Mathematics Seminars Abstracts

13 December 2016 (PDE Seminar)

Enea Parini (Aix-Marseille, France)
The eigenvalue problem for the fractional p-Laplacian

In this talk I will present some results on the eigenvalue problem for the fractional p-­laplacian, a nonlinear, nonlocal differential operator. In particular, the minimization of the first and the second eigenvalues will be discussed, as well as the stability of the variational spectrum when the operator tends to its local counterpart, the p-­laplacian. The results were obtained in collaboration with Lorenzo Brasco (Ferrara), Erik Lindgren (Stockholm), Marco Squassina (Brescia).

22 November 2016 (PDE Seminar)

Birzhan Ayanbayev (Reading)
A pointwise characterisation of the PDE system of vectorial calculus of variation in L∞

After introducing the main objects of vectorial Calculus of Variations in L∞ in an accessible way, I will describe a new result which establishes that generalised solutions to the relevant PDE system describing critical points can be characterised via local affine variations of the energy functional. This is talk is based on recent joint work with N. Katzourakis.

8 November 2016 (PDE Seminar)

Andrew Comech (Texas A&M University and IITP, Moscow)
On stability of solitary waves in the nonlinear Dirac equation

We consider the point spectrum of non-selfadjoint Dirac operators which arise as linearizations at solitary wave solutions to the nonlinear Dirac equation. We show that in the model with the Soler-type nonlinearity, in the nonrelativistic limit (small-amplitude solitary waves with frequency near m), the spectral stability and linear instability results essentially parallel the case of the nonlinear Schrodinger equation. Besides analytic results, we present numerical computations of the spectrum in dimensions up to three. Results are partially based on the preprint "On spectral stability of the nonlinear Dirac equation" (with Nabile Boussaid), http://arxiv.org/abs/1211.3336 to appear in Journal of Functional Analysis.

1 November 2016 (PDE Seminar)

Alessia Kogoj (Salermo, Italy)
Liouville theorems for Hypoelliptic Partial Differential Operators on Lie Groups

We present several Liouville-type theorems for caloric and subcaloric functions on Lie groups in R^{N}. Our results apply in particular to the heat operator on Carnot groups, to linearized Kolmogorov operators and to operators of Fokker-Planck-type like the Mumford operator. An application to the uniqueness for the Cauchy problem is also showed. The results presented are obtained in collaboration with A. Bonfiglioli, E. Lanconelli, Y.Pinchover and S. Polidoro.

18 October 2016 (PDE Seminar)

Andrea Moiola (Reading)
Sobolev spaces on non-Lipschitz sets with application to boundary integral equations on fractal screens

The scattering of a time-harmonic acoustic wave by a planar screen with Lipschitz boundary is classically modelled by boundary integral equations (BIEs). If the screen is not Lipschitz, e.g. has fractal boundary, the correct Sobolev space setting to pose the problem is not obvious, because many of the relations between the standard definitions of Sobolev spaces on subsets of Euclidean space (e.g. restriction, completion of spaces of smooth functions, interpolation...) that hold in the Lipschitz case, fail to hold in general.

To extend the BIE framework to general screens, we study properties of the classical fractional Sobolev spaces (or Bessel potential spaces) on general non-Lipschitz subsets of Rn. In particular, we extend results about duality, s-nullity (whether a set with empty interior can support distributions with given Sobolev regularity), and about the equivalence or not between alternative space definitions, providing several examples. An interesting application is the approximation of variational (integral or differential) problems posed on fractal sets by problems
posed on prefractal approximations.

This is a joint work with S.N. Chandler-Wilde (Reading) and D.P. Hewett (UCL).

11 October 2016 (PDE Seminar)

Igor Velcic (Zagreb, Croatia)
Homogenization of thin structures in nonlinear elasticity - periodic and non-periodic

We will give the results on the models of thin plates and rods in nonlinear elasticity by doing simultaneous homogenization and dimensional reduction. In the case of bending plate we are able to obtain the models only under periodicity assumption and assuming some special relation between the periodicity of the material and thickness of the body. In the von K\'arm\'an regime of rods and plates and in the bending regime of rods we are able to obtain the models in the general non-periodic setting. In this talk we will focus on the derivation of the rod model in the bending regime without any assumption on periodicity.

10 May 2016

Sarah Filippi (Oxford)
A Bayesian nonparametric approach to testing for dependence between random variables

Nonparametric and nonlinear measures of statistical dependence between pairs of random variables are important tools in modern data analysis. In particular the emergence of large data sets can now support the relaxation of linearity assumptions implicit in traditional association scores such as correlation. Here we describe a Bayesian nonparametric procedure that leads to a tractable, explicit and analytic quantification of the relative evidence for dependence vs independence. Our approach uses Polya tree priors on the space of probability measures which can then be embedded within a decision theoretic test for dependence. Polya tree priors can accommodate known uncertainty in the form of the underlying sampling distribution and provides an explicit posterior probability measure of both dependence and independence. Well known advantages of having an explicit probability measure include: easy comparison of evidence across different studies; encoding prior information; quantifying changes in dependence across different experimental conditions, and; the integration of results within formal decision analysis.

6 May 2016

Joint Departmental and Analysis Seminar:

Nilima Nigam (Simon Fraser University, Vancouver)
Numerical approximation of the Laplace eigenvalues with mixed boundary data

Eigenfunctions of the Laplace operator with mixed Dirichet-Neumann boundary conditions may possess singularities, especially if the Dirichlet-Neumann junction occurs at angles $\geq \frac{\pi}{2}$. This suggests the use of boundary integral strategies to solve such eigenproblems. As with boundary value problems, integral-equation methods allow for a reduction of dimension, and the resolution of singular behaviour which may otherwise present challenges to volumetric methods.

In this talk, we present a novel integral-equation algorithm for mixed Dirichlet-Neumann eigenproblems. This is based on joint work with Oscar Bruno and Eldar Akhmetgaliyev (Caltech).

For domains with smooth boundary, the singular behaviour of the eigenfunctions at Dirichlet-Neumann junctions is incorporated as part of the discretization strategy for the integral operator. The discretization we use is based on the high-order Fourier Continuation method (FC).

For non-smooth (Lipschitz) domains an alternative high-order discretization is presented which achieves high-order accuracy on the basis of graded meshes.

In either case (smooth or Lipschitz boundary), eigenvalues are evaluated by examining the minimal singular values of a suitably stabilized discretesystem. This is in the spirit of the modification proposed by Trefethen and Betcke in the modified method of particular solutions.

The method is conceptually simple, and allows for highly accurate and efficient computation of eigenvalues and eigenfunctions, even in challenging geometries.

26 April 2016

Colva Roney-Dougal (St Andrews)
Groups, diagrams and geometries

The study of finitely-presented groups has been ongoing since the work of Hamilton in the 1850s - almost as long as group theory itself! This talk will be a gentle introduction to finitely-presented groups, with an emphasis on algorithms. I'll describe some finite diagrams, and some potentially infinite geometries, that are naturally associated with any finitely-presented group, and show how results about the diagrams and geometries prove structural results about the group, and vice versa.

