A Comprehensive Program in Rigidity

The focus of this research project mainly lies within the field of hyperbolic dynamical systems, a subfield of mathematics which encompasses a wide range of real-world systems that undergo changes over time. These systems can vary in complexity, ranging from the free motion of a simple mechanical system to the behavior of gas in a closed vessel. Hyperbolic dynamics studies the properties and long-term behavior "on average" of such systems. The project will address fundamental questions in hyperbolic dynamics that could potentially impact areas such as physics, chemistry, and engineering, where chaotic dynamical systems naturally arise. Additionally, the project has an educational component and provides support for the training of graduate students. The rigidity program in rank one dynamics aims to establish a stronger form of equivalence (such as smooth conjugacy) from a priori weaker forms of equivalence (such as orbit equivalence or topological conjugacy), assuming the vanishing of obstructions that are usually associated with periodic orbits (such as periods of periodic orbits or eigenvalue data at periodic points). Recent ideas proposed by the PI and collaborators open up a lot of new ground, especially for higher dimensional systems. The PI will explore these new avenues for rigidity in dynamics. Even more importantly, some of these ideas also enable progress in neighboring areas. Accordingly, the PI plans to expand the rigidity program beyond smooth dynamics questions, where tools of smooth dynamics could potentially be transferred. For example, the PI plans to pursue marked and weighted marked length spectrum rigidity on negatively curved manifolds. 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.