The Mathematics Department's Research Experience for Undergraduates (REU) program continues to yield impressive results, with students making significant contributions to dynamical systems and ergodic theory research.
Summer 2023: Pioneering Work in Symbolic Dynamics
Under the guidance of Professor Dan Thompson, a team of talented undergraduates successfully tackled an advanced project in symbolic dynamics. In the summer of 2023, Cristian Ramirez (UC-Berkeley) and Amy Somers (UC-Irvine) began a ten-week visit to Ohio State, initiating a project on a generalization of S-gap shifts. Their research focused on developing the theoretical framework for a new family of symbolic dynamic systems called (w, S)-gap shifts.
These systems generalize the classic S-gap shifts—binary sequences where the length of a run of consecutive 1's between any two 0's belongs to a prescribed set of natural numbers, S (e.g., the prime numbers). Ramirez and Somers considered a further generalization where the length of the run of 1's belongs to S, and the location of the run is determined by some infinite sequence w.
In the ten weeks available, Cristian and Amy achieved a complete solution to the problem. They rigorously established the definitions and basic properties of (w, S)-gap shifts and, most notably, proved a natural entropy formula for these systems. Their findings were subsequently published in the REU Journal PUMP.
Summer 2024: Extending the Variational Principle
For the Summer 2024 REU, Amy Somers returned to Ohio State, while Alex Paschal (UNC-Chapel Hill) joined the team (Cristian Ramirez having graduated and taken a position at a federal governmental agency). With this powerful team in place, the project scope expanded to adapting a proof of a fundamental result in thermodynamic formalism—the Variational Principle.
While the Variational Principle is a well-established result for compact dynamical systems, the goal of this project was to extend its validity to certain non-compact systems. The students' task was to understand a complex, unfinished working draft of a proof by Thompson and co-authors—originally developed for geodesic flows—and adapt it to the context of symbolic dynamics.
Despite the high level of difficulty and the challenging source material, the team made significant headway. They defined a new, potentially foundational class of systems: "countable-state shifts with the specification property." This class is believed to be the first set of countable-state shifts beyond Markov shifts that is analytically tractable for study, a result that is poised to make a significant impact within the subfield of countable-state symbolic dynamics.
Amy and Alex have already co-authored a first paper on this topic for an REU journal and are returning to OSU this summer to collaborate with Professor Thompson on a research paper suitable for publication in a high-level dynamics journal.
Recognition and Outreach
The outstanding quality of this undergraduate research has already earned the students notable recognition. Alex and Amy have been presenting their work at high-profile conferences, including the Joint Mathematics Meetings, the Midwest Dynamical Systems Conference at Northwestern, and the major international conference "Beyond Uniformly Hyperbolic Dynamics" in Trieste, Italy.
These achievements are truly remarkable for an undergraduate research program. By proving results of interest to the international research community, Amy Somers and Alex Paschal are serving as fantastic ambassadors for Mathematics at Ohio State