## Analysis Seminars 2012 - 2013

### Programme

All seminars are held at 3:10pm in Room M/2.06, Senghennydd Road, Cardiff unless stated otherwise.

Programme Organiser and Contact: Dr Federica Dragoni

##### 8 October 2012

Speaker: Sergei Favorov (Kharkov National University).

Title: Spectral Perturbation Theory, Generalized Convexity, and Blacshke Condition in Unbounded Domains.

##### 15 October 2012

Speaker: Komil Kuliev (Cardiff University)

Title: Half-linear Sturm-Liouville problem with weights.

Abstract: We prove a necessary and suffcient conditions for discreteness of the set of all eigenvalues (with the usual Sturm-Liouville properties) of half-linear eigenvalue problem with locally integrable weights. Our conditions appear to be equivalent to the compact embedding of certain weighted Sobolev and Lebesgue spaces. Every eigenvalue allows the variational characterization of Ljusternik-Schnirelmann type.

##### 22 October 2012

Speaker: Jean-Claude Cuenin (Imperial College London).

Title: Eigenvalue estimates for non-selfadjoint Dirac operators.

Abstract: Recent years have seen an increasing interest in the spectral theory of non-selfadjoint di erential operators. In particular, eigenvalue estimates for Schrodinger operators with complex potentials have been investigated by various authors. On the other hand, corresponding results for non-selfadjoint Dirac operators are not known.

In this talk we present eigenvalue bounds for Dirac operators on the real line with non-selfadjoint potentials.

##### 29 October 2012

Speaker: Peter Topping (University of Warwick).

Title: Ricci flow on surfaces.

Abstract: We will present a survey of Ricci flow on surfaces, which had its first wave of development in the 1980s, in particular with the beautiful results of Hamilton and Chow, and is now undergoing something of a revival. The theory has a number of applications concerning uniformisation of Riemann surfaces, and compactness properties of Ricci flows, and there are possible future applications to spectral theory. We will also see how geometric ideas from the theory lead to a better understanding of the so-called logarithmic fast diffusion equation.

##### 5 November 2012

Speaker: Yiannis Petridis (UCL)

Title: Quantum Unique Ergodicity of Eisenstein Series and Scattering states.

Abstract: An important problem of quantum chaos is to describe the behaviour of eigenfunctions of the Laplacian, as the eigenvalue parameter tends to infinity. We concentrate on the modular surface, which is a hyperbolic surface with ergodic geodesic flow. At the same it is rich with arithmetic structure. Luo and Sarnak proved that for generalised eigenfunctions (non-holomorphic Eisenstein series) the quantum limit is unique and is the Liouville measure. We study the quantum limits of scattering states through the study of quantum measures of Eisenstein series away from the critical line. We also examine the situation of Eisenstein series of half-integral weight, where we identify a condition on multiple Dirichlet series that imply QUE.
This is joint work with Morten S. Risager and Nicole Raulf.

##### 12 November 2012

Speaker: Lucia Scardia (University of Glasgow)

Title: Multiscale problems in dislocation theory.

Abstract:
Dislocations are defects in the crystal lattice of metals, and their collective motion gives rise to macroscopic permanent or plastic deformations. Since the typical number of dislocations is very large, keeping track of each dislocation is too costly and therefore upscaled models must be derived. The derivation of such models is one of the hard open problems in mechanical engineering.

This talk will address the rigorous derivation of mesoscopic dislocation models from discrete models using Gamma-convergence.

This is based on work in collaboration with Marc Geers, Stefan Mueller, Ron Peerlings, Mark Peletier and Caterina Zeppieri.

##### 19 November 2012

Speaker: Sergey Naboko (St Petersburg State University)

Title: Jacobi matrices with gaps in the essential spectrum: the eigenvectors structure.

Abstract: The asymptotic behaviour of the eigenvectors and generalized eigenvectors for the spectral parameter belonging to a gap in the essential spectrum to be considered. We also plan to present some applications of the eigenvectors analysis.

##### 26 November 2012

Speaker: Florian Theil (University Of Warwick)

Title: Optimality of periodic arrangements for particle systems with Wasserstein interaction.

Abstract: TBC

##### 3 December 2012

Speaker: Grigorios A. Pavliotis (Imperial College London)

Title: Convergence to equilibrium for nonreversible diffusions.

Abstract: The problem of convergence to equilibrium for diffusion processes is of theoretical as well as applied interest, for example in nonequilibrium statistical mechanics and in statistics, in particular in the study of Markov Chain Monte Carlo (MCMC) algorithms. Powerful techniques from analysis and PDEs, such as spectral theory and functional inequalities (e.g. logarithmic Sobolev inequalities) can be used in order to study convergence to equilibrium. Quite often, the diffusion processes that appear in applications are degenerate (in the sense that noise acts directly to only some of the degrees of freedom of the system) and/or nonreversible. The study of convergence to equilibrium for such systems requires the study of non-selfadjoint, possibly non-uniformly elliptic, second order differential operators. In this talk we will prove exponentially fast convergence to equilibrium for such diffusion processes using the recently developed theory of hypocoercivity. Furthermore, we will show how the addition of a nonreversible perturbation to a reversible diffusion can speed up convergence to equilibrium. This is joint work with M. Ottobre, K. Pravda-Starov, T. Lelievre and F. Nier.

