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.
More recently, I have 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. I am continuing to work on this project, along with additional connections between cutting sequences and continued fraction maps.
For Summer 2023, I am working with a 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.
Papers
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 ]
Continued fractions connecting number theory, dynamical systems, and hyperbolic geometry. Expository. Accepted by the editorial board for Research Connections, an in progress peer-reviewed volume that aims to introduce math majors to the mathematical interests and career paths of early career researchers.
Other undergraduate reseach mentoring
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
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]
Univeristy of Wisconsin, From modular surfaces to continued fractions, Junior Geometry and Topology Conference, October 26, 2019
Mathfest, Cutting Sequences of Lehner and Farey Expansions, Great Talks for a General Audience, August 3, 2019.
University of Maryland, Cutting sequences of Lehner and Farey expansions, Workshop on Dynamical Systems and Related Topics, April 13, 2019.
Joint Math Meetings, Coding geodesic flows for various continued fractions, AMS Special Session: Geometry and Dynamics of Continued Fractions, January 18, 2019.
Summer Illinois Math Camp Unfolding Billiards Tables, SIM Camp Epsilon (rising 8th and 9th graders), June 29, 2018.
George Mason University, Increasing outreach activities in the Illinois Geometry Lab, Student Research Talks (StReeTs) at George Mason University, September 21, 2018. [Slides]
Agnes Scott College, Using modular surfaces to generate continued fractions, Carolina Dynamics Symposium, April 14, 2018.
University of Maryland, Coding continued fractions as cutting sequences, Workshop on Dynamical Systems and Related Topics, April 7, 2018.