Research
My main interest is connecting geometry and number theory with ergodic theory. I am working on problems connecting the dynamics of certain classes of continued fractions to flows on modular surfaces. My thesis advisor was Florin Boca.
I am currently mentoring three undergraduate students on a Mason Experimental Geometry Lab (MEGL) project on the Geometry of Nearest Integer Continued Fractions. This project connects modular surfaces, symbolic dynamics, and dynamical systems to study nearest integer continued fractions. The project started in Fall 2025 and is expected to continue through Fall 2026.
I have also been studying the effect changing the dynamical system that generates the odd-denominator continued fractions has on the various analytic properties. Specifically, I worked with Yusuf Hartono, Cor Kraaikamp, and Niels Langeveld on the relationship between the continued fraction domain and entropy of the dynamical system.
Papers
Continued fractions connecting number theory, dynamical systems, and hyperbolic geometry. Expository. Chapter 7 in Research Connections Career and Research Journeys from the SMP Community, 2025. Each self-contained chapter contains a short mathematical biography of the authrot and an undergraduate friendly introduction to one of the author's research.
Geodesic flows and slow downs of continued fraction maps.
(With Yusuf Hartono, Cor Kraaikamp, and Niels Langeveld) Natural extensions and entropy of α-continued fraction expansions with odd partial quotients. Discrete and Continuous Dynamical Systems, August 2023. [ arXiv version]
Geodesic flows and the mother of all continued fractions. International Journal of Number Theory, November 2021.[ arXiv version]
(with Florin Boca) α -expansions with odd partial quotients. Journal of Number Theory, June 2019.[arXiv version]
(with Florin Boca) Coding of geodesics on some modular surfaces and applications to odd and even continued fractions. Indagationes Mathematicae, October 2018. [arXiv version ]
Other undergraduate reseach mentoring
In Summer 2023, I worked with a Davisdon college student in RISE, an intensive summer research fellowship program to introduce students to undergraduate research after their freshman year. This project introduces the student to measure theory and continued fractions, with a goal of eventually exploring thhe connections between geometry and nearest integer continued fractions.
I served as a graduate mentor on four undergraduate research projects through the Illinois Geometry Lab.
Spring 2019, faculty mentor: Florin Boca, undergraduate students: Xavier Ding, Gustav Jennetten, and Joel Rozhon, Natural extension domains of α-odd continued fractions used numerical methods to generate natural extension domains for various values of α. We also generated some animations of the domain changing as α decreases from 1 to 0.
Fall 2018, faculty mentor: Philipp Hieronymi, graduate student mentors: Simone Sisneros-Thiry and myself, undergraduate students: Yasir Badillo Acosta, Kelly Jezior, Alex Lehan, Kagen Quiballo, Iris Tong. Developing Exciting Outreach Material formalized a hands-on platonic solids and Euler characteristic activity for fourth and fifth grade students and created a hands-on four color theorem activity for second and third grade students.
Spring 2016, faculty mentors: Jenya Sapir and Cary Malkiewich , graduate student mentors: Ruth Lou and myself, and undergraduate studets: Cameron Dart, Tyler Mulick, Zesheng Wang, Jessica Chen. Interactive Learning Tools for Linear Algebra. The game can be found at http://p415x.xyz/.
Fall 2014, faculty mentors: Jayadev Athreya and Francesco Cellarosi, and undergraduate students: Abigail Turner, Ananya Uppal, Peng Xu. Random Subsets of Discrete Sets looked at the gap distribution and pair correlation properties of random subsets of sequences, such as the Farey seqence. The undergraduate students posted a paper on arXiv of their work from Fall 2013-Spring 2014. It does not appear to have been submitted anywhere.
Select Past Talks
ICERM, Visualizing domains and surfaces to study continued fractions Illustrating Mathematics: Reunion/Expansion, Providence, RI [Slides][Video]
Carleton College, What do fractions, dynamical systems, and geometry have to do with each other? Summer Math Program for Women 30 Reunion Conference, June
Joint Mathematics Meetings, Natural extensions and entropy of α-odd continued fraction, AMS Special Session on Ergodic Theory, Symbolic Dynamics, and Related Topics, January 4, 2024.
George Mason Univeristy, Geodesic flows and slow downs of continued fraction maps, Geometry MMA: Metrics, Measures, and Algorithms, October 10, 2023.
AMS Spring Southeastern Sectional Meeting, Natural extensions and entropy of α-odd continued fraction, Special Session on Fractal Geometry and Dynamical Systems, September 17, 2022.
AMS Fall Central Sectional Meeting, Natural extensions and entropy of α-odd continued fraction, Special Session on The Intersection of Number Theory and Combinatorics, September 17, 2022.
Pacific Lutheran University, What do fractions, dynamical systems, and geometry have to do with each other?, Colloquium, March 24, 2021
One World Numeration Seminar, α-odd continued fractions, January 5, 2021. [Slides][ Video]
Expanding Dyanmics VII–Creative Online Ventures in Dynamics, α-odd continued fractions, December 8, 2020.[Slides][ Video]
Midwest Dynamical Systems Early Career Conference, Changing the rules for continued fraction expansions and their natural extension domains, November 14, 2020.[Slides][Video]
ICERM, Hyperbolic surfaces, cutting sequences, and continued fractions, October 21-25, 2019 [slides without animations][Video]