Complex Dynamics and Moduli Spaces

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From particle physics to finance, from evolution to climate change, the world is full of dynamical systems. Simple algebraic transformations already exhibit many of the features of these natural phenomena, such as phase transitions and tipping points that signal the onset of new regimes. These universal patterns may be revealed through the rigorous study of so-called moduli spaces, their compactifications, and their stratifications by dynamical invariants. This research project appeals to a broad range of mathematical disciplines to both deepen our understanding of dynamical systems and to sharpen our mathematical and computational methods. Its methods have already led to the discovery of new and unexpected algebraic regimes, through a combination of theoretical tools that narrow the domain of search, and experimental methods such as the simulation of billiard flows in idealized polygons. Moduli spaces of lattices, Riemann surfaces, rational maps and other algebraic structures exhibit rich geometry, often accompanied by rigidity and a connection with arithmetic. These spaces also have a dynamical nature -- they support natural flows or group actions with complicated orbits, or they classify such actions. This research project investigates moduli spaces from a dynamical and geometric perspective. In the setting of Riemann surfaces, the project aims to reveal the mechanisms within dynamics, algebraic geometry, and number theory that underlie the existence of unexpected, recently-discovered primitive, totally geodesic complex surfaces in moduli space. The investigator also aims to develop the L^p geometry of Teichmueller space, to interpolate between and go beyond the Teichmueller and Weil-Petersson metrics (which represent the cases p=1 and p=2). The case p=infinity in particular should lead to the sharpest bounds on the hyperbolic 3-manifold that fiber over the circle. In the setting of homogeneous spaces, the investigator plans to establish Ratner-like rigidity theorems for suitable open hyperbolic 3-manifolds. Additionally, in the setting of proper holomorphic maps on the unit disk, the investigator aims to develop a dynamical analogue of the theory of simple closed curves and stretch maps, enhancing the dictionary between rational maps and Kleinian groups.