Refined Weyl Law for the Perturbed Harmonic Oscillator
Montag, 3.5.21, 16:15-17:15, Anderssen (BBB)
We consider the quantum harmonic oscillator \(H_0=(1/2)(-\bDelta+|x|^2)\). The underlying classical flow is periodic with period \(2\bpi\). By an explicit calculation one can see that the solution operator to the dynamical Schrödinger equation of \(H_0\) is the identity (modulo a sign) at \(2\bpi\bmathbb{Z}\) and locally smoothing otherwise. This periodicity is related to a sharp remainder estimate for the\ncounting function of the eigenvalues of \(H_0\). If we perturb the operator by a pseudodifferential operator of lower order, then we break the symmetry and could hope for an improved remainder estimate. We will present results on recurrence of singularities for these operators as well as an improved remainder estimate.\n\nThis is based on joint work with Oran Gannot, Jared Wunsch, and Steve\nZelditch.
Maurer-Cartan elements, twisting and homotopy
Montag, 10.5.21, 16:15-17:15, Anderssen (BBB)
The globalisation of Kontsevich's formality to smooth manifolds depends on choices, namely of a torsion-free covariant derivative and some section of a pro-finite dimensional vector bundle. In my talk, I explain that even if the globalised formality changes with different choices, its homotopy class does not. The idea of the proof relies on some basic knowledge of strong homotopy Lie algebras, their morphisms, Maurer-Cartan elements and the so-called twisting procedure, which I recall in an introductory part. This talk is based on arXiv:2102.10645 joint with Andreas Kraft. \n\n
t.b.a
Montag, 17.5.21, 16:15-17:15, Anderssen (BBB)
Narrow escape problem on Riemannian manifolds
Montag, 31.5.21, 10:15-11:15, ZOOM (link in the email)
We use geometric microlocal methods to compute an asymptotic expansion of the mean first arrival time for Brownian particles on Riemannian manifolds. This approach provides a robust way to treat this problem, which has thus far been limited to very special geometries. (Joint work with Justin Tzou and Leo Tzou)