19 April 2016

Athanasios Pantelous (Liverpool)
Stochastic response determination of linear and nonlinear dynamical systems with singular matrices

A generalization of random vibration techniques is developed for determining the stochastic response of linear and nonlinear dynamical systems with singular matrices. This system modelling can arise when a greater than the minimum number of coordinates is utilized, and can be advantageous, for instance, in cases of complex multi-body systems where the explicit formulation of the equations of motion can be a nontrivial task. In such cases, the introduction of additional/redundant degrees of freedom can facilitate the formulation of the equations of motion in a less labour intensive manner [1-2]. First, relying on the generalized matrix inverse theory and on the Moore-Penrose (M-P) matrix inverse, standard concepts, relationships, and equations of the linear random vibration theory are extended and generalized herein to account for systems with singular matrices. It was shown in [3] that adopting a state-variable formulation, equations governing the system response mean vector and covariance matrix are formed and solved. In this paper, an alternative frequency domain approach for stochastic response determination of linear systems with singular matrices is developed as well. Further, the above solution framework is generalized to account for nonlinear systems. To this aim, the potent statistical linearization technique [4] is generalized to account for systems with singular matrices.

Co-Authors: Vasileios C. Fragkoulis (University of Liverpool, UK) and Ioannis A. Kougioumtzoglou (Columbia University, USA)

Keywords: Random Vibration, Stochastic Dynamics, Nonlinear Systems, Singular Matrix, Moore-Penrose Generalized Inverse.

SELECTED REFERENCES

[1] de Falco, D., Pennestrì, E. and Vita, L. (2009), "Investigation of the influence of pseudoinverse matrix calculations on multibody dynamics simulations by means of the Udwadia-Kalaba formulation", J. AEROSP. ENG., 22(4), 365-372.

[2] Udwadia, F. E. and Schutte, A. D. (2010), "Equations of motion for general constrained systems in Lagrangian mechanics", ACTA Mech., 213(1), 111-129.

[3] Fragkoulis, V. C., Kougioumtzoglou, I. A. and Pantelous, A. A. (2016), "Linear random vibration of structural systems with singular matrices", ASCE J. Eng. Mech., 142(2), 04015081.

[4] Roberts, J. B., Spanos, P. D. (2003), Random vibration and statistical linearization, Dover Publications, New York.

15 March 2016

Kristoffer Van Der Zee (Nottingham)
Optimal discretisation in Banach spaces

Is it possible to obtain near-best approximations to solutions of linear operator equations in a general Banach space setting? Can this be done with guaranteed stability?

In this seminar, I address these questions by considering nonstandard, nonlinear, Petrov-Galerkin discretizations.

I build on ideas of residual minimization and the recent theory of optimal Petrov-Galerkin methods in Hilbert-space settings. I will demonstrate that the implementable version of the Petrov-Galerkin formulation is naturally related to a mixed method with a monotone nonlinearity. In the setting of certain special Banach spaces, I will prove optimal a priori error estimates (a la Cea / Babuska), with constants depending on the geometry of the involved Banach spaces.

-This is joint work with Ignacio Muga of Pontificia Universidad Catolica de Valparaiso.

08 March 2016

Paul Ledger (Swansea)
Describing Metal Detection Signals using Polarizability Tensors

Recent work has demonstrated that the response of a conductive object to a low frequency alternating magnetic field, in which the eddy current approximation is valid, can be approximated using a rank-2 tensor dependent only on the object's shape, size, conductivity and permeability.

We will present the connection between an asymptotic expansion of the perturbed magnetic field, as the object's size tends to zero, and the response that is predicted by engineers. Furthermore, we will discuss the properties of these tensors and their connection with the Polya-Szego tensor, which is associated with the presence of inclusions in electrical impedance tomography and characterising electromagnetic scattering from objects. The talk will comprise of new analysis, as well as numerical and experimental examples of these polarizability tensors, and the relevance of these findings to the location of low-metallic content anti-personnel land mines, and distinguishing them from other buried objects.

01 March 2016

Peter Ashwin (Exeter)
Design and quantification of noisy network attractors

Networks of coupled nonlinear systems may possess attractors that themselves are networks in phase space. In the context of neural systems, these attractors may be used to model internal states of a computational system that can change state according to time-dependent inputs which typically include noise as well as signals. In this talk I will describe how one can construct systems with arbitrary networks in phase space, as well as aspects of their response to noise.
(Joint work with Claire Postlethwaite, Auckland)

23 February 2016

Jozsef Lorinczi (Loughborough)
Regime change in the decay rates of ground states of a class of non-local Schrödinger operators

Non-local operators and related jump processes are currently attracting much attention at the interface area across functional analysis, PDE and probability, as well as in applications to mathematical physics and the natural sciences. Non-local Schrödinger operators include a potential term beside a kinetic component described by a non-local (pseudo-differential) operator. Feynman-Kac-type formulae allow to study evolution semigroups and spectral properties of such operators by running a suitable random process and averaging over its paths, thereby turning the operator problem into a problem of stochastics. I will consider a class of operators determined by Lévy jump processes with the property that the intensity of single large jumps dominates the intensity of all multiple large jumps (jump-paring class). Then I discuss the spatial decay properties of ground states (lowest lying eigenfunctions) obtained under a perturbation through a confining or decaying potential. It will be seen that for decaying potentials a sharp qualitative transition occurs in the fall-off patterns of ground states as the Lévy intensity is varied from sub-exponential to exponential order and beyond.

16 February 2016

Leandro Farina (Porto Alegre, Brazil)

Mapping submerged flat plates onto circular plates in the presence of water waves

The problem of interaction of water waves with a thin rigid plate can be reduced to a hypersingular (finite part) integral equation, when the plate is submerged. If the plate is circular, this equation can be solved for the jump in the velocity potential by a semi-analytical spectral method. The unknown is expanded in terms of certain orthogonal functions over the unit disc: a Fourier series in the azimuthal angle, with the coefficients expanded in terms of Gegenbauer polynomials. This approach has two virtues: all hypersingular integrals are evaluated analytically and the edge condition on the plate is satisfied automatically. In this talk, we establish a convergence proof and show how this method can be extended to the case where the plate is non-circular. Numerical results are presented for a class of nearly circular plates. Extensions of the method to the cases of porous plates and of a plate under an ice cover will be proposed.

09 February 2016

Carola-Bibiane Schönlieb (Cambridge)
PDE constrained optimisation for learning the optimal image de-noising model

We will discuss the concept of an optimal image de-noising model made up of a data discrepancy term and an appropriate regularisation. Starting with a generic, parametrised variational approach consisting of different regularisation functionals and discrepancy terms, we phrase the quest for an optimal model as seeking those parameters that return an optimal subset of the generic model, optimal for a training set of exemplar images. Formally, this approach constitutes a bilevel optimisation with a nonlinear PDE as constraint. The analysis and numerical realisation of this approach will be presented, furnished with various examples.

