Around Non-Positive Curvature

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Geometry is concerned with quantitative features of a space, while topology studies qualitative aspects of a space. As an example, a teacup and a donut are topologically the same (they both have a single “hole”), but geometrically different. Non-positive curvature is a geometric property, and roughly means that on small scales the space does not curve back on itself. These spaces have an interesting associated dynamical system: one can try to imagine how a billiard ball in such a space would move over time. This proposal is focused on studying these spaces from all three points of view (geometry, topology, and dynamics), providing a rich melting pot for different approaches and techniques. While the notion of non-positive curvature might seem strange, these spaces are in fact pervasive, both in mathematics and in nature. For this reason, they are a fundamental class of spaces for mathematicians to study and understand. This project will address a series of projects that are loosely centered around the notion of non-positive curvature. This project will provide opportunities for graduate student research and training. In addition the PI will help organize the EDGE (Enhancing Diversity in Graduate Education) program in 2022, a recruitment program with the goals of strengthening the ability of women to pursue careers in mathematical research and education and placing more women in visible leadership roles in the mathematics community.The Principal Investigator (PI) will work on various projects in non-positive curvature that fall into four broad categories. (1) Projects on locally symmetric spaces: the PI will consider the isomorphism problem for lattices in higher rank semisimple Lie groups, will construct new non-arithmetic lattices in PO(n,1), will produce divisible integral homology classes, and will study branched covers over immersed submanifolds. (2) Projects in coarse geometry: the PI will show that groups obtained via the Charney-Davis hyperbolizations are linear, will develop new QI-invariants of pairs of spaces, and study QI rigidity for manifolds with boundary. (3) Projects of a dynamical nature: the PI will work on a version of Khinchine’s theorem for continued fractions with constraints, and will study the decay of correlations for geodesic flows on compact locally CAT(-1) spaces. (4) Projects on K-theory: the PI will produce distinct algebraic number fields with isomorphic integral K-theory, and will show that all finitely generated Q-linear groups have finite VC geometric dimension.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.