Authors: 

Sung-Yi Liao @ Department of Mathematics, Virginia Tech

Prof. Jose Madrid @ Department of Mathematics, Virginia Tech

Prof. Eyvindur Palsson @ Department of Mathematics, Virginia Tech

Dr. Julian Weigt @ Math Section, International Centre for Theoretical Physics, Italy

Title:

Discrete maximal operators on characteristic functions (in progress)

Authors: 

Sung-Yi Liao @ Department of Mathematics, National Taiwan University 

Prof. Thang Pham @ Department of Mathematics, Vietnam National University

Prof. Chun-Yen Shen @ Department of Mathematics, National Taiwan University

Title: On L^2 estimates for quadratic images of product Frostman measures

Abstract: 

Let $f\in\mathbb R[x,y,z]$ be a fixed non-degenerate quadratic polynomial. Given an $\alpha$-Frostman probability measure $\mu$ supported on $[0,1]$ with $\alpha\in(0,1)$, consider the pushforward measure $\nu=f_{\#}(\mu\times\mu\times\mu)$ on $\mathbb R$. We prove the following $L^2$ energy estimate: for a fixed nonnegative Schwartz function $\varphi$ with $\int\varphi=1$ and $\varphi_\delta(t)=\delta^{-1}\varphi(t/\delta)$, there exist $\epsilon>0$ and $\delta_{0}>0$ (depending only on $\alpha$ and the coefficients of $f$) such that
\[\int_{\mathbb R}(\varphi_\delta*\nu(t))^{2}\,dt \ \lesssim\ \delta^{\alpha+\epsilon-1}\text{ for all } \delta\in(0,\delta_{0}].\]
The proof expands the $L^2$ energy into a weighted six-fold coincidence integral and reduces the main contribution to a planar incidence problem after a controlled change of variables. The key new input is an incidence estimate, which is of independent interest, for point sets that arise as bi-Lipschitz images of a Cartesian product $M\times M$ of a $\delta$-separated and non-concentrated set $M$, yielding a power saving beyond what is available from separation and non-concentration alone. We also give examples showing that bounded support and Frostman-type hypotheses are necessary for such $L^2$ control.

Authors: 

Mr. Sung-Yi Liao @ Department of Mathematics, National Taiwan University 

Prof. Kwok-Wing Tsoi @ Department of Mathematics, National Taiwan University 

Title:

Waring's Problem with Reciprocal Exponents (available upon request)

Abstract:

Motivated by classical Waring's Problem and Descartes' Theorem, in this article, we characterize all rational solutions of the equation

\[{(x_1)}^{\frac{1}{m_1}}+\dotsb+{(x_h)}^{\frac{1}{m_h}}={(y_1)}^{\frac{1}{n_1}}+\dotsb+{(y_k)}^{\frac{1}{n_k}}\]

where $m_1,\dotsc,m_h,n_1,\dotsc,n_k$ and $h,k$ are positive integers. Our result also gives a complete rational parametrization to the equation when all the exponents are equal. As an application, we address a modified Waring's problem in which reciprocals of varying positive integers replace the exponentials.

Advisors: 

Mr. Cheng-Hua Tsai @ Taichung First Senior High School (Feb. 2017 - June 2018)

Prof. Ming-Hsuan Kang @ National Chiao-Tung University (July 2018 - June 2019)

Title:

The Extension of Fermat Polygonal Number Theorem

Awards: 

2019 Shing-Tung Yau High School Mathematics Award (Gold medal)

2019 alternative member in the national representative team for the international science fairs