02 February 2016

Filip Rindler (Warwick)
On the structure of PDE-constrained measures and applications

In many physically interesting situations, singularities arise in solutions to PDEs or minimization problems of the calculus of variations. I will discuss how sometimes one can analyse the "shape" of such singularities, which often yields both physically and mathematically interesting conclusions. More specifically, I will investigate the singular part of vector-valued measures satisfying a linear PDE constraint and present a recent general structure theorem for such singularities. As applications, we obtain a simple proof of Alberti's rank-one theorem on the shape of derivatives of functions with bounded variation (BV), its extensions to functions of bounded deformation (BD), which are important in elasto-plasticity theory, and a structure theorem for the singular part of measures whose divergence is also a measure. Via some further geometric consequences of our theorem, we can also prove the conjecture that if every Lipschitz function is differentiable almost everywhere with respect to a measure ("generalized Rademacher theorem"), then this measure has to be absolutely continuous. This is joint work with Guido De Philippis (ENS-Lyon).

19 January 2016

Richard Norgate (Lloyds)
Some mathematical applications in banking

Professor Richard Norgate is the Group Director of Analytics and Modelling for Lloyds Banking Group. Richard will give a brief overview of Lloyds Banking Group, and highlight some of the major areas where maths (in its broadest sense) is used across the Group. Richard will then present some case studies of real examples, including credit risk and fraud analytics.

12 January 2016

Jose Antonio Carillo (Imperial College)
Minimizing interaction energies

I will start by reviewing some recent results on qualitative properties of local minimizers of the interaction energy to motivate the main topic of my talk: to discuss global minimizers. We will show the existence of compactly supported global miminizers under quite mild assumptions on the potential in the complementary set of classical H-stability in statistical mechanics. A strong connection with the classical obstacle problem appears very useful when the singularity is strong enough at zero.

08 December 2015

Bas Lemmens (Kent)
Isometries of Hilbert geometries and Jordan Banach algebras

Hilbert geometries are metric spaces that generalise Klein's model of the real hyperbolic space. They were introduced by Hilbert in finite dimensional spaces, but can also be defined in infinite dimensions. In finite dimensions it is known that the structure of the isometry group of these metric spaces is closely linked to the theory of symmetric cones, i.e. Euclidean Jordan algebras. In infinite dimensions, however, the isometries are not so well understood. In this talk I will give a survey of this area and discuss some recent progress.

01 December 2015

Dmitri Finkelshtein (Swansea)
Travelling waves and long-time behaviour in a doubly nonlocal Fisher-KPP equation

We consider a Fisher-KPP-type equation, where both diffusion and nonlinear parts are nonlocal, with anisotropic probability kernels. Under minimal conditions on the coefficients, we prove existence, uniqueness, and uniform space-time boundedness of a positive solution. We investigate existence, uniqueness, and asymptotic behaviour of monotone traveling waves for the equation. We also describe the existence and main properties of the front of propagation. The talk is based on a joint paper with Yu.Kondratiev and P.Tkachov (Bielefeld, Germany), arXiv:1508.02215.

24 November 2015

Beatrice Pelloni (Reading)
The semi-geostrophic system: analytical tools and a survey of results

I will discuss a particular reduction of the Euler equation of fluid dynamics, known as the semi-geostrophic system. This system has been around for a long time and was first derived by physicists as a model for the formation of fronts. In the last twenty years, the availability of new rigorous tools to solve it analytically has concentrated the attention also of mathematicians. I will describe these tools briefly, and survey the results obtained in recent years, as well as their importance for applications.

17 November 2015

Johannes Zimmer (Bath)
Nonlinear diffusion: From particles to thermodynamics

In many applications, one has a 'small-scale' model which is simple but computationally intractable, and wishes to derive an effective (computable) 'large-scale' description. This talk will study this problem of scale-bridging from a thermodynamic perspective. In particular, gradient flows will be explained; they can be thought of as steepest descent/ascent of an entropy. Over the last two decades the associated theory of metric gradient flows has flourished and the talk will summarise some now-classic key results. We will then discuss the interplay between particle models and their thermodynamic description at hand of a class of nonlinear diffusion equations. It will first be shown how an underlying particle model can reveal an underlying (geo-)metric structure of the governing PDE, notably a gradient flow setting for a class of nonlinear diffusion equations. We will also sketch the derivation of a more general thermodynamic structure (a separation of conservative and dissipative elements of the governing equations) from a different particle model.

10 November 2015

Jan Christensen (Oxford)
The Morse-Sard theorem, generalized Luzin property and level sets for Sobolev functions

Many classical results from multivariate calculus can be generalized to suitable Sobolev functions that need not even be everywhere differentiable. In this talk we discuss some new results that have been obtained in joint work with Jean Bourgain (Princeton) and Mikhail Korobkov(Novosibirsk).

03 November 2015

Nikolai Saveliev (Miami, USA)
End-periodic index theory and non-Kahler surfaces

We extend the Atiyah-Patodi-Singer index theorem for Dirac type operators from the context of manifolds with product ends to that of manifolds with periodic ends. Our theorem expresses the index in terms of a new periodic eta-invariant, which equals the Atiyah-Patodi-Singer eta-invariant in the product end setting. I will spend most of the talk discussing a family of non-product end examples whose ends are modeled on the (non-Kahler) Inoue surfaces. Calculating the periodic eta-invariant of the Dirac-Dolbeault operator in these examples reduces to solving a family of Schrodinger equations on the real line. This is a joint project with T. Mrowka and D. Ruberman.

27 October 2015

Ali Taheri (Sussex)
From Weyl chambers to Weyl's law

The spectral counting function of an elliptic operator on a compact manifold can be seen as the trace of its spectral measure and thus relates directly to the functional calculus of the operator. In the case of the Laplace-Beltrami operator, the wave equation approach pioneered by Hormander provides deep insights into the relation between the clustering of the spectrum on the one hand and the periodicity of the geodesic flow on the other. In this talk I will discuss some new results for the case of compact Lie groups through the study of their root systems and highest weight theory and describe the
connection with some geometric problems in the calculus of variations. The talk is based on joint work with Charles Morris at Sussex.

20 October 2015

Anna Cherubini (visiting Imperial College London)
A study of the stochastic resonance as a random dynamical system

We study a standard model for the stochastic resonance from the point of view of dynamical systems. We present a framework for random dynamical systems with nonautonomous deterministic forcing and we prove the existence of an attracting random periodic orbit for a class of one-dimensional systems with a time-periodic component. In the case of the stochastic resonance, we can derive an indicator for the resonant regime.

