Geometry, Topology, and Dynamics of Spaces of Non-Positive Curvature

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Geometry is concerned with quantitative aspects of spaces. Negatively curved spaces locally look like the surface of a saddle (in every pair of directions). While this might be hard to visualize, spaces with this property are quite prevalent, both in mathematics, and in nature. For these spaces, there is a tight link between the geometry of the space and the topology (qualitative aspects of the shape) of the space. There is also a natural associated dynamical system, which records how a particle inside the space will move, and which exhibits hyperbolic behavior - extreme sensitivity to initial conditions. For example, if you put a marble on a saddle and see how it rolls, then two nearby marbles can roll in very different directions. Again, hyperbolicity is a very common type of behavior for dynamical systems. This research proposes to study how the geometry, topology, and dynamics of these spaces interact and influence each other.The Principal Investigator (PI) plans to work on a series of projects that are loosely centered around the notion of non-positive curvature. These projects fall into three broad categories. (1) Branched coverings: the PI plans to systematically study branched covers of non-positively curved (or negatively curved) locally symmetric manifolds, ramified over a codimension two totally geodesic submanifold. The main goal is to understand whether or not these covers support Riemannian non-positive (or negative) curvature metrics. (2) Complex geometry: the PI plans to study several questions concerning complex manifolds. He plans to create new closed negatively curved Kaehler manifolds. He wants to investigate the topology of the space of almost complex, complex, and Kaehler structures on a manifold. He also wants to give high dimensional examples of almost complex manifolds that do not support complex structures. (3) Geometry/topology problems: the PI wants to show that hyperbolizations always produce spaces whose fundamental groups are linear. He would like to study the Singer conjecture for Gromov-Thurston manifolds. He is interested in studying non-arithmetic lattices in SO(n; 1), particularly the structure of totally geodesic submanifolds inside nonarithmetic hyperbolic manifolds. He would also like to find new obstructions for manifolds to support Anosov diffeomorphisms.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.