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)
(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