Undergraduate Research
Undergraduate research mentorship, projects, and publications.
Undergraduate research mentorship is an important part of my academic work. I am committed to providing students with opportunities to explore mathematical problems, develop independent research skills, and communicate their results.
At the University of Michigan, I have mentored undergraduate students through the Laboratory of Geometry at Michigan (LoG(M)), Research Experiences for Undergraduates (REU), and independent studies. Several of these projects have resulted in research manuscripts and publications.
Laboratory of Geometry at Michigan (LoG(M))
From 2024 to 2026, I served as director of the Laboratory of Geometry at Michigan (LoG(M)) at the University of Michigan, an undergraduate research program that introduces students to mathematical research through collaborative projects.
As director, I coordinated research activities, recruited faculty and graduate student mentors, participated in the selection of undergraduate researchers, and organized research presentations and outreach activities.
In addition to my administrative responsibilities, I taught Math 440 (Geometry Lab), a course focused on developing students’ mathematical communication skills. Through this course, I guided students in presenting their research findings through written reports, oral presentations, and posters, helping them communicate mathematical ideas clearly and effectively.
Selected Research Projects
Why Does the Kid Swing Alone in the Park?
A project investigating parametric resonance and the mathematical mechanisms underlying the motion of a playground swing.
Students: Pranavi Pratapa, Wenjie Zhou, Qingyao Zhu, Ziqian Zhu.
Euler, Lagrange, and Kovalevskaya Tops
A project exploring classical integrable systems in rigid-body dynamics, with particular emphasis on the Euler, Lagrange, and Kovalevskaya cases.
Students: Ramon Diego, Haley Gipson, Huaidian Hou, Anna Huang.
Research Experiences for Undergraduates (REU)
I served as a research mentor in the Research Experiences for Undergraduates (REU) program at the University of Michigan during the summers of 2024 and 2025.
Through these projects, I guided undergraduate students in investigating open problems in sub-Riemannian geometry and geometric control theory. Both projects resulted in research manuscripts.
Summer 2025: Metric Lines in the Space of Curves
Students: Daniela Catalá and Miriam Vollmayr-Lee.
This project investigated the classification of globally minimizing geodesics in the space of curves, using the sequence method to establish their global minimality.
The project resulted in the following preprint:
Metric Lines in the Space of Curves
Summer 2024: Metric Lines in the Special Euclidean Group
Students: Yu Wang and Sean Ku.
This project investigated globally minimizing geodesics in the special Euclidean group of the plane, SE(2), a fundamental example in sub-Riemannian geometry and geometric control theory.
The resulting paper was published in Involve in 2026:
Metric Lines in the Special Euclidean Group on the Plane
Independent Studies
I have supervised independent study projects that allow undergraduate students to explore advanced mathematical topics and develop their own research questions.
These projects emphasize mathematical reasoning, independent problem-solving, and scientific communication. Several have resulted in research manuscripts.
Marcus T. C. Lau — Optimal Synthesis on the Special Linear Group
This project investigated the Riemannian geometry of the special linear group $\mathbb{SL}_2(\mathbb{R})$, focusing on the determination of cut times, the cut locus, and the injectivity radius.
The project resulted in a research manuscript submitted to Differential Geometry and its Applications.
Research manuscript::
Optimal Synthesis on the Special Linear Group
Yoshihiro Paris — Irrationality of Infinite Products
This independent study originated from Yoshihiro’s interest in irrational numbers and number theory. During the Winter 2025 semester, he approached me with ideas concerning the irrationality of infinite products. His enthusiasm and persistence inspired me to supervise this project, which provided an opportunity to support undergraduate research beyond my primary area of expertise.
The project investigated the irrationality of infinite products of the form
\[\prod_{n=1}^{\infty}\left(1+\frac{r}{A^n}\right),\]where $A\in\mathbb{Z}$ and $r\in\mathbb{Q}$ satisfy certain technical conditions.
Yoshihiro developed an elementary proof of irrationality for this family of products. Although more general results were previously established by Bundschuh in 1969, his approach provides an accessible alternative using techniques suitable for undergraduate students.
My role was to help Yoshihiro formalize his arguments, strengthen the mathematical rigor of his proof, and develop the manuscript into a research paper.
Read the manuscript (PDF) The Irrationality of $\;0.8\cdot 0.98\cdot 0.998\cdot 0.9998\cdots$, submitted to the Rose-Hulman Undergraduate Mathematics Journal.
The Integrability of a Knife-Edge Billiard in a Disk
Huaidian Hou developed the numerical simulation code, which played a fundamental role in the development of our research. The simulation code is publicly available on GitHub and accompanies our research manuscript
Numerical Simulation