## Analysis Seminars 2011 - 2012

### 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

##### 10 October 2011

**Speaker:** Anne-Maria Ernvall-Hytonen.

**Title:** On short exponential sums connected with cusp forms.

**Abstract:** I will briefly tell about the history of exponential sums involving Fourier coefficients of cusp forms, and how they connect to the exponential sums involving values of the divisor function. I will then turn to the short sums, and tell what we know about them, what we believe about them, how they connect to the values of individual coefficients, and what kind of computer data we have.

##### 24 October 2011

**Speaker:** Vitaly Moroz (Swansea).

**Title: **Asymptotic properties of ground states of a semilinear elliptic problem with a vanishing parameter.

**Abstract: **We consider a semilinear elliptic problem on the entire space with a double-well type nonlinearity. When one of the parameters vanishes the ground state of the problem converges after a suitable rescaling to the ground state of a limiting problem, however the form of the limiting equation depends on whether the original problem contains a subcritical, critical or supercritical term. We establish the precise asymptotic rate of the limiting rescaling and derive asymptotic behaviour of the ground states at the origin and near infinity.

This is a joint work with Cyrill Muratov (NJIT, USA).

##### 31 October 2011

**Analysis and Applied Mathematics Octoberfest Cardiff 2011**.

This event will start with a lunch at 12:00 in Room M/1.04 and will be followed by the following 3 talks, the first of which starts at 14:10. Please note all talkes will take place in M2.06.

**13:10-14:00**

**Speaker:** Elaine Crooks (Swansea).

**Title: **Highly nonlinear large-competition limits of elliptic systems.

**Abstract:** In this talk, I will discuss the effect of different types of competitive
interaction terms on the large-interaction limit of nonlinear elliptic systems
modelling the steady states of populations that compete in some region. As
the competition rate tends to infinity, non-negative solutions of quite simple-looking
systems converge to the positive and negative parts of solutions of a scalar limit
problem which may be much more strongly nonlinear than the original system,
possibly with quadratic growth in the gradient of the limit function. This is joint
work with Norman Dancer.

**14:30-15:20**

**Speaker:** Alan Haynes (Bristol).

**Title:** Separated nets which are bounded displacement from a lattice.

**Abstact:** A subset Y of a metric space is called a separated net if there are constants r, R>0 such that every ball of radius R intersects Y and every ball of radius r contains at most one point of Y. There is a technique in tiling theory called the `cut and project method' which provides an abundant source of separated nets in Euclidean space. Until recently it was an open problem to determine whether separated nets coming from this method can be deformed in a uniform way onto a lattice. In this talk we will describe these ideas in more detail and explain our proof that almost all separated nets coming from the cut and project method can be translated onto a lattice by moving each point by at most some fixed constant distance. This is joint work with Barak Weiss and Michael Kelly.

**15:40-16:30 **

**Speaker:** Martin Sieber (Bristol).

**Title:** Semiclassical approach to discrete symmetries in quantum chaos.

**Abstract:** We use semiclassical methods to evaluate spectral statistics
of quantum systems that are classically chaotic and possess discrete
spatial symmetries. The energy spectra of these systems fall into
subspectra that are associated to irreducible representations of the
corresponding symmetry group. We show that for (spinless) time reversal
invariant systems the spectral two-point correlation function for
subspectra associated to real and complex representations, respectively,
coincide with predictions from the Gaussian Orthogonal Ensemble (GOE)
and the Gaussian Unitary Ensemble (GUE) of Random Matrix Theory.
For systems without time reversal invariance all subspectra show
GUE statistics.

##### 7 November 2011

**Speaker:** Federica Dragoni (Cardiff).

**Title:** X-convexity and applications.

**Abstract:**We introduce a new notion of convexity, namely X -convexity, which applies to any given family of vector fields and in particular to the sub-Riemannian case. We then show a PDE-characterization for X-convex functions and we investigate some properties of these functions. (Joint work with Martino Bardi).

##### 9 November 2011 at 15:10 in Room M/2.06

**Speaker:** Marco Marletta (Cardiff).

**Title:** Weak stability for inverse Sturm-Liouville problems with finite data.

**Abstract:**The talk will be based on a paper I published in 2005 with Rudi Weikard.

It shows that if one knows, up to error \epsilon, the first N

Dirichlet-Dirichlet and Dirichlet-Neumann eigenvalues for a Sturm-Liouville

problem in Liouville normal form with square summable potential then

the potential is determined up to an error

const.(\epsilon \log(N) + N^{-1/2})

in a W^{-1,\infty} norm.

The method of proof is by transformators, which I will introduce and

discuss, and generalized eigenfunction expansions.

##### 14 November 2011

**Speaker:** Ralf Rueckriemen (Cardiff).

**Title:** Isospectrality on quantum graphs.

**Abstract:** I will discuss the two main methods to build isospectral quantum graphs. The first method is inherited from combinatioral graphs, one uses Seidel switching to build isospectral combinatorial graphs, and then applies investigates under what circumstances these give rise to isospectral quantum graphs. The second method is called Sunada's method. It was originally designed for manifolds but there are versions for quantum graphs.

