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DTSTAMP:20250710T033536Z
DESCRIPTION:Title: Lyapunov exponents\, Schrödinger cocycles\, and Avila's 
 global theory\n\n \n\nAbstract:\n\nIn the 1950s Phil Anderson made a predi
 ction about the effect of random impurities on the conductivity properties
  of a crystal. Mathematically\, these questions amount to the study of sol
 utions of differential or difference equations and the associated spectral
  theory of self-adjoint operators obtained from an ergodic process. With t
 he arrival of quasicrystals\, in addition to random models\, nonrandom lat
 tice models such as those generated by irrational rotations or skew-rotati
 ons on tori have been studied over the past 30 years. By now\, an extensiv
 e mathematical theory has developed around Anderson's predictions\, with s
 everal 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 a decade ago that the Lya
 punov exponent is piecewise affine in the imaginary direction after comple
 xification of the circle. In fact\, the slopes of these affine functions a
 re integer valued. This is easy to see in the uniformly hyperbolic case\, 
 which is equivalent to energies falling into the gaps of the spectrum\, du
 e 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 co
 ntinuity property of the Lyapunov exponent in both energy and frequency. A
 vila built his global theory (Acta Math. 2015) on this quantization proper
 ty. I will present some recent results with Rui HAN (Louisiana) connecting
  Avila's notion of acceleration (the slope of the complexified Lyapunov ex
 ponent in the imaginary direction) to the number of zeros of the determina
 nts of finite volume Hamiltonians relative to the complex toral variable. 
 This connection allows one to answer questions arising in the supercritica
 l case of Avila's global theory concerning the measure of the second strat
 um\, Anderson localization on this stratum\, as well as settle a conjectur
 e on the Hölder regularity of the integrated density of states.\n\n \n\nRo
 om: 5340\, Pavillon André-Aisenstadt\, Université de Montréal\n\nor\n\nZoo
 m link: https://umontreal.zoom.us/j/83118539851?pwd=bk5IOXBLNDRNSnR4dEcrSU
 FJVWhPZz09\n\nMeeting ID: 831 1853 9851\n\nPasscode: 215516\n\n \n
DTSTART:20230324T180000Z
DTEND:20230324T190000Z
SUMMARY:Wilhelm Schlag (Yale University)
URL:/mathstat/channels/event/wilhelm-schlag-yale-unive
 rsity-347035
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