##### 10 December 2012

Speaker: Ilia Kamotsky (UCL)

Title: On the linear water wave problem in the presence of a critically submerged body

Abstract: We study the two-dimensional problem of propagation of linear water waves in deep water in the presence of a critically submerged body (i.e. the body touching the water surface). Assuming uniqueness of the solution in the energy space, we prove the existence of a solution which satisfies the radiation conditions at infinity as well as, additionally, at the cusp point where the body touches the water surface. This solution is obtained by the limiting absorption procedure.
Next we introduce a relevant scattering matrix and analyse its properties. Under a geometric condition introduced by V. Maz'ya, 1978, we prove an important property of the scattering matrix, which may be interpreted as the absence of total internal reflection. This property also allows us to obtain uniqueness and existence of a solution in some function spaces (e.g. $H^2_{loc}\cap L^\infty$ ) without use of the radiation conditions and the limiting absorption principle, provided a spectral parameter in the boundary conditions on the surface of the water is large enough. The talk is based on the joint work with V. Maz'ya.

##### 28 January 2013

Speaker: Arieh Iserles (University Of Cambridge)

Title: Computing the Schrödinger equation with no fear of commutators.

Abstract: In this talk I report  recent work on the solution of the linear Schrödinger equation (LSE) by exponential splitting in a manner that separates different frequency scales. The main problem in discretizing LSE originates in the presence of a very small parameter, which generates exceedingly rapid oscillation in the solution. However, it is possible to exploit the features of the graded free Lie algebra spanned by the Laplacian and by multiplication with the interaction potential to split the evolution operator in a symmetric Zassenhaus splitting so that the arguments of consecutive exponentials constitute an asymptotic expansion in the small parameter. Once we replace the Laplacian by an appropriate differentiation matrix, this results in a high-order algorithm whose computational cost scales like O(N log N), where N is the number of degrees of freedom and whose error is uniform in the small parameter.

##### 04 February 2013

Speaker: Alexander Kiselev (St.Petersburg State University and Moscow State University)

Title: Inverse topology problems for graph Laplacians

Abstract: For a wide class of non-Kirchhoff matching conditions we derive an infinite series of trace formulae and discuss how these can be utilized in the study of inverse problems. The approach is based on the theory of boundary triples.

##### 11 February 2013

Speaker: Eugene Shargorodsky (King's College London)

Title: Negative eigenvalues of two-dimensional Schroedinger operators

Abstract: I will discuss estimates for the number of negative eigenvalues of a two-dimensional Schroedinger operator in terms of "L log L" type Orlicz norms of the potential. The obtained results prove a conjecture by N.N. Khuri, A. Martin and T.T. Wu (2002).

##### 18 February 2013

Speaker: Arghir Zarnescu (University of Sussex)

Title: Mathematical problems of the Q-tensor theory of liquid crystals.

Abstract: The challenge of modeling the complexity of nematic liquid crystals through a model that is both comprehensive and simple enough to manipulate efficiently has led to the existence of several major competing theories.

One of the most popular (among physicists) theories was proposed by Pierre Gilles de Gennes in the 70s and was a major reason for awarding him a Nobel prize in 1991. The theory models liquid crystals as functions defined on a two or three dimensional domains with values in the space of Q-tensors (that is symmetric, traceless, three-by-three matrices).

Despite its popularity with physicists the theory has received little attention from mathematicians until a few years ago when John Ball initiated its study.Nowadays it is a fast developing area, combining in a fascinating manner topological, geometrical and analytical aspects. The aim of this talk is to survey this development.

##### 25 February 2013

Speaker: Vitalij Moroz (Swansea University)

Title: Existence and qualitative properties of grounds states to the non-local Choquard-type equations.

Abstract: The Choquard equation, also known as the Hartree equation or nonlinear Schrodinger-Newton equation is a stationary nonlinear Schrodinger type equation where the nonlinearity is coupled with a nonlocal convolution term given by an attractive gravitational potential.

We present sharp Liouville-type theorems on nonexistence of positive
supersolutions of such equations in exterior domains. We also discuss
existence, positivity, symmetry and optimal decay properties of ground state solutions under various assumptions on the decay of the external potential and the shape of the nonlinearity. In particular, we obtain a sharp decay estimate of the ground state solution which was discovered by E.Lieb in 1977.