##### 15 November 2011 from 10:00am in Room M/1.25

**WIMCS/Leverhulme Postdoctoral Fellowships**

**10:00-10:35**

**Speaker: **Shane Cooper (University of Bath).

10:35-11:10

**Speaker: **Veronique Fischer (Imperial College London).

**11:10-11:45 **

**Speaker: **Michael Strauss (Heriot-Watt University).

##### 21 November 2011

**Speaker:** Gabor Kiss (Exeter).

**Title:** Currency exchange rate fluctuations by delay differential equations.

**Abstract:** In many applications the rate of change of state variables depends on
their states at prior times. When these processes are assumed to be
deterministic, they are modelled by delay differential equations. In the simplest cases only one, time invariant time lag is considered. However, equations with multiple delays offer richer dynamics, thus they are of mathematical interest and potential models of real-world problems with complex oscillations. We present results on the coexistence of periodic solutions to an equation with two point delays; a novel rigorous
computational tool which allows us deriving valuable existence results
for these infinite dimensional dynamical systems is also described. Furthermore, we report on the existence of pullback attractors to
equations with multiple time-varying delays. A model for exchange-rate
fluctuations is considered as a motivating example.

##### 28 November 2011

**Speaker:** Nicolas Dirr (Cardiff).

**Title:** Minimizers of random functionals.

**Abstract: **We consider a class of functionals on Sobolev spaces which consist of a part penalizing
spatial oscillations of the function (i.e. the $L62$-norm of the gradient or a nonlocal version thereof)
a double-well potential which prefers the function to be close to either $+1$ or $-1,$ and a random field,
which, while neutral in average, prefers locally the $+1$ or $-1$ state. The competition of these three terms
leads to very different behaviour of the minimizers, depending on the space dimension and choice of parameters.

##### 30 November 2011

**Speaker:** Sergey Naboko (St.Petersburg State University).

**Title: **On the boundary version of the Analytic Fredholm Theorem and its applications.

**Abstract:** The classical Analytic Fredholm Theorem determines the invertibility of the analytic operator -valued function
in an open domain provided it point wise differs from Identity operator by a compact one.The talk deals with the problem of its
invertibility almost everywhere on the boundary important in many problems of perturbation theory including the scattering theory.
A counterexample proving the result sharpness to be presented.

##### 5 December 2011

**Speaker:** Sergei Fedotov.

**Title:** Fractional reaction-transport equations.

**Abstract:** I will talk about how to incorporate the nonlinear terms into non-Markovian master equations corresponding to random walks with non-exponential waiting time distributions. I will show how to derive nonlinear fractional partial differential equations for subdiffusive transport. I apply these equations to the problem of front propagation in the reaction-transport systems.

##### 6 February 2012 at 15:10 in Room M/2.06

**Speaker:** Yaroslav Kurylev.

**Title:** Mathematical Aspects of Invisibility.

**Abstract: **We consider so me recent results regarding the mathematical
theory of invisiblity, in particular, the problem os convergence of the
systems described by the "realistic", i.e, isotropic, non-singular
conductivities to the idealised ones. We also discuss some new physical
phenomena due to cloacking, e.g. cloaked sensors, etc.

This is a joint work with A. Greenleaf, M.Lassas and G.Uhlmann.

##### 20 February 2012 at 15:10 in Room M/2.06

**Speaker:** Sanju Velani (York).

**Title:** Diophantine approximation, fractal sets and lacunary sequences.

**Abstract:** The metric theory of Diophantine approximation on fractal sets is developed
in which the denominators of the rational approximates are restricted to lacunary
sequences. The case of the standard middle third Cantor set and the sequence
{3^n : n \in N} is the starting point of our investigation. Our metric results for this
simple setup answers a problem raised by Mahler. As with all ‘good’ problems – its
solution opens up a can of worms.

##### 27 February 2012 at 15:10 in Room M/2.06

**Speaker:** Laura Caravenna (Oxford).

**Title: **Continuous solutions to a balance equation.

**Abstract:** I will consider continuous solutions to a scalar, 1D balance law with bounded source term. I will most focus on the simple equation $u_{t}+ [u^{2}/2]_{x}=w$ with $w$ bounded. Even when this equation is satisfied only in distributional sense, and $u$ is not either Sobolev or BV, we show that the continuity allows to reduce the PDE to ODEs along characteristics. We gain still correspondence between Eulerian and Lagrangian formulation. This is strictly related to a characterization of intrinsic Lipschitz graphs in the sub-Riemannian Heisenberg groups, as it will be mentioned. The talk will be mainly based on a collaborations with G. Alberti, S. Bianchini, F. Bigolin, F. Serra Cassano.

##### 5 March 2012

**Spectral Analysis Day. More information to follow...**

##### 19 March 2012 at 15:10 in Room M/2.06

**Speaker:** Ralf Rueckriemen (Cardiff).

**Title:** The Bloch spectrum of a quantum graph.

**Abstract:** I will introduce quantum graphs and give some history about their spectra and inverse spectral problems on quantum graphs. I will then explain the notion of Bloch spectrum and present some results on what properties of a quantum graph are determined by the Bloch spectrum.