 

09 July 2015

Distribution of Resonances in Scattering by Thin Barriers
by Jeffrey Galkowski (Berkeley, USA)

We consider resonances for operators of the form − Δ + V ⊗ δ∂Ω and − Δ − V ⊗ ∂v δ∂Ω where Ω ⊂ ℝd is a bounded domain. These operators are models for quantum corrals as well as other systems with thin barriers. We give a bound on the size of the resonance free region for very general Ω. In the case that ∂Ω is strictly convex, we give a dynamical characterization of the resonance free region that is generically sharp and can be thought of as a Sabine Law. The key technical tool in the analysis is a precise understanding of boundary layer operators at high energy.

02 June 2015

Gaussian solitary waves in granular chains
by Dmitry Pelinovsky (McMaster University, Hamilton, Canada)

I will review recent results on a class of fully-nonlinear Fermi-Pasta-Ulam (FPU) lattices, consisting of a chain of particles coupled by fractional power nonlinearities. This class of systems incorporates a classical Hertzian model describing acoustic wave propagation in chains of touching beads in the absence of precompression.

We analyze the propagation of localized waves in the asymptotic limit, when solutions of the FPU lattice varying slowly in space and time are described by the logarithmic KdV equation. The logarithmic KdV equation possesses linearly orbitally stable Gaussian solitary wave solutions. In particular, I will discuss well-posedness of the logarithmic KdV equation, nonlinear stability of the Gaussian solitary waves, and the justification of the KdV approximation for the Hertzian-FPU lattice.

26 May 2015 (2-3pm)

Random Schroedinger Operators on the Lattice
by Raffael Hagger (Hamburg)

After the introduction of random operators to nuclear physics by Eugene Wigner in 1955, random quantum systems have grown in popularity, one of the more popular models being the one studied by Anderson in 1977. Anderson was interested in spin diffusion (also known as "hopping quantum particles") on multidimensional lattices with random potentials. The Hamiltonian of this system can be considered as an infinite tridiagonal matrix with a randomly distributed main diagonal and constant sub- and superdiagonals. In this talk we are going to consider more general (usually non-self-adjoint) random operators on the one-dimensional lattice. In particular, we show that the numerical range of an arbitrary tridiagonal random operator can be computed quite easily. While for self-adjoint operators this result is rather boring, it is all the more surprising in the non-self-adjoint case and has interesting consequences. We also want to consider one particular random operator, which is called the Feinberg-Zee random hopping matrix. This operator can be used to describe a quantum particle hopping on a one-dimensional lattice and randomly changing its spin. For this particular model we present some recent results about upper and lower bounds to the spectrum (in addition to the upper bound provided by the numerical range).

26 May 2015 (12-1pm)

Exactly-solvable non-Markovian dynamic network
by Enrico Scalas (Sussex)

Non-Markovian processes are widespread in natural and human-made systems, yet explicit modelling and analysis of such systems is underdeveloped. In this letter we consider a dynamic network with random link activation and deletion (RLAD) with non-exponential inter-event times. We study a semi-Markov random process when the inter-event times are heavy tailed Mittag-Leffler distributed, thus considerably slowing down the corresponding Markovian dynamics and study the system far from equilibrium. We derive an analytically and computationally tractable system of forward equations utilizing the Caputo derivative for the probability of having a given number of active links in the network. As an example showing the effects of non-Markovianity, the dynamic network is coupled with a susceptible-infected-susceptible (SIS) spreading dynamics leading to more persistent epidemics. The convergence to equilibrium is discussed in terms of the mixing time of the embedded chain and the difference with the Markovian case is highlighted. The novelty of our approach lies in showing a rigorous route from a non-Markovian model to the corresponding Kolmogorov-like equations and their analytical treatment. This is joint work with Nikos Georgiou and Istvan Z. Kiss.

19 May 2015

On Elliptic Hybrid Inverse Problems
by Yves Capdeboscq (Oxford)

In this talk I will discuss some analysis questions arising in hybrid inverse problems, which consist in parameter identification in (elliptic) PDEs using some local information about the solutions. The focus of this talk will be of the determination of suitable boundary conditions a priori, from general  properties of elliptic second order boundary value problems, rather than the resolution of the inverse problem itself.

12 May 2015

Everywhere discontinuous anisotropy of thin periodic composite plates
by Mikhail Cherdantsev (Cardiff)

We consider an elastic periodic composite plate in the full bending regime, i.e. when the displacement of the plate is of finite order. Both the thickness of the plate h  and the period of the composite structure ε are small parameters. We start from the non-linear elasticity setting. Passing to the limit as h, ε → 0 we carry out simultaneous dimension reduction and homogenisation to obtain an effective limit elastic functional which describes the asymptotic properties of the composite plate. We show, in particular, that in the regime h << ε2 the limit elastic functional is discontinuously anisotropic in every direction of bending. This remarkable property (suggesting that the corresponding composite plate can be referred to as metamaterial) is due to the in-limit linearisation of the bending deformations and the multi-scale interaction.

05 May 2015

The black hole stability problem
by Gustav Holzegel (Imperial College)

A fundamental open problem in general relativity is to establish the non-linear stability of the Kerr-family of black holes. After discussing the definitive decay results obtained in the context of the free scalar wave equation on Kerr-spacetimes (and the mechanisms that underlie them) I will outline our recent proof of the linear stability of the Schwarzschild solution under gravitational perturbations. The latter is joint work with Dafermos and Rodnianski

28 April 2015

Regularity and sufficient conditions for strong local minimality
by Judith Campos-Cordero (Augsburg)

An outstanding question in the Calculus of Variations is that of finding sufficient conditions on extremals to ensure that they furnish strong local minimizers. Grabovsky and Mengesha showed that, for quasiconvex integrands, assuming an a priori C1-regularity on the extremals is enough. In this talk we present an alternative proof of this fact and we discuss related regularity results for local and global minimizers. Part of this work has been done in collaboration with Jan Kristensen.

21 April 2015

The logarithmic fast diffusion equation from a geometric viewpoint
by Peter Topping (Warwick)

The logarithmic fast diffusion equation is a heavily studied nonlinear heat equation that originally arose in physics. I will explain how this also turns out to be an important equation in geometry. This geometric picture gives us a lot of intuition about the PDE, and leads to a resolution of the well-posedness problem, as I will explain.

24 March 2015

Stabilised finite element methods for non symmetric, non coercive and ill-posed problems
by Erik Burman (University College London)

In numerical analysis the design and analysis of computational methods is often based on, and closely linked to, a well-posedness result for the underlying continuous problem. In particular the continuous dependence of the continuous model is inherited by the computational method when such an approach is used. In this talk our aim is to design a stabilised finite element method that can exploit continuous dependence of the underlying physical problem without making use of a standard well-posedness result such as Lax-Milgram's Lemma or The Babuska-Brezzi theorem. This is of particular interest for inverse problems or data assimilation problems which may not enter the framework of the above mentioned well-posedness results, but can nevertheless satisfy some conditional stability results. First we will discuss non-coercive elliptic and hyperbolic equations where the discrete problem can be ill-posed even for well posed continuous problems and then we will discuss the linear elliptic Cauchy problem as an example of an ill-posed problem where there are conditional stability results available that are suitable for the framework that we propose.

