Speaker: Ben Lowe from University of Chicago
Time: August 9,12,13 14, 9am-10am.
Zoom Meeting ID:944 1729 4293 (Passcode: 185048)
Abstract: Discrete subgroups of semisimple Lie groups, or equivalently nonpositively curved locally symmetric spaces, are central objects in most areas of modern mathematics. The first part of the minicourse will treat the Mostow-Margulis (super-)rigidity theorems, which give a structural understanding of nonpositively curved finite volume locally symmetric spaces and the possible maps between them. A great achievement of this theory is Margulis’s arithmeticity theorem, which states that finite volume higher rank locally symmetric spaces (e.g., corresponding to discrete subgroups of SL(n,R) for n>2) must be arithmetic, or must correspond to discrete subgroups obtained by a procedure analogous to taking the integer points of a group of matrices. Although this statement fails in rank one, recently Bader-Fisher-Miller-Stover and Margulis-Mohamaddi proved an analogue of the Margulis arithmeticity theorem in the rank one setting of finite volume real hyperbolic manifolds: namely, if such a manifold contains infinitely many maximal finite volume totally geodesic submanifolds of dimension greater than one, then it must be arithmetic. I will give a gentle overview of this area taking care to explain necessary background and definitions.
In the second part of the minicourse, I will describe a research program that attempts to prove analogues of the above results in the broader setting of differential geometry. Work in this direction has drawn on several different parts of mathematics, including dynamical systems, representation theory, and geometric analysis. After surveying some of the foundational results including work by Mok-Siu-Yeung and Besson-Courtois-Gallot, I will describe recent work of myself and work joint with Filip-Fisher that proves versions of the Bader-Fisher-Miller-Stover and Margulis-Mohamaddi arithmeticity theorem for finite volume negatively curved manifolds. I will also describe some of the many open problems and directions for future research in this area.
About the speaker:
Ben is a Dickson Instructor and NSF Postdoc at the University of Chicago. Before that he was a graduate student at Princeton where my advisor was Fernando Coda Marques. His interest is in Differential Geometry, Minimal Surfaces, Hyperbolic Geometry, Dynamical Systems, and Scalar Curvature etc.
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