Speaker: Jakob Hedicke and Stefan Suhr
Title: The space of Null Geodesics
Abstract: In the first half of this mini course we will review the constructions of the natural topology, smooth structure and contact structure on the space of all future pointing null geodesics of a Lorentzian manifold. We will discuss various known criteria that ensure that the space of null geodesics is a smooth manifold. Here the main focus lies on the case of causally simple spacetimes, where the topology on space of null geodesics can be connected to the conformal extendibility of the spacetime.
In the second half we will discuss examples of compact spacetimes with a smooth space of null geodesics (“Zollfrei”) and the important class of Kerr* spacetimes. For the later the space of null geodesics is known not to be Hausdorff. But it is an open problem whether it can be turned into a Hausdorff space by a finite quotient contraction. We will discuss indications for this.
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Title: Lyapunov exponents, Schrödinger cocycles, and Avila’s global theory.
Abstract: In the 1950s Phil Anderson made a prediction about the effect of random impurities on the conductivity properties of a crystal. Mathematically, these questions amount to the study of solutions of differential or difference equations and the associated spectral theory of self-adjoint operators obtained from an ergodic process. With the arrival of quasicrystals, in addition to random models, nonrandom lattice models such as those generated by irrational rotations or skew-rotations on tori have been studied over the past 30 years.
By now, an extensive mathematical theory has developed around Anderson’s predictions, with several questions remaining open. This talk will attempt to survey certain aspects of the field, with an emphasis on the theory of SL(2,R) cocycles with an irrational or Diophantine rotation on the circle as base dynamics. In this setting, Artur Avila discovered about 15 years ago that the Lyapunov exponent is piecewise affine in the imaginary direction after complexification of the phase. In fact, the slopes of these affine functions are integer valued. This is easy to see in the uniformly hyperbolic case, which is equivalent to energies falling into the gaps of the spectrum, due to the winding number of the complexified Lyapunov exponent. Remarkably, this property persists also in the non-uniformly hyperbolic case, i.e., on the spectrum of the Schrödinger operator. This requires a delicate continuity property of the Lyapunov exponent in both energy and frequency. Avila built his global theory (Acta Math. 2015) on this quantization property. I will present some results with Rui HAN (Louisiana State) connecting Avila’s notion of acceleration (the slope of the complexified Lyapunov exponent in the imaginary phase direction) to the number of zeros of the determinants of finite-volume Hamiltonians relative to the complex phase. This connection, which builds on the machinery developed by Michael Goldstein and the author (GAFA 2008, Annals of Math. 2011), allows one to answer questions arising in the supercritical case of Avila’s global theory concerning the measure of the second stratum, Anderson localization on this stratum, as well as settle a conjecture on the Hölder regularity of the integrated density of states. We will also describe applications to the block Jacobi matrix setting which shed light on almost reducibility questions via duality.
Speaker: Jakob Hedicke and Stefan Suhr
Title: The space of Null Geodesics
Abstract: In the first half of this mini course we will review the constructions of the natural topology, smooth structure and contact structure on the space of all future pointing null geodesics of a Lorentzian manifold. We will discuss various known criteria that ensure that the space of null geodesics is a smooth manifold. Here the main focus lies on the case of causally simple spacetimes, where the topology on space of null geodesics can be connected to the conformal extendibility of the spacetime.
In the second half we will discuss examples of compact spacetimes with a smooth space of null geodesics (“Zollfrei”) and the important class of Kerr* spacetimes. For the later the space of null geodesics is known not to be Hausdorff. But it is an open problem whether it can be turned into a Hausdorff space by a finite quotient contraction. We will discuss indications for this.
Speaker: Michael Kiessling
Title: Quest for the Relativistic Version of Hilbert’s 6th Problem
Abstract: By the end of the 19th century the atomistic explanation of the material universe had reached such a widespread acceptance among physicists that Hilbert, in 1900 at the International Congress of Mathematicians, proposed as his 6th problem to lay the rigorous mathematical foundations of the macroscopic continuum laws of physics in terms of the Newtonian dynamics of a huge number of atoms. (Obviously this is an open-ended project, not a clearly limited problem like Hilbert’s problem 8a: The Riemann Hypothesis.) In the 125 years hence, in particular most recently, mathematical physicists have made impressive progress on Hilbert’s 6th problem with hard-sphere atoms by deriving the Maxwell-Boltzmann kinetic equation for dilute gases, and from it the Navier–Stokes equations of fluids! The spirit of Hilbert's 6th problem is not limited to its pre-relativistic formulation, and in this presentation I argue that the relativistic version of Hilbert's 6th problem is an underappreciated frontier of mathematical physics research.
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| Title: | Spring break: no talk this week |