CAREER: Group actions, rigidity, and dynamics in low dimensions

The research component of this project lies at the intersection of geometry, dynamical systems, and group theory. A dynamical system is concerned with the long-term behavior of evolving physical or geometric objects, such as planets orbiting the sun or water flowing through a pipe. Mathematicians have gained deep insights about geometric shapes by studying the dynamics of their symmetries. This information is naturally recorded in a group, and algebra and geometry become intimately connected. This project will exploit the dynamics of such symmetries to unravel intrinsic properties of low-dimensional geometric objects. The project's goals in both intellectual merit and broader impacts are deeply intertwined, and the proposed research will directly inform and enhance the educational components of the project through the involvement of undergraduates in research and outreach, and graduate students in research, mentoring, and outreach. The project has four main research goals: (1) to prove the singularity conjecture for Cannon-Thurston maps, (2) to study the dynamics of free group automorphisms to obtain rigidity results in free and free-by-cyclic groups, (3) to classify hyperbolic extensions of free groups, and (4) to study dynamics in flat surfaces to reveal exotic number-theoretic behavior and obtain curve counting results. The educational component of this project includes enhancing the infrastructure of the Madison Experimental Mathematics Lab (MXM), expanding MXM through carefully designed outreach initiatives, and creating and implementing a Research Incubator for graduate students and postdocs. Additionally, the PI will continue their engagement in other professional activities through organizing conferences, workshops, and seminars; developing a new graduate course; and mentoring of undergraduates, graduate students, and postdocs. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.