Research

Current Research

Solving Poisson equation with incomplete noisy information

Current research project advised by Dr. Andrea Bonito:

• Algorithmic Development: Engineered C++ code to implement complex minimization algorithms, utilizing the MOSEK C++ FUSION API for high-performance optimization.
• Numerical Experiments: Conducted numerical simulations to validate algorithm accuracy and performance, utilizing ParaView for data visualization, Docker for containerization and tmux for efficient CLI-based workflows.
• Software Rigor: Applied best practices in memory safety, debugging, and version control to ensure the reliability of mathematical software.
• Theory Translation: Translated mathematical frameworks into computationally efficient code, bridging the gap between theoretical research and practical application.

Past Research

Applying Combinatorial Nullstellensatz to no-3-in-a-line problem

Repeatedly applying the Combinatorial Nullstellensatz for Zero-sum Grids to Martin Gardner’s minimum no-3-in-a-line problem
Published Version: https://doi.org/10.1016/j.ejc.2024.104095;
ArXiv Link:
https://arxiv.org/abs/2401.03119;\

• Seunghwan Oh, John R. Schmitt, Xianzhi Wang

Large sets avoiding patterns

Large sets avoiding infinite arithmetic / geometric progressions, Real Analysis Exchange.
Published Version:
DOI: 10.14321/realanalexch.48.2.1668676378;
ArXiv Link: https://arxiv.org/abs/2210.09284;

• Alex Burgin, Samuel Goldberg, Tamás Keleti, Connor MacMahon, Xianzhi Wang
• We investigate some problems that’s related to the Erdős similarity conjecture.
• We construct a compact subset E of the real line such that zero is a Lebesgue density point of E, but E does not contain any (non-constant) infinite geometric progression.
• We give some small improvements on some known related results.
• We give a sufficient condition that guarantees that a given Cantor-type set contains at least one infinite geometric progression with any quotient between 0 and 1, and this condition could be used to prove that symmetric Cantor sets of
  positive measure contain infinite geometric progressions.

Planar Turán number

Extremal Planar Graphs with no Cycles of Perticular Lengths, under review.
ArXiv Link:
https://arxiv.org/abs/2208.13477;

• Ervin Győri, Xianzhi Wang, Zeyu Zheng
• We give a new proof of the planar Turán number of C5 and a nicer extremal construction.
• We applied “contribution method” to determine the maximal number of edges in a bipartite/triangle-free planar graph without small, even cycles.