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SUMMARY:Craig Sutton (Dartmouth College)
DTSTART:20230125T160000Z
DTEND:20230125T170000Z
DTSTAMP:20260423T021223Z
UID:VSGS/61
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/VSGS/61/">Ge
 neric properties of Laplace eigenfunctions in the presence of torus action
 s</a>\nby Craig Sutton (Dartmouth College) as part of Virtual seminar on g
 eometry with symmetries\n\n\nAbstract\nA result of Uhlenbeck (1976) states
  that for a generic Riemannian metric $g$ on a closed manifold $M$ of dime
 nsion at least two the real eigenspaces of the associated Laplace operator
  $\\Delta_g$ are each one-dimensional and the nodal set (i.e.\, zero set) 
 of any $\\Delta_g$-eigenfunction is a smooth hypersurface. Now\, let $T$ b
 e a non-trivial torus acting freely on a closed manifold $M$ with $\\dim M
  > \\dim T$. We demonstrate that a generic $T$-invariant metric $g$ on $M$
  has the following properties: (1) the real $\\Delta_g$-eigenspaces are ir
 reducible representations of $T$ and\, consequently\, are of dimension one
  or two\, and (2) the nodal set of any $\\Delta_g$-eigenfunction is a smoo
 th hypersurface. The first of these statements is a mathematically rigorou
 s instance of the belief in quantum mechanics that non-irreducible eigensp
 aces are ``accidental degeneracies.'' \n\nRegarding the second statement\,
  in the event the non-trivial quotient $B = M/T$ satisfies a certain topol
 ogical condition\, we show that\, for a generic $T$-invariant metric $g$\,
  any orthonormal basis $\\langle \\phi_j \\rangle$ consisting of $\\Delta_
 g$-eigenfunctions possesses a density-one subsequence $\\langle \\phi_{j_k
 }\\rangle$ where the nodal set of each $\\phi_{j_k}$ is a smooth hypersurf
 ace dividing $M$ into exactly two nodal domains\, the minimal possible num
 ber of nodal domains for a non-constant eigenfunction. This observation st
 ands in stark contrast to the expected behavior of the nodal count in the 
 presence of an ergodic geodesic flow\, where examples suggest one should a
 nticipate the nodal count associated to a ``typical'' sequence of orthogon
 al Laplace eigenfunctions will approach infinity. \n\nThis is joint work w
 ith Donato Cianci (GEICO)\, Chris Judge (Indiana) and Samuel Lin (Oklahoma
 ).\n
LOCATION:https://researchseminars.org/talk/VSGS/61/
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