17 March 2015

Spectral problems for interior transmission eigenvalues
by Vesselin Petkov (Université de Bordeaux)

Given a bounded domain with smooth boundary the interior transmission eigenvalues (ITE) are related to the scattering operator of the wave equation with index of refraction n(x). The inverse scattering sampling method of Colton and Kress breaks down if the far-field operator F(k) determined by the scattering amplitude has a non trivial kernel or cokernel. In the later case the real number k is an (ITE). The (ITE) are eigenvalues of a non self-adjoint operator which is not parameter-elliptic and the problems with (ITE) have been examined very intensively in the last few years. In this talk we will present results concerning two problems: (A) eigenvalues-free regions in the complex plan, (B) Weyl formula with remainder for the counting function of the complex (ITE). Both problems are related and the proofs are based on a fine semi-classical analysis. This is a joint work with G. Vodev.

3 March 2015

Multilevel Monte Carlo methods
by Mike Giles (Oxford)

Multilevel Monte Carlo is an approach to reducing the cost of Monte Carlo simulations by using a sequence of cheap coarse approximations as control variates for more costly fine approximations. I will introduce the basic ideas, emphasising their simplicity and generality, and give a couple of numerical examples coming from mathematical finance. I will then give an overview of its use in a wide variety of other application areas, before talking about one of my current projects on the stochastic simulation of long-chain FENE molecules in a solvent.

24 February 2015

Stochastic Eigenvalue Problems
by Harri Hakula (Aalto University)

Given the recent success of stochastic finite element methods (SFEM) it is natural to consider analogous stochastic or multiparametric eigenvalue problems. The aim of this talk is to illuminate the problems arising from the parameter-dependence of the eigenpairs, such as ambiguity of the smallest eigenvalue. Both Galerkin and collocation methods are considered. First the state-of-the-art in stochastic eigensolution of the Laplacian is reviewed. Finally, a special problem of linear elasticity, vibrations of shells of revolution, is used to discuss the problem of multiple smallest eigenvalues. The approach is numerical with emphasis on well-designed experiments.

17 February 2015

Multiple domination parameters in graphs and their applications
by Anush Poghosyan (Reading)

Nowadays domination is an area in graph theory with an extensive research activity and various real-world applications. Dominating sets are sets of vertices that "are near" (dominate) all the vertices of a graph. Domination problem is known as NP-complete for arbitrary graphs. In the first part of my talk I will focus on introducing the concept of domination and multiple domination. I will then present the probabilistic method and results for upper bounds for multiple domination parameters. In the second part of my talk I will discuss the problem of weighted domination and its application in social networks.

10 February 2015

On the positivity preservation of finite element based discrete schemes for the heat equation
by Panagiotis Chatzipantelidis (University of Crete)

We consider the initial boundary value problem for the homogeneous heat equation, with homogeneous Dirichlet boundary conditions. By the maximum principle the solution is nonnegative for positive time if the initial data are nonnegative. Our purpose is to discuss analogues of this property for some finite element methods, based on piecewise linear finite elements, including, in particular, the Standard Galerkin (SG) method, the Lumped Mass (LM) method, and the Finite Volume Element (FVE) method. We also provide numerical examples that illustrate and complement our findings.

03 February 2015

Trefttz type methods for the acoustic wave equation
by Lehel Banjai (Heriot-Watt University, Edinburgh)

In recent years there has been much interest in using non-polynomial basis functions to discretize wave propagation problems in the frequency domain.

In particular the Trefttz method uses local solutions of the Helmholtz equation as basis functions. The motivation behind this is to reduce the number of degrees of freedom per wavelength required to obtain accurate results for high-frequency problems. As these methods have proved very successful in practice it is a natural question to ask whether they can be extended and whether they can be equally successful in the time-domain.

In this talk we will present two possible ways of extending Trefttz methods to the time-domain. A natural way of doing this is by a time-space discontinuous Galerkin method. We will describe one such scheme, discuss the analysis of it, and provide the results of some preliminary numerical experiments. We will also introduce a promising alternative method of lines approach.

27 January 2015

Efficient solvers for unsteady incompressible flow problems: hydrodynamic stability and UQ
by David Silvester (Manchester)

The first part of the talk reviews recent developments in the design of robust solution methods for the Navier-Stokes equations modelling incompressible fluid flow. There are two building blocks in our solution strategy. First, an implicit time integrator that uses a stabilized trapezoid rule with an explicit Adams-Bashforth method for error control, and second, a robust Krylov subspace solver for the spatially discretized system.

The focus of the second part of the talk is on uncertainty quantification. We discuss two problems: a) stochastic Galerkin approximation of a fluid with an uncertain viscosity (or more generally, an uncertain Reynolds number), and b) stochastic collocation approximation of critical eigenvalues of the linearised operator associated with the transition from steady flow to vortex shedding behind an obstacle in a channel. Our results confirm that classical linear stability analysis is an effective way of assessing the stability of such a flow.

20 January 2015

Local spectral asymptotics for second-order perturbations of the Landau Hamiltonian
by Georgi Raikov (PUC, Santiago de Chile)

I will consider perturbations of the Landau Hamiltonian by second-order differential operators. I will discuss the asymptotic behaviour of the discrete spectrum of the perturbed operator near the Landau levels, for perturbations whose coefficients admit power-like or exponential decay at infinity, or have a compact support.

This is a joint work with Tomas Lungenstrass (PUC).

13 January 2015

Fast algorithms for compressed sensing and matrix completion
by Jared Tanner (Oxford)

The essential information in large data sets is typically contained in a low dimensional subspace of the space in which the data is observed or collected. For instance, natural images are highly compressible in standard wavelet bases and many matrices are well approximated by a low rank approximation. Prior knowledge that the data has an underlying simplicity allows data to be acquired at a rate proportional to the information content as opposed to the ambient space in which the data is represented. Namely, signals/images which are known to be compressible in a given basis can be fully recovered from their inner product with few random vectors, and matrices which are low rank can be determined from a subset of their entries.

We discuss recent advances in these topics of compressed sensing and matrix completion, focusing on maximally sparse sampling operators and recovery using algorithms that directly solve the natural non-convex formulation of these questions. In particular, we present a new algorithm for compressed sensing and matrix completion based on classical numerical linear algebra algorithms.

This work is joint with Jeffrey D. Blanchard (Grinnell College) and Ke Wei (HKUST).

09 December 2014

Scaling Limits in Computational Bayesian Inversion
by Claudia Schillings (Warwick)

In this talk, preconditioning strategies for sparse, adaptive quadrature methods for computational Bayesian inversion of operator equations with distributed uncertain input parameters will be presented. Based on sparsity results of the posterior, error bounds and convergence rates of dimension-adaptive Smolyak quadratures can be shown to be independent of the parameter dimension, but the error bounds depend exponentially on the inverse of the covariance of the additive, Gaussian observation noise. We will discuss asymptotic expansions of the Bayesian estimates, which can be used to construct quadrature methods combined with a curvature-based reparametrization of the parametric Bayesian posterior density near the (assumed unique) global maximum of the posterior density leading to convergence with rates independent of the number of parameters as well as of the observation noise variance.

