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SUMMARY:On the role of the SCI hierarchy in spectral theory and PDEs — D
 etermining uncertainty principles and the boundaries of mathematics in an
 alysis - Anders Hansen (University of Cambridge)
DTSTART:20260204T093000Z
DTEND:20260204T103000Z
UID:TALK239620@talks.cam.ac.uk
DESCRIPTION:The problem of finding algorithms for computing spectra of ope
 rators and solutions to PDEs has both fascinated and frustrated mathematic
 ians since the seminal work by H. Goldstine\, F. Murray\, and J. von Neuma
 nn in the 1950s. &nbsp\;W. Arveson pointed out in the 1990s that despite t
 he plethora of papers on the subject of computing spectra\, the general co
 mputational spectral problem remained unsolved. We will discuss the soluti
 on to the computational spectral problem and show how it can only be solve
 d through the Solvability Complexity Index (SCI) hierarchy. Moreover\, the
  theory of the SCI hierarchy also implies uncertainty principles in spectr
 al theory and PDEs.&nbsp\;In particular\, for the wave equation\, given ap
 propriate Sobolev norms\, one can prove that the forward operator taking t
 he initial data to the solution of the PDE at time T=1 is bounded. Yet\, t
 here is an epsilon > 0 such that &mdash\; under the same conditions &mdash
 \; the question: `does the solution at T = 1 intersect the open ball centr
 ed at zero with radius epsilon?&rsquo\; becomes undecidable (non-provable)
 . That is\, determining the location of the solution is independent of the
  mathematical axioms. This implies that there is an uncertainty principle 
 for the wave equation. This phenomenon is linked to T. Tao&rsquo\;s progra
 m on blow-up of PDEs and undecidability of long term behaviour of PDEs. Po
 tentially surprisingly\, our results imply that a similar undecidability p
 henomenon can happen even for finite fixed time for well-posed problems. F
 inally\, uncertainty principles exist for spectra of operators\, even for 
 the self-adjoint cases: one can prove that there are self-adjoint operator
 s &nbsp\;where the spectrum depends continuously on the operator\, yet\, t
 here is an epsilon\, so that statements about the location of the spectrum
  to epsilon accuracy becomes undecidable.&nbsp\;
LOCATION:Seminar Room 1\, Newton Institute
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