• J. Leng, C. E. Silva, and Y. Wu. On Examples of Rank-Two Symbolic Shifts. https://arxiv.org/abs/2010.05165

• J. Leng. Moser's Method and Conservative Extensions of Diffeomorphisms. https://arxiv.org/abs/2006.14542

• H. Drillick, A. Espinoza-Dominguez, J. N. Jones-Baro, J. Leng, Y. Mandelshtam, and C. E. Silva. Non-rigid Rank-One Infinite Measures on the Circle. https://arxiv.org/abs/1810.11095

Talks:

• JMM 2019

• UCLA Participating Analysis Seminar: Spring 2021, Fall 2021, Winter 2022, Spring 2022, Winter 2023, Spring 2023

• Caltech Discrete Analysis Seminar Spring 2023

• AMS Southeastern Sectional on Discrete Analysis Spring 2023.

• UCLA Analysis and PDE Seminar Fall 2023

• ETA Seminar Fall 2023

• JMM 2023 AIM Special Session on Multiplicative Number Theory and Additive Combinatorics

I have occasionally been mistaken (by the UC Berkeley architecture department, in fact) as James Leng, a lecturer of architecture at Berkeley.