##### 21 March 2012 at 15:10 in Room M/2.06

**Speaker:** Svetlana Pastukhova (Moscow)

**Title:** Homogenisation of elasticity theory on this periodic structures.

**Abstract: **I will briefly introduce the theory of homogenisation for high-contrast periodic PDE and the related spectral analysis. I will then discuss recent progress in the analysis of problems where the associated periodic measure is supported by a structure whose volume tends to zero in the homogenisation limit. The related limit spectrum has a striking resemblance to its analogue in the high-contrast case.

##### 26 March 2012 at 15:10 in Room M/2.06

**Speaker:**Alexander Kiselev

**Title:** Recovering coupling conditions for a class of quantum graphs.

**Abstract:** A series of trace formulae can obtained for a quantum graph with delta-type coupling by utilizing the boundary triples based approach. These can be shown to be useful in particular for recovering coupling conditions at the vertices based on the knowledge of the graph Laplacian spectrum. A couple of open problems will be discussed, time permitting.

##### 26 March 2012 at 15:10 in Room M/2.06

**Speaker:** Matthew Lettington (Cardiff)

**Title:** Recent Progress in the Distribution of Lattice Points near to Convex Surfaces and Polytopes.

**Abstract:** In 1925 V. Jarnik linked convex curves to convex polygons to obtain an upper bound for the number of lattice points lying on a curve. Since then, much progress has been made with the generalised problem of the number of lattice points close to convex hypersurfaces. I will give a brief outline of the history of this subject, focusing on recent developments and explaining how in tackling this problem, Jarnik's method has been evolved to link convex polytopes to convex hypersurfaces. To conclude, I will compare these developments with recent results by Iosevich, giving an overview of the current understanding of this subject.

##### 23 April 2012 at 15:10 in Room M/2.06

**AFN Distinguished Analysis Day**

**Time: **15:10.

**Speaker:** Leonid Parnovski (UCL)

**Title:** Integrated density of states of Schroedinger operators with periodic or almost-periodic potentials

**Abstract:** I will discuss the asymptotic behaviour for large energies of

the integrated density of states (IDS) of periodic or almost-periodic

Schroedinger operator. Under certain broad assumptions on the potential,

I will show that there is a complete asymptotic expansion of the IDS.

This is a joint result with R.Shterenberg.

**Time:** 16:10.

**Speaker:** Karl Michael Schmidt (Cardiff).

**Title:** Spectral properties of a q-Sturm-Liouville operator

**Abstract:** The talk presents a second-order divided difference operator which arises in connexion with q^{-1} Hermite polynomials and resembles a Sturm-Liouville operator in structure. It turns out to have curious spectral properties and allows a transparent construction of examples with exotic behaviour such as dense point or singularly continuous spectrum. The question of which spectral type is generic will also be discussed.

This is joint work with BM Brown and JS Christiansen.

##### 30 April 2012 at 15:10 in Room M/2.06

**Speaker:** Jonathan Bennett (Birmingham).

**Title: **Weighted norm inequalities for oscillatory integrals.

**Abstract:** The theory of weighted norm inequalities has been enormously successful for broad classes of important operators in euclidean harmonic analysis, such as maximal averaging operators, fractional integral operators, Calder\'on-Zygmund singular integral operators and Littlewood--Paley square functions. The purpose of this talk is to present some general weighted $L^2$-norm inequalities for \textit{oscillatory} integral operators. Our setting is that of convolution operators on the line with phases of finite type..

##### 2 May 2012 at 15:10 in Room M/2.06

**Speaker:** Johannes Zimmer (Bath).

**Title:** Phase transitions in solids: from atomistic waves to a continuum picture.

**Abstract:** The equations of elasticity in one space dimension, $u_{tt} =

\sigma(u_x)_x$, become ill-posed if the potential energy density is

nonconvex, or equivalently if $\sigma$ is non-monotone. This

complication necessarily arises in the theory of so-called martensitic

phase transitions, which are diffusionless solid-solid transformations

where several stable phases can coexist.

Different regularisations of this ill-posed problem have been

proposed; we will here focus on so-called kinetic relations, which

relate the velocity of a moving interface to a driving

force. Phenomenological kinetic relations have been proposed, but a

natural question is whether they can in simple situations be derived

from first principles, namely atomistic considerations.

To investigate this question, we study the simplest one-dimensional

chain model of martensitic materials, where neighbouring atoms are

coupled by a spring with bi-quadratic potential. We present existence

results for travelling waves and discuss non-uniqueness of microscopic

solutions. This non-uniqueness will be discussed in light of the

macroscopic kinetic relation.

##### 28 May 2012 at 15:00 in Room M/2.06

**Speaker:** Luis Silva (IIMAS, UNAM, Mexico City)

**Title: **Inverse spectral problems for infinite mass-spring systems: necessary and sufficient conditions.

**Abstract:** TBC

##### 15 August 2012 at 14:10 in Room M/2.06

**Speaker:** Jussi Behrndt (TU Graz, Austria).

**Title:** Inverse problems of Calderon Type

**Abstract:** TBC