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Sefydliad Gwyddorau Cyfrifiadurol a Mathemategol Cymru SGCMC WIMCS Wales Institute of Mathematical and Computational Sciences. Analysis Cluster. Research interests include:. Nodal length of random spherical harmonics.

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SefydliadGwyddorauCyfrifiadurol a MathemategolCymru SGCMCWIMCS

Wales Institute of Mathematical and Computational Sciences

Analysis Cluster

Research interests include:

Nodal length of random spherical harmonics

Geometric methods for image processing

  • Method-parametrized convexity-based transforms : for functions f: RN R such that
  • |f(x)| Cf|x|2 + C and  > Cf, define (i) lower transform: L(f) f (ii) upper transform: U(f) f
  • key “tight” approximation property:lim L(f) = f, limU(f) = f
  •  and for every point xat whichfis C1,1 in a neighbourhood of x,
  • L(f(x)) = f(x) = U(f(x))
  • when is greater than a constant depending on f and the size of the neighbourhood
  • singularity detection : small neighbourhoods of singular points of f can thus be captured using the differences f - L(f) and U(f) – f. Different combinations of transforms and parameters enable specific types of singularities to be targetted.
  • Applications

Motivation

Let M be a compact surface, and - Laplacian on M. We are interested in the eigenvalues and eigenfunctions of : .

It is well-known that the spectrum is discrete: , and

span .

The nodal line of is simply its zeros set . We are interested in the geometry of the nodal lines as , most basically, in lj, the nodal length of .

Yau conjectured that is commensurable with in the sense that

. This conjecture was settled by Bruning & Donnelly-Fefferman in real analytic case (still open in the smooth case). Berry proposed

to model using random planewaves (RWM) with wavenumber .

Zelditch (?) conjectured that the nodal lines of generic chaotic manifolds are equidistributed. In particular, . The latter is wrong for completely integrable systems, such as the sphere or the torus.

Aim

We study the nodal length distribution of random spherical harmonics. Let Enbe the space of spherical harmonics of degree n (eigenvalue n(n+1)) – its dimension 2n+1, and pick any L2-orthonormal basis . We define the random spherical harmonic

, where ak are standard Gaussian i.i.d. Let

be the nodal length of the random spherical harmonic. We want to study the distribution of for large n. It is easy to compute that its expected value is of order n, consistent to Yau. Also, it follows from Yau that is bounded from below and above. However, does this it really fluctuate between two different numbers?

We may infer further information about the equidistribution conjecture using the following principle: Suppose X holds for generic eigenfunctions on completely integrable M. Then X also holds for all eigenfunctions on generic chaotic M.

Outcome

Our primary focus is the variance of the nodal length. We proved the following:

Theorem (2008): .

It already answers our first inquiry: since the variance of the normalized length, , vanishes, it does not fluctuate for typical spherical harmonics, so that typically, we have . However, it is also important to evaluate the variance. We proved:

Theorem (2009): . The constant 65/32 in the theorem is different than the one predicted by Berry (1/64), based on the RWM.

The cover of the book "Entdeckungenüber die Theorie des Klanges" by Ernst F.F. Chladni (Discoveries concerning the theory of sound; Leipzig, 1787)

Removal of high-density noise from large images

Left: Lena (1229x1229) with 99.5% salt and pepper noise

Right: recovered image

Identification of multiscale

medial axis

Left: multiscale medial axis of a Chinese character including fine branches

Right: main branches of medial axis

The nodal structure of a random spherical harmonic. The blue and red connected components are positive and negative nodal domains respectively. The nodal lines are the domain boundaries.

Feature detection

Left: detection of crossing points

Right: detection of end points

Patent application“Image processing” GB 0921863.7, filed December 2009

Joint project Kewei Zhang (Maths, Swansea), Antonio Orlando (Engineering, Swansea), Elaine Crooks (Maths, Swansea)

Past Cluster Workshops

  • LMS South West and South Wales Regional Meeting and Workshop on Calculus of Variations and Nonlinear PDEs, Swansea, September 15th-17th, 2008
    • Organizers : Niels Jacob, VitaliLiskevich, Kewei Zhang, Elaine Crooks and VitalyMoroz (Swansea)
    • Meeting speakers : Nicola Fusco (Naples), IstvanGyongy (Edinburgh), Bert Peletier (Leiden)
    • Workshop speakers : Marie-Françoise Bidaut-Véron (Tours), Georg Dolzmann (Regensburg), Daniel Faraco (Madrid), Marek Fila (Bratislava), UgoGianazza (Pavia),
    • Bernd Kirchheim (Oxford), Jan Kristensen (Oxford), Antonio Orlando (Swansea), IreneoPeral (Madrid), LászlóSzékelyhidi Jr. (Bonn), Laurent Véron (Tours)
  • Young Researchers Workshop on Spectral Theory, Quantum Chaos and Random Matrices, Cardiff, June 29th-July 1st, 2009
    • Organizers : Michael Levitin (Cardiff), UzySmilansky (Cardiff and Weizmann Institute of Science)
    • Workshop speakers: Maha Al-Ammari (Manchester), AmitAronovich (Weizmann), Ram Band (Weizmann), Sabine Burgdorf (Konstanz), Remy Dubertand (Bristol), AlexandreGirouard (Cardiff), Eva-Maria Graefe (Bristol), Maria Korotayeva (Humboldt-Universität), Hillel Raz (Cardiff), Sebastian Wilffeuer (Aberystwyth) and more.

Future Cluster Activities

  • Planned Wales Analysis Workshops in 2010-11, funded by WIMCS and LMS (pending)
    • Metamaterials and high-contrast homogenisation: analysis, numerics and applications (Cardiff: organizer – KirillCherednichenko)
    • Analysis of fractional elliptic operators (Swansea: organizer – Niels Jacob)
    • Calculus of variations and Nonlinear PDEs (Swansea: organizers -- Kewei Zhang, Elaine Crooks, VitalyMoroz)
    • The Malliavin calculus in the Fock Space (Aberystwyth: organizer – John Gough)

Poster Presenters: Dr I Wigman, Dr E Crooks