Research Experience

Cooperation with Prof. T. Pham & Prof. C-Y. Shen      (2022.09-2023.10)

Authors: 

Mr. 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 the dimension of polynomial images in three variables (available upon request)

Vietnam Polymath REU       (2022.09-2023.10)

Advisors: 

Prof. Chun-Yen Shen @ National Taiwan University

Prof. Thang Pham @ Vietnam National University

Teammate: 

Mr. Dung Ha Minh @ Hanoi University of Science and Technology

Title:  

Exploration on Incidence Geometry and Sum-Product Phenomena (available at https://arxiv.org/abs/2310.07964)

Abstract:

In this dissertation, we investigate the Erdos-Szemeredi Conjecture and its relationship with several well-known results in incidence geometry, such as the Szemeredi-Trotter Incidence Theorem. We first study these problems in the setting of real numbers and focus on the proofs by Elekes and Solymosi on sum-product estimates. After introducing these theorems, our main focus is the Erdos-Szemeredi Conjecture in the setting of $\mathbb{F}_p$. We aim to adapt several ingenious techniques developed for real numbers to the case of finite fields. Finally, we obtain a result in estimating the number of bisectors over the ring $\mathbb{Z}/p^3\mathbb{Z}$ with $p$ a $4n+3$ prime.

Award:

Global Undergraduate Awards: Highly Recommended (Top 10% among all applicants)

Short-Term Cooperation with Prof. K-W. Tsai                (2022.08)

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.

Post-Quantum Cryptography Internship Program     (2021.07-2021.09)

Advisor: 

Prof. Bo-Yin Yang @ Institution of Information Science, Academia Sinica (July 2021 - Sept. 2021)  

Topic:

Post-Quantum Cryptography System Implementation: NTRU (stand for "Number Theory Research Unit") Prime

Mathematical Reseach in High School
      (2017.02-2019.06)

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 Taiwan representative team for international science fair