Month: May 2021

By awb1472 on Uncategorized

05/17: David Fisher

Totally geodesic submanifold, superrigidity and arithmeticity

Abstract: I will discuss some recent work showing that finite volume real and complex hyperbolic manifolds having infinitely many maximal totally geodesic submanifolds are arithmetic.  I will put this in the context of what we do and (mostly) don’t know about real  and complex hyperbolic manifolds, particularly in dimension at least 4.  Key ingredients in the proofs depend on homogeneous dynamics and algebraic dynamics and it is tempting to believe these might be relevant to some remaining mysteries.  This is based on joint work with Bader, Miller and Stover and also some earlier joint work with Lafont, Miller and Stover.

Slides

 

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05/24: Amir Algom

Pointwise normality and Fourier decay for self-conformal measures

Abstract: A real number is called p-normal if its orbit under the (times p) map equidistributes for the Lebesgue measure. It is a fundamental problem, motivated by Borel’s normal number Theorem, to study which singular measures are supported on normal numbers. In this talk we will survey the classical approach of Davenport-Erdos-LeVeque, and the recent innovative approach of Hochman-Shmerkin. We will then introduce a new dynamical method to attack this problem for self-conformal measures, that also allows us to estimate their Fourier transform.

Joint work with Federico Rodriguez Hertz and Zhiren Wang.

 

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05/10: Pratyush Sarkar

Generalization of Selberg’s 3/16 theorem for convex cocompact thin subgroups of \mathrm{SO}(n, 1)

Abstract: Selberg’s 3/16 theorem for congruence covers of the modular surface is a beautiful theorem which has a natural dynamical interpretation as uniform exponential mixing. Bourgain-Gamburd-Sarnak’s breakthrough works initiated many recent developments to generalize Selberg’s theorem for infinite volume hyperbolic manifolds. One such result is by Oh-Winter establishing uniform exponential mixing for convex cocompact hyperbolic surfaces. These are not only interesting in and of itself but can also be used for a wide range of applications including uniform resonance free regions for the resolvent of the Laplacian, affine sieve, and prime geodesic theorems. I will present a further generalization to higher dimensions and some of these immediate consequences.

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05/03: Carlangelo Liverani

Projective cones and Billiards

Abstract:  The study of the statistical properties of billiards has seen a rapid evolution in the last 25 years and many new ideas have been developed to further their study. Yet there are many interesting models (such as open systems, time varying billiards and, most of all, the Random Lorentz gas) for which the current techniques are not optimal. To overcome this  state of affairs we have attempted to adapt to billiards the projective cones techniques that has proven very effective in related problems. I will first describe such a technique in a simple case, then I will discuss its application to billiards. (work in collaboration with Mark Demers).