02 December 2014

Bifurcations of random dynamical systems
by Martin Rasmussen (Imperial College London)

Despite its importance for applications, relatively little progress has been made towards the development of a bifurcation theory for random dynamical systems. In this talk, I will demonstrate that adding noise to a deterministic mapping with a pitchfork bifurcation does not destroy the bifurcation, but leads to two different types of bifurcations. The first bifurcation is characterized by a breakdown of uniform attraction, while the second bifurcation can be described topologically. Both bifurcations do not correspond to a change of sign of the Lyapunov exponents, but I will explain that these bifurcations can be characterized by qualitative changes in the dichotomy spectrum and collisions of attractor-repeller pairs.

This is joint work with M. Callaway, T.S. Doan, J.S.W Lamb (Imperial College) and C.S. Rodrigues (MPI Leipzig).

25 November 2014

Time stepping in weather and climate models
by Paul Williams (Reading)

The leapfrog (second-order centred-difference) time-stepping scheme is commonly used in models of weather and climate.

The Robert-Asselin filter is used in conjunction with it, to damp the computational mode. Although the leapfrog scheme makes no amplitude errors when integrating linear oscillations, the Robert-Asselin filter introduces first-order amplitude errors. The RAW filter, which was recently proposed as an improvement, eliminates the first-order amplitude errors and yields third-order amplitude accuracy. This development has been shown to significantly increase the skill of medium-range weather forecasts. However, it has not previously been shown how to further improve the accuracy by eliminating the third- and higher-order amplitude errors.

This presentation will show that leapfrogging over a suitably weighted blend of the filtered and unfiltered tendencies eliminates the third-order amplitude errors and yields fifth-order amplitude accuracy. It will also show that the use of a more discriminating (1, -4, 6, -4, 1) filter instead of a (1, -2, 1) filter eliminates the fifth-order amplitude errors and yields seventh-order amplitude accuracy. Other related schemes are obtained by varying the values of the filter parameters, and it is found that several combinations offer an appealing compromise of stability and accuracy.

The proposed new schemes are shown to yield substantial forecast improvements in a medium-complexity atmospheric general circulation model. They appear to be attractive alternatives to the filtered leapfrog schemes currently used in many weather and climate models.

18 November 2014

The plasmonic eigenvalue problem
by Daniel Grieser (Oldenburg)

Plasmonics is the study of the interaction of electromagnetic waves (e.g. light) and free electrons in a metal. Mathematically this leads to a number of interesting problems involving the Laplace equation and Dirichlet-Neumann operators. I will explain the problem and some results obtained with the help of microlocal analysis, which I will also explain in the talk.

11 November 2014

Concentrated englacial shear over rigid basal ice, West Antarctica: implications for modelling and ice sheet flow
by Martin Siegert (Imperial College)

Basal freeze-on, deformation and ice crystal fabric re-organisation have been invoked to explain thick, massive englacial units observed in the lower ice column of the Antarctic and Greenland ice sheets. Whilst recognised as having very different rheological properties to overlying meteoric ice, studies assessing the impact of basal units on the large-scale flow of an ice sheet have so far been limited.

We report the discovery of a previously unknown, extensive (100 km long, >25 km wide, and up to 1 km thick) englacial unit of near-basal ice beneath the onset zone of the Institute Ice Stream, West Antarctica. Using radio-echo sounding, we describe the form and physical characteristics of this unit, and its impact on the stratigraphy and internal deformation of overlying ice. The lower englacial unit, characterised by a highly-deformed to massive structure, is inferred to be rheologically distinct from the overlying ice column. The overlying ice contains a series of englacial 'whirlwind' features, which are traceable and exhibit longitudinal continuity between flow orthogonal radar lines. Whirlwinds are the representation of englacial layer buckling, so provide robust evidence for enhanced ice flow. The interface between the primary ice units is sharp, abrupt and 'wavy'. Immediately above this interface, whirlwind features are deformed and display evidence for flow-orthogonal horizontal shear, consistent with the deformation of the overlying ice across the basal ice unit. This phenomenon is not a local process, it is observed above the entirety of the basal ice, nor is it dependent on flight orientation, direction of shear is consistent regardless of flight orientation.

These findings have clear significance for our understanding and ability to realistically model ice sheet flow. Our observations suggest that, in parts of the onset zone of the Institute Ice Stream, the flow of the ice sheet effectively ignores the basal topography. Instead, enhanced ice flow responds to a pseudo-bed, with internal deformation concentrated and terminating at an englacial rheological interface between the upper ice sheet column and the massive basal ice.

Our results demonstrate that we may need to: (i) adapt numerical models of those parts of the ice sheet with extensive and thick basal ice units; and (ii) carefully reconsider existing schematic models of ice flow, to incorporate processes associated with concentrated englacial shear.

28 October 2014

Multicentric calculus: still another look at φ(A)  
by Nevanlinna Olavi (Aalto University, Finland)

Abstract (pdf)

21 October 2014

Exploiting approximation properties: Improving filtering and discontinuity detection
by Jennifer Ryan (East Anglia)

Much work goes into creating effective numerical approximations for modeling different physical phenomena such as gas dynamics, climate change, materials, etc. These approximations are usually tuned to the model. However, an important question to ask is how to improve existing numerical approximations. In this talk, we present a generalized discussion of the numerical method concentrating on a basic concept: exploiting the existing approximation properties. We focus on the discontinuous Galerkin (DG) method and show how rewriting the existing approximation can be useful. The discontinuous Galerkin method uses a piecewise polynomial approximation to the variational form of a PDE. It uses polynomials up to degree k for a k+1 order accurate scheme.

In the first portion of the talk, we focus on better discontinuity detection during time integration by rewriting the approximation. The DG method can be related to a multi-wavelet decomposition. We demonstrate that this multi-wavelet expansion allows for more accurate detection of discontinuity locations. Further, there are many areas in which the multi-wavelet relation to the DG approximation could be useful. For example, in determining mesh adaptivity, adaptive time-stepping, or approximations with multi-scale phenomena.

The second portion of the talk will concentrate on useful superconvergence. Superconvergence is the phenomena of a method to converge faster than expected. For example, discontinuous Galerkin (DG) methods are known theoretically to have order k+1 convergence. However, at specific points within an element, the method has order 2k+1 convergence. Further, DG has the same order of convergence in a negative-order norm, which is achieved through the superconvergent fluxes. Using this information we can construct solutions with reduced errors. We concentrate on a B-spline convolution filters known as a Smoothness-Increasing Accuracy-Conserving (SIAC) filters which mprove both smoothness and accuracy of the approximation. Specifically, for linear hyperbolic equations it can improve the order of accuracy of a DG approximation from k+1 to 2k+1 and smoothness to k-1. This portion of the talk will discuss the barriers in making superconvergence extraction techniques useful for applications, specifically focusing on SIAC filtering for discontinuous Galerkin approximations. We discuss how this property is underutilized, and how it could help is such areas as filtering for visualization.

