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SUMMARY:What Can Be Computed in Infinite-Dimensional Spectral Problems? - 
 Matthew Colbrook (University of Cambridge)
DTSTART:20260417T091500Z
DTEND:20260417T101500Z
UID:TALK244045@talks.cam.ac.uk
DESCRIPTION:Spectral computation in infinite dimensions is fundamentally d
 ifferent from its finite-dimensional counterpart. Standard discretisation 
 strategies\, including finite sections\, domain truncation\, and eigenvalu
 e solvers\, can fail spectacularly\, producing spectral pollution\, spectr
 al invisibility\, and ill-conditioned limits even for normal or self-adjoi
 nt operators. This talk presents a unified framework that brings together 
 foundational results and practical algorithms for understanding and overco
 ming these failures.&nbsp\;The central theme is a resolvent-based\, infini
 te-dimensional perspective on spectral computation\, coupled with the Solv
 ability Complexity Index (SCI) hierarchy\, which precisely characterises w
 hat spectral information can be computed\, how many limiting processes are
  required\, and when certified error control is impossible. I will explain
  why multi-limit algorithms are unavoidable\, how injection moduli and pse
 udospectra lead to pollution-free and invisible-free methods\, and how the
 se ideas extend beyond spectra to spectral measures\, spectral types\, ess
 ential spectra\, and nonlinear operator families. The programme is illustr
 ated\, where time permits\, with examples ranging from Schr&ouml\;dinger a
 nd Dirac operators to quasicrystals\, non-normal stability problems\, and 
 data-driven Koopman operators\, highlighting both sharp impossibility resu
 lts and practical\, provably convergent algorithms.
LOCATION:Seminar Room 1\, Newton Institute
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