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30 March 2017

Spectral geometry: a meeting in Leeds



One-day workshop supported by LMS conference grant:
celebrating new appointments

Organiser: Gerasim Kokarev

The lectures will hold in MALL, Level 8, School of Mathematics, The University of Leeds. Refreshements and buffet lunch will be served on Level 9.
Speakers:

Yaroslav Kurylev (University College London) *

Alessandro Savo (Sapienza Universita Di Roma)

Alexander Strohmaier (University of Leeds)

Igor Wigman (Kings College London)


Programme:

11.00 Coffee and Reception

11.30 Alexander Strohmaier
Scattering theory for hyperbolic surfaces and Dirichlet to Neumann maps
Abstract. I will explain the connection between the Dirichlet to Neumann map for a hyperbolic surface with horocyclic boundary and the scattering matrix for the associated surface with cusp. This can be used to compute the scattering matrix for perturbed surfaces numerically. I will look in particular at how scattering resonances behave for perturbations of the modular domain. (Joint work with Michael Levitin, Reading)

12.30 Buffet Lunch

13.30 Alessandro Savo
Geometric rigidity of constant heat flow
Abstract. We discuss geometric rigidity of overdetermined boundary value problems, in the context of Riemannian Geometry. More precisely, on a compact Riemannian manifold with boundary we consider the solution u=u(t,x) of the heat equation with constant unit initial data and Dirichlet boundary conditions. If at every fixed time t the normal derivative of u(t,.) is a constant function on the boundary, we say that the manifold has the "constant flow property". We give a classification result for such manifolds which shows that this property is an analytic counterpart of the isoparametric property, very much studied in Riemannian geometry. We also relate the constant flow property with other overdetermined PDE's; in particular, the classical Serrin problem, and the Schiffer overdetermined eigenvalue problem, which is open, and known to be equivalent to the Pompeiu problem.

14.30 Yaroslav Kurylev
Geometric Whitney problem and machine learning.
Abstract. This is a joint result with Ch. Fefferman, S. Ivanov, M. Lassas and H. Narayanan. We consider the following approximation problem problem which is the geometric version of the Whitney's extension problem: Let X be a compact metric space. What are the conditions on X so that it could be approximated in the Gromov-Hausdorff sense by a Riemannian manifold bounded sectional curvature. Is it possible a construct, in an almost optimal way, an approximating manifold from X?

15.30 Tea Break

16.00 Igor Wigman
Nodal intersections of random toral eigenfunctions against a test curve
Abstract. This talk is based on joint works with Zeev Rudnick, and Maurizia Rossi. We investigate the number of nodal intersections of random Gaussian Laplace eigenfunctions on the standard 2-dimensional flat torus ("arithmetic random waves") with a fixed reference curve. The expected intersection number is universally proportional to the length of the reference curve, times the wavenumber, independent of the geometry. Our first result prescribes the asymptotic behaviour of the nodal intersections variance for generic smooth curves in the high energy limit; remarkably, it is dependent on both the angular distribution of lattice points lying on the circle with radius corresponding to the given wavenumber, and the geometry of the given curve. For these curves we can prove the Central Limit Theorem. We then construct some examples of exceptional "static" curves where the variance is of smaller order of magnitude, and the limit distribution is non-Gaussian.

17.00 Informal discussions

18.00 Leaving from the School for dinner


Registration:

If you are planning to attend the workshop please register by sending an email to
G "dot" Kokarev "at" leeds "dot" ac "dot" uk.

There is no registration fee charged, but the registration is necessary for catering purposes.

Travel:

Leeds is easily accessible by train and has direct inter-city links with major destinations in the UK. In particular, if you are travelling from London, there is a direct high-speed train from King's Cross railway station with average journey time of 140 minutes.

From the railway station, the University campus is within walking distance of approximately 15-20 minutes.

The Google map of the university campus can be found here; on the campus map from the university web-pages the School of Mathematics is located in the building number 84


Financial support:

The workshop is supported by an LMS conference grant. We have some limited funds for the travel expenses of young UK-based researchers. Please contact Gerasim Kokarev for details.
* To be confirmed.
http://www1.maths.leeds.ac.uk/~pmtgk/seminar/winter2017.html
Last modified: 9 March 2017