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UID:69d9fc39bc3b2
DTSTAMP:20260411T034601
DTSTART:20190123T150000
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DTEND:20190123T150000
URL:https://murmitoyen.com/events/vanille/udem/detail/861215-can-one-hear-t
 he-shape-of-a-drum-and-deformational-spectral-rigidity-of-planar-domains
LOCATION:Université de Montréal - Pavillon André-Aisenstadt\, 2920\, che
 min de la Tour\, Montréal\, QC\, Canada\, H3T 1N8
SUMMARY:Can one hear the shape of a drum and deformational spectral rigidit
 y of planar domains
DESCRIPTION:Conférence Nirenberg en analyse géométrique
\nConférence de
  Vadim Kaloshin\, titulaire de la chaire Michael Brin en mathématiques de
  l'Université du Maryland. Il a obtenu son doctorat de l'Université de P
 rinceton en 2001 sous la supervision de John Mather. Le professeur Kaloshi
 n a apporté des contributions fondamentales à la théorie des systèmes 
 dynamiques\, notamment à l'étude de la diffusion d'Arnold\, du problème
  à n corps et de la conjecture de Birkhoff pour le billard convexe.
\nRé
 sumé:Kac popularized the following question 'Can one hear the shape of a 
 drum?' Mathematically\, consider a bounded planar domain Ω ⊆ R2 with a 
 smooth boundary and the associated Dirichlet problem
\nΔu + λu=0\, u|∂
 Ω=0.
\nThe set of λ's for which this equation has a solution is called t
 he Laplace spectrum of Ω. Does the Laplace spectrum determine Ω up to is
 ometry? In general\, the answer is negative. Consider the billiard problem
  inside Ω. Call the length spectrum the closure of the set of perimeters 
 of all periodic orbits of the billiard inside Ω. Due to deep properties o
 f the wave trace function\, generically\, the Laplace spectrum determines 
 the length spectrum. We show that a generic axially symmetric domain is dy
 namically spectrally rigid\, i.e. cannot be deformed without changing the 
 length spectrum. This partially answers a question of P. Sarnak. The talk 
 is based on two separate joint works with J. De Simoi\, Q. Wei and with J.
  De Simoi\, A. Figal.
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