Sefydliad Gwyddorau Cyfrifiadurol a Mathemategol Cymru SGCMC WIMCS

Wales Institute of Mathematical and Computational Sciences

Analysis ClusterResearch interests include:

Nodal length of random spherical harmonics

MotivationLet 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 proposedto 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.

AimWe study the nodal length distribution of random spherical harmonics. Let En be 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.

OutcomeOur 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.

Past Cluster Workshops

Future Cluster Activities

Poster Presenters: Dr I Wigman, Dr E Crooks

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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 x at which f is 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 Removal of high-density noise from large imagesLeft: Lena (1229x1229) with 99.5% salt and pepper noiseRight: recovered image

Identification of multiscale medial axisLeft: multiscale medial axis of a Chinese character including fine branchesRight: main branches of medial axis

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

•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, Vitali Liskevich, Kewei Zhang, Elaine Crooks and Vitaly Moroz (Swansea)• Meeting speakers : Nicola Fusco (Naples), Istvan Gyongy (Edinburgh), Bert Peletier (Leiden)• Workshop speakers : Marie-Françoise Bidaut-Véron (Tours), Georg Dolzmann (Regensburg), Daniel Faraco (Madrid), Marek Fila (Bratislava), Ugo Gianazza (Pavia),

Bernd Kirchheim (Oxford), Jan Kristensen (Oxford), Antonio Orlando (Swansea), Ireneo Peral (Madrid), László Székelyhidi 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), Uzy Smilansky (Cardiff and Weizmann Institute of Science)• Workshop speakers: Maha Al-Ammari (Manchester), Amit Aronovich (Weizmann), Ram Band (Weizmann), Sabine Burgdorf (Konstanz), Remy Dubertand (Bristol),

Alexandre Girouard (Cardiff), Eva-Maria Graefe (Bristol), Maria Korotayeva (Humboldt-Universität), Hillel Raz (Cardiff), Sebastian Wilffeuer (Aberystwyth) and more.

Planned Wales Analysis Workshops in 2010-11, funded by WIMCS and LMS (pending)

• Metamaterials and high-contrast homogenisation: analysis, numerics and applications (Cardiff: organizer – Kirill Cherednichenko)• Analysis of fractional elliptic operators (Swansea: organizer – Niels Jacob)• Calculus of variations and Nonlinear PDEs (Swansea: organizers -- Kewei Zhang, Elaine Crooks, Vitaly Moroz)• The Malliavin calculus in the Fock Space (Aberystwyth: organizer – John Gough)

Feature detectionLeft: detection of crossing pointsRight: detection of end points

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

analytic and computational spectral theory

function spaces and integral operators

inverse problems stochastic partial differential equations

semi-classical methods Dirichlet to Neumann maps

spectral theory of differential operators on domains, manifolds and graphs, including operators arising in mathematical physics and non-self-adjoint problems

qualitative theory of elliptic and parabolic second order linear, semi-linear and quasi-linear partial differential equations

applications of semigroup theory to partial differential equations, Markov semi-groups and perturbation theory of generators of semi-groups

vectorial calculus of variations and its applications to material micro-structure models, forward-backward diffusion equations and their applications to image processing, compensated convexity

reaction-diffusion and reaction-diffusion-convection equations and systems, their singular limits, and their applications to population dynamics, travelling waves and mathematical aspects of phase transitions

systems of fully nonlinear PDEs, studied particularly using convex analysis, variational methods, and concepts from optimal mass transportation

functional equations of the Wiener-Hopf type

waves in inhomogeneous lattices the theory and applications of the singular equations with fixed point singularities

analytic number theory and applications in numerical analysis, mathematical bioinformatics, internet security and risk management, archaeology, image recognition, oceanology, plasticity and visco-plasticity, and many others