PRIMES: Geometry, Topology, and Dynamics of Higher Rank Character Varieties.

At very small scales, a surface looks like a flat, two-dimensional plane. But when you zoom out, it can take on much more complicated shapes, like the curved surface of the Earth or a pretzel. This topological complexity is described mathematically by an algebraic object called the fundamental group of the surface. In mathematics, as well as in physics, studying how this fundamental group can act as symmetries of geometric objects often leads to the discovery of new structures and phenomena. The PI will study surfaces and their symmetries using tools from geometry, algebra, and dynamical systems. The PI will also lead vertically integrated research projects for undergraduate and graduate students at Sam Houston State University, organize a Junior Colloquium series for math majors and graduate students at his institution, and a conference for early career mathematicians based in Southeast Texas. The central goal of the project is to better understand the relationship between the geometry of certain mathematical objects, called Hitchin representations, and a dynamical way to measure their similarities. Specifically, Hitchin representations generalize holonomies of hyperbolic structures and form a rich family of discrete and faithful representations of the fundamental group of a surface into a higher rank Lie group. To compare two Hitchin representations, the project uses a number called the correlation number, which ranges from 0 to 1, where a value closer to 1 means the two representations are more similar. In previous joint work, the PI described geometrically interesting diverging sequences of Hitchin representations whose correlation numbers stay uniformly bounded away from zero, as well as diverging sequences of Hitchin representations whose correlation numbers decay to zero. Here, the PI will apply methods coming from the Thermodynamic Formalism of Anosov dynamical systems to study and systematically relate the geometry of diverging sequences of pairs of Hitchin representations to the end-behavior of their correlation numbers. 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.