This is a joint work with Jean Van Schaftingen (Louvain-la-Neuve, Belgium)

##### 4 March 2013

Speaker: Yuri Safarov (King's College London)

Title: Ergodicity of branching billiards..

Abstract: If the Riemannian metric has a jump discontinuity on a surface of codimension one then a geodesic hitting the surface admits two possible continuations called the reflected and retracted rays. In this situation, instead of the usual geodesic flow, one has to deal with a branching (ray-splitting) billiard dynamics. It is not immediately clear how to define ergodicity for such billiard systems. On the other hand, quantum ergodicity of eigenfunctions of the corresponding Laplacian can be defined in the usual way. The talk will discuss the link between ergodic properties of eigenfunctions and characteristics of branching billiards and introduce a new notion of ergodicity in terms of a dynamical system on the space of functions on the cotangent bundle.

This is a joint work with D. Jakobson and A. Strohmaier.

##### 11 March 2013 - 15:10 - 16:10

Speaker: Olga Maleva (University of Birmingham)

Title: Differentiability inside null sets.

Abstract: The general question we are interested in is given the space X to find subsets S as small as possible with the property that every Lipschitz function on X has a point of differentiability in S. In finite-dimensional spaces Rademacher theorem says that Lipschitz functions are differentiable almost everywhere. Hence every subset S of positive Lebesgue measure will have this property. It turns out that not only there exist null subsets maintaining this property but one can choose them to be compact null sets of upper box-counting dimension 1. In this talk, I will speak about the geometry of such sets, the general approach to constructing such sets and applications to open problems.

##### 11 March 2013 - 16:30-17:30

Speaker: Uzy Smilansky (Cardiff University and Weizmann Institute)

Title: The spectra of (regular) tournament graphs.

Abstract: Tournament graphs are directed graphs with an (asymmetric) adjacency matrix D which summarizes the result of regular round-robin tournaments between N players: Every player plays against all the others, if i wins against j then D(i,j)=1 and D(j,i)=0. Clearly D(i,i)=0. If N is odd then a tournament can be regular – each player wins exactly half the times. The spectrum of D for regular tournaments consists of one point on the real axis, the rest are in the complex plane, all with real part = 1/2. After writing a trace formula for the spectral density, I shall prove that for large N the spectral density approaches the semi-circle law. Moreover, the spectral statistics is consistent with Random Matrix GUE statistics. The construction of regular tournaments requires the introduction of a random walk process in the space of tournament matrices which is interesting in its own right and will be discussed as well.

##### 18 March 2013 - CANCELLED

Speaker: Jonathan Bevan (University of Surrey)

Title: N-covering stationary points and constrained variational problems.

##### 15 April 2013

Speaker: Christoph Ortner (University of Warwick)

Title: Atomistic Models of Dislocations.

Abstract: Dislocations are possibly the most important type of crystal defect as they give rise to plasticity. I will explain how to describe dislocations within atomistic models, and motivate some interesting analytical questions. For example, it is very difficult in some cases to gauge from numerical experiments whether dislocations are stable or meta-stable effects.

Moreover, some of the ideas developed to answer this question naturally lead to a resolution of the question how to naturally define the dislocation core energy and core radius, which are important parameters in dislocation dynamics, and also provide techniques for their numerical evaluation. (joint work with Tom Hudson, Oxford)

##### 22 April 2013

Speaker: Sergey Naboko (St Petersburg State University and University of Kent)

Title: Jacobi matrices with gaps in the essential spectrum: the eigenvectors structure. Part II.

Abstract: The asymptotic behaviour of the eigenvectors and generalized eigenvectors for the spectral parameter belonging to a gap in the essential spectrum to be considered. We also plan to present some applications of the eigenvectors analysis. The role of the discrete spectrum in the gaps of the essential spectrum will be discussed.

##### 29 April 2013

Speaker: Yves Capdeboscq (University of Oxford)

Title: Regularity Estimates in High Conductivity Homogenization.

Abstract: It is known that homogenizing (or averaging) linear elliptic PDE with bounded $L^\infty$ coefficients is stable under small perturbations of the coefficients in $L^1$-norm. However, the magnitude of the perturbation depends on the smoothness of the sequence of solutions. In a recent work with Marc Briane and Luc Nguyen, we considered the case of a periodic micro-structure with highly conducting fibres. Fenchenko and Khruslov showed 30 years ago that for a particular scaling range, the effective problem includes a non-local term. We show that:

1. From a homogenization corrector result, one can deduce a lower bound on all norms $W^{1,p}_{loc}$ of the solution for p>2, and this bound blow-ups like $exp(C/\epsilon^2)$, a given power of the inverse of the radius of the rods.

2. This is not a surface effect : the blow-up occurs also outside the fibres. 3. Everywhere but at a distance less than $\epsilon^{1+\delta}$ from the fibres, the solution is uniformly $C^{1,\alpha}$ smooth. The measure of the forbidden domain tends to zero with a given rate in epsilon.