UW-Milwaukee Department of Mathematical Sciences presents,
Dr. Xianghong Chen;
Visiting Assistant Professor
Friday, April 15, 2016
3:30pm in EMS E495
Please note the change in time from 2pm to 3:30pm due to a scheduling conflict with a second colloquium. This change was made on Monday, April 11.
*Refreshments will be served at 1:30pm in EMS E424A*
Fourier analysis of fractals: decay and restriction
Fractals arise in many different contexts in mathematics, physics, and biology. According to Mandelbrot, the coiner of the word “fractal”, they are “beautiful, damn hard, increasingly useful.” (An example of such a fractal is illustrated here.)
A fractal often carries a natural measure quantifying its distribution in the ambien space. One can then study the behavior of its Fourier transform as the frequency tends to infinity. Such behavior interacts with various geometric/arithmetic aspects of the fractal such as dimensionality, self-similarity and pseudorandomness.
In this talk I will focus on pointwise Fourier decay estimates for certain random-like Cantor sets, as well as average Fourier decay estimates for functions supported on them. The latter is the dual formulation of the Fourier restriction problem, which has been extensively studied in the context of surfaces, but is relatively new for fractals.
I will give some background on this subject, survey some recent results, and discuss some open problems.