Collaborative Research: NSF-UEFISCDI: Higher Dimensional Dynamics

The field of dynamics studies how processes evolve over time, such as the motion of planets, population growth, or the behavior of digital networks. The mathematical theory of dynamical systems offers a powerful language to describe these changes, uncover patterns, predict future behavior, and identify when systems may become chaotic. One-dimensional holomorphic dynamics is a mature field of mathematics, rooted in the famous work of Fatou and Julia on fractal sets. In contrast, higher-dimensional holomorphic dynamics is a newer but rapidly developing area, marked by fundamentally different behavior and rich phenomena absent in the one-dimensional setting. The PIs will advance understanding of dynamical systems in several complex variables by bridging this gap between dimension one and higher dimensions. The project will also provide training opportunities for graduate students and postdoctoral researchers. This is a project funded jointly by the National Science Foundation's Division of Mathematical Sciences, in the Directorate for Mathematical and Physical Sciences, and the Romanian Executive Agency for Higher Education, Research, Development and Innovation Funding (UEFISCDI), in accordance with the Memorandum of Understanding between the NSF and UEFISCDI. The PIs will investigate the dynamics of higher-dimensional germs of holomorphic diffeomorphisms, particularly those with neutral fixed points, which pose unique challenges. A key goal is to characterize the structure of the dynamical system near the fixed points and to extend concepts like hedgehogs—intricate invariant sets from one-dimensional dynamics—to higher dimensions, especially in the setting of conservative holomorphic germs. The PIs will also analyze the dynamics and bifurcations of polynomial automorphisms of two-dimensional complex space, with particular attention to the relationship between Julia sets and critical loci—sets of tangencies between dynamically defined foliations. The research activity conducted under this award will generate pioneering techniques in higher-dimensional dynamics, with impact in other areas of mathematics such as topology and geometry. 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.