14 October 2014

Toeplitz determinants with merging singularities
by Igor Krasovsky (Imperial College London)

We discuss the asymptotic behaviour of a Toeplitz determinant whose symbol possesses 2 singularities on the unit circle which can approach each other as the dimension of the determinant grows.

We devote a special attention to some interesting applications. The applications we discuss involve asymptotic estimation of integrals of Toeplitz determinants with respect to the position of singularities on the unit circle. We prove a conjecture of Dyson about a Bose gas and a conjecture of Fyodorov and Keating on the behaviour of the second moment of the characteristic polynomials of random matrices.

The talk is based on a joint work with Tom Claeys.

07 October 2014

Mathematics and Climate: A New Partnership
by Hans G. Kaper (Georgetown University, Washington, DC)

Climate is an emerging area of research in the mathematical sciences, part of a broader portfolio that addresses issues of complexity and sustainability. The stakes are high, decision makers have more questions than we can answer, and mathematical models and statistical arguments play a central role in assessment exercises. All this to indicate that we better get involved and apply our disciplinary expertise to the major challenge of our time. In this talk I will identify some problems of current interest in climate science and indicate how, as mathematicians, we can find inspiration for new applications.

30 September 2014

Preconditioners for two-sided eigenvalue problems and applications to model order reduction
by Melina Freitag (Bath)

Convergence results are provided for inexact two-sided inverse and Rayleigh quotient iteration. Moreover, the simultaneous solution of the forward and adjoint problem arising in two-sided methods is considered and the successful tuning strategy for preconditioners is extended to two-sided methods.

This is joint work with Patrick Kuerschner (MPI Magdeburg, Germany).

03 June 2014

Dispersionless integrable systems in 3D and Einstein-Weyl geometry
by Evgeny Ferapontov (Loughborough)
(based on joint work with B Kruglikov)

For several classes of second order dispersionless PDEs, we show that the symbols of their formal linearizations define conformal structures which must be Einstein-Weyl in 3D (or self-dual in 4D) if and only if the PDE is integrable by the method of hydrodynamic reductions.
This demonstrates that the integrability of dispersionless PDEs can be seen from the geometry of their formal linearizations.

20 May 2014

Systematic Coarse Graining Soft Matter Dynamics
by Patrick Ilg (ETH Zurich and Reading)

Soft matter systems typically respond easily to external perturbations, leading to strong anisotropic properties. On a microscopic level, these systems are successfully described by interacting many-particle systems obeying Newtonian or Langevin dynamics for translational and rotational degrees of freedom. In order to capture the long-time, large-length scale behavior, coarse-grained descriptions of the orientational ordering and dynamics are urgently needed due to the huge computational effort of solving the high-dimensional microscopic dynamics. Here, we discuss a general method based on projection operators and demonstrate some steps in order to systematically and consistently connect microscopic and coarse-grained levels of description.

13 May 2014

Random infinite graphs and applications to quantum gravity
by Georgios Giasemidis (Reading)

In the first part of of the talk I will speak about random walks on random infinite graphs, such as the random comb, the generic random tree and the uniform infinite causal triangulation. The two latter examples are also related to a critical Galton-Watson process conditioned to non-extinction. Their properties focusing on their dimensionality, e.g. the spectral and Hausdorff dimensions, will be presented. In the second part of the talk I will introduce the problem of quantum gravity and explain how the previous ideas can be applied to it

 

06 May 2014

Summary statistics for approximate Bayesian computation
by Dennis Prangle (Reading)

Parameter inference is a central problem within statistics. Given a family of probability distributions indexed by a vector of parameters, which are the most plausible to have produced some observed data? There are well developed methods based on calculating the "likelihood function", the probability of the observations as a function of the parameters, for various parameter choices. However, increasing model complexity means such calculations are not always possible, while simulation from the model given parameters is often still possible.

"Likelihood-free" methods, such as approximate Bayesian computation, exploit this to perform approximate inference by finding parameters which produce simulations sufficiently similar to the observations.

This talk introduces the area and discusses research I've been involved in on how best to define "sufficiently similar" based on summary statistics.

29 April 2014

When do polynomial equations possess integer solutions?
by Sean Prendiville (Reading)

The existence/non-existence of integer solutions to polynomial equations is a notoriously difficult problem in full generality, including, as it does, Fermat's last theorem and Hilbert's tenth problem. A somewhat naive approach to such a problem would be to first consider the (easier) task of solving the corresponding congruence modulo every positive integer, then try and 'stitch' these congruence solutions together to form an integer solution. We will discuss the successes of this approach over the last 100 years, meeting colourful characters such as Minkowski, Hardy, Littlewood and Ramanujan. For a wide class of equations we will see how the seemingly unrelated theory of Fourier analysis allows one to prove that infinitely many integer solutions exist, without ever actually constructing one.

11 March 2014

Harnack's inequality for nonhomogeneous elliptic equations 
by Vesa Julin (Jyvaskyla, Finland)

Harnack inequality is one of the cornerstones of the theory of elliptic partial differential equations. It was proven by Moser for linear divergence form operators and by Krylov and Safonov for nondivergence form operators. In my talk I introduce a generalization of Harnack inequality for equations which are not homogeneous.

04 March 2014

Some diffuse interface models for compressible liquid vapor flows and their relations
by Jan Giesselmann (Stuttgart)
Abstract (PDF)

25 February 2014

Epidemics and elections: the importance of demographic noise in adaptive networks
by Tim Rogers (Bath)

Adaptive networks are models of complex systems in which the structure of the interaction network changes on the same time-scale as the status of the nodes. For instance, consider the spread of a disease over a social network that is changing as people try to avoid the infection. In this talk I will try to persuade you that demographic noise (random fluctuations arising from the discrete nature of the components of the network) plays a major role in determining the behaviour of these models.

18 February 2014

Mathematical Methods for Multiscale Modelling in Molecular, Cell and Population Biology
by Radek Erban (Oxford)

I will discuss methods for spatio-temporal modelling in molecular, cell and population biology. Three classes of models will be considered:

(i) microscopic (molecular-based, individual-based) models which are based on the simulation of trajectories of molecules (individuals) and their localized interactions (for example, reactions);

(ii) mesoscopic (lattice-based) models which divide the computational domain into a finite number of compartments and simulate the time evolution of the numbers of molecules (numbers of individuals) in each compartment; and

(iii) macroscopic (deterministic) models which are written in terms of mean-field reaction-diffusion-advection partial differential equations (PDEs) for spatially varying concentrations.

