Research

Some key words of my research: Fourier restriction estimates, decoupling, local smoothing estimates, maximal operator along curves, cinematic curvature, resolution of singularity, Kakeya, Polynomial method, Hörmander problem, Polynomial Wolff Axioms, kakeya compression, oscillatory integrals, counting rational points near manifold, diophantine approximation.

6. Sharp bounds for rational points near space curves

with Andreas Seeger, Rajula Srivastava and Niclas Technau.  (arXiv)

5. Restriction and Kakeya maximal estimates in R^4

with Tainara Borges, Tiklung Chan, Diankun Liu, Yakun Xi and Yufei Zhan, submitted. (arXiv).

4. Rational points near planar flat curves

submitted. (arXiv)

3. Oscillatory integral operators and variable Schrödinger propagators: beyond the universal estimates

with Shengwen Gan, Shaoming Guo, Jonathan Hickman, Marina Iliopoulou and James Wright, GAFA(2025). (journal, arXiv).  

2. The dichotomy of Nikodym sets and local smoothing estimates for wave equations

with Shaoming Guo, submitted. (arXiv).

1. A multi-parameter cinematic curvature

with Shaoming Guo and Tongou Yang, submitted. (arXiv).

0. Ph.D. Thesis: Planar maximal operators

This is a combination of  1 and 2. (pdf)

Collaborators

 Tainara Borges, Tiklung Chan, Shengwen Gan, Shaoming Guo, Jonathan Hickman, Marina Iliopoulou, Diankun Liu, Minxing Shen, Andreas Seeger, Rajula Srivastava, Niclas Technau, James Wright, Yakun Xi, Tongou Yang, Xuerui Yang, Joshua Zahl, Yufei Zhan.

Slides and poster

• Rational points near planar flat curves (slides)

• Improved bounds for curved restriction and kakeya problem and its application (slides)

• Restriction estimates using decouping theorems and two-ends Furstenberg inequalities, after Wang and Wu (slides)

• Planar maximal operator (slides)

• The dichotomy of Nikodym sets and local smoothing estimates for wave equations (poster) 

Recent travel and talks

• 2026/08 Harmonic Analysis conference, Nanjing university, Nanjing.

• 2026/07 Chern Institute of mathematics, Nankai university, Tianjin.

• 2026/07 Analysis seminar, Capital normal university.

• 2026/06 Analysis seminar, Sun Yat-sen University, Guangzhou.

• 2026/06 Chern Institute of mathematics, Nankai university, Tianjin.

• 2025/12 Analysis and PDE seminar, UC Berkeley.

• 2025/11 MIT.

• 2025/10 Analysis and PDE seminar, UCLA.

• 2025/10 AMS Sectional Meeting, “Harmonic Analysis, Geometric Measure Theory and Fractals”, Saint Louis, Missouri. 

• 2025/08 Kopp Summer School: Discrete Directions in Harmonic Analysis, Kopp, Germany.

• 2025/08 Reading and Research Groups in Analysis 2025, University of Pennsylvania, Philadelphia, PA.