Mingfeng Chen's(陈铭峰) Homepage

About me

Hi, I am Mingfeng Chen(陈铭峰), a second year Phd student at University of Wisconsin-Madison, department of mathematics. My research interests lie in Harmonic analysis,  Geometric measure theory and Combinatorics. My phd advisor is Shaoming Guo. Before that, I was an undergraduate at Nanjing University.


Contact Information

Email: mchen454@wisc.edu or mingfengchennju@gmail.com


Research

1.A multi-parameter cinematic curvature

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

We state a multi-parameter cinematic curvature condition and prove L^p bounds for related maximal operators. This is a generalization of Sogge’s classical result and has applications in geometric measure theory. In particular, we can prove L3 bound for ellipse maximal operator, which solves a conjecture of Erdogan. Combine with Joshua Zahl’s recent work, we can prove sharp L^d-bound for a family of (d-1)-parameter planar curves. The proof relies on decoupling inequality and local smoothing type estimate. This is the joint work with Shaoming Guo and Tongou Yang.

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

with Shaoming Guo, submitted. (arXiv).

We classify the planar maximal average over curves, that is: we find all the curves such that Nikodym sets exist, thus the corresponding maximal operator is not bounded on L^p for any p<∞; for other curves, we prove sharp L^p bound for the maximal operator.


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

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


We consider a class of Hörmander-type oscillatory integral operators in R^n for n odd with real analytic phase. We derive weak conditions on the phase which ensure L^p bounds beyond the universal p≥2(n+1)/(n-1) range guaranteed by Stein's oscillatory integral theorem. This expands and elucidates pioneering work of Bourgain from the early 1990s. We also consider a closely related class of variable coefficient Schrödinger propagator-type operators, and show that the corresponding theory differs significantly from that of the Hörmander-type operators. The main ingredient in the proof is a curved Kakeya/Nikodym maximal function estimate. This is established by combining the polynomial method with certain uniform sublevel set estimates for real analytic functions. The sublevel set estimates are the main novelty in the argument and can be interpreted as a form of quantification of linear independence in the real analytic category.


Slides and poster

Travel and talks