In the first part of my talk, I will discuss connections between the modelling frameworks (i)-(iii). I will consider chemical reactions both at a surface and in the bulk. In the second part of my talk, I will present hybrid (multiscale) algorithms which use models with a different level of detail in different parts of the computational domain.

The main goal of this multiscale methodology is to use a detailed modelling approach in localized regions of particular interest (in which accuracy and microscopic detail is important) and a less detailed model in other regions in which accuracy may be traded for simulation efficiency. I will also discuss hybrid modelling of chemotaxis where an individual-based model of cells is coupled with PDEs for extracellular chemical signals.

11 February 2014

Modelling paleoclimate with a simple box model
by Andrew Fowler (Oxford)

We argue that, while Milanković variations in solar radiation undoubtedly have a major influence on the timing of the Quaternary ice ages, they are partly incidental to their underlying causes. Based on observations of the significance of CO2, we propose a conceptually simple (but complicated in detail) energy balance type model which has the ability to explain the underlying oscillatory nature of ice  ages. We are led to develop amodel which combines ice sheet growth and atmospheric energy balance with ocean carbon balance. In order to provide results which mimic the basic features of the observations, we develop novel hypotheses as follows. The succession of the most recent ice ages can be explained as being due to an oscillation due to the interaction of the growing northern hemisphere ice sheets and proglacial lakes which form as they migrate south. The CO2 signal which faithfully follows the proxy temperature signal can then be explained as being due to a combination of thermally activated ocean biomass production, which enables the rapid CO2 rise at glacial terminations, and enhanced glacial carbonate weathering through the exposure of continental shelves, which enables CO2 to passively follow the subsequent glacial cooling cycle. Milanković variations provide for modulations of the amplitude and periods of the resulting signals.

04 February 2014

Multivariate extremes value methods for univariate and spatial flood risk assessment
by Jonathan Tawn (Lancaster)

The talk will cover two distinct problems in flood risk assessment: the estimation of the distribution of flood peaks at a site and the estimation of the distribution of "financial loss" over a region from flooding. Approaches based on univariate extreme value theory exist for each of these, with the one for flood peaks being very widely used. Both of these problems are essentially multivariate problems. In this talk I will present a multivariate extreme value approach to each of the two problems that offers substantial improvements over the existing methods.

21 January 2014

Many-particle quantum systems on graphs and Bose-Einstein condensation
by Jens Bolte (RHUL)

Quantum graphs are popular models for one-dimensional systems
with a complex topology, both in quantum chaos and in spectral geometry.
In this talk I will briefly review quantum graphs and then introduce singular
two-particle interactions. These interactions allow to construct
Lieb-Liniger models as well as Tonks-Girardeau gases on graphs. I will then
discuss whether or not Bose gases on graphs display Bose-Einstein
condensation. For non-interacting gases a complete classification will be
achieved. I will then show that a Tonks-Girardeau gas shows no condensation,
even if a corresponding free gas does.

14 January 2014

A theory of regularity structures
by Martin Hairer (Warwick)

Heuristically, one can give arguments why the fluctuations of many classical models of statistical mechanics near criticality are typically expected to be described by nonlinear stochastic PDEs. Unfortunately, in most examples of interest, these equations seem to make no sense whatsoever due to the appearance of infinities or of terms that are simply ill-posed.

I will give an overview of a new theory of "regularity structures" that allows to treat such equations in a unified way, which in turn leads to a number of natural conjectures. One interesting byproduct of the theory is a new (and rigorous) interpretation of "renormalisation group techniques" in this context.

At the technical level, the main novel idea involves a complete rethinking of the notion of "Taylor expansion" at a point for a function or even a distribution. The resulting structure is useful for encoding "recipes" allowing to multiply distributions that could not normally be multiplied. This provides a robust analytical framework to encode renormalisation procedures.

10 December 2013

Flow control in the presence of shocks
by Enrique Zuazua (Bilbao)

Flow control is an topic for its many applications and its potential economic, social and environmental impact: water quality and supply, oil recovery, pollution, traffic, aeronautic design, ... From a mathematical viewpoint this is a challenging topic in between the theory of Partial Differential Equations (PDE), Control Theory, Optimal Design and Numerical Analysis.

The problem consists on controlling or optimizing parameters on the relevant models of Fluid Mechanics.

We focus on scalar conservation laws.

The first key issue is the linearization of the system around solutions presenting shock discontinuities, that is developed using the point of view of multi-physics, understanding the system as the coupling of the motion of the shock with the equations describing the dynamics of the fluid to both sides of it.

We then explain how this linearization, together with an alternating descent method can be implement to build descent algorithms which turn out to be more efficient that classical continuous or discrete descent ones.

We conclude with some comments on possible directions for future research.

This lecture summarizes joint work with C. Castro, R. Lecarós, F. Palacios and N. Allahverdi.

26 November 2013

Towards Time-Parallel Simulation Methods: From PARAREAL to Multilevel Integration
in Space and Time
by Rolf Krause (Università della Svizzera Italiana in Lugano)

The ever increasing parallelism of modern supercomputers will demand million- to billion-concurrency in the near future. This can be seen as a challenge for the numerical methods and simulation tools we are currently using, but it can also be exploited as a possibility to develop new and inherently parallel algorithms. Here, a particularly promising new approach is parallelization in time. Once spatial parallelization saturates, parallelization in time can help to make use of the massive concurrency available nowadays. Significant progress has been made during the last years in time-parallel methods, but still there are many technological and methodological questions eft open. In this talk, we will present recent developments in time-parallel methods and discuss the related difficulties. Starting from the now classical idea of PARAREAL, we will discuss more elaborated methods such as PFASST, with a particular focus on their scalability. As an important aspect within this framework, we will also comment on multilevel-in-time integration, which in one way or the other forms an important ingredient of time parallel methods. Numerical examples including parallel-in-time runs up to 448 k cores will be presented.

19 November 2013

Jacobi operators: spectral analysis and the asymptotic of the orthogonal polynomials
by Sergey Naboko (Kent)

Some basic things of the spectral theory of Hermitian Jacobi operators to be considered.

The analysis is based on the asymptotic properties of the related orthogonal polynomials at infinity for the fixed values of the spectral parameter. Some particular examples of bounded and unbounded Jacobi matrices, classical and more recent as well, will be discussed.

Hopefully the talk will be introductory to the theory of Jacobi matrices and orthogonal polynomials.

5 November 2012

Singularities of the Navier-Stokes equations at the moving contact-line: a view from the standpoint of macroscale and nanoscale
by Alex Lukyanov (Reading)

29 October 2013

Many-particle systems and kinetic theory
by Clément Mouhot (Cambridge)

We will discuss a program set up by Kac in the late 1950s for understanding the asymptotic of Boltzmann-like PDEs in terms of the scaling limits of some simple Markov collision processes. Then we will present answers to some of his questions, together with the new method we have developed to this purpose, and example of further applications of the method. The talk is based on joints work with Mischler and Wennberg.

22 October 2013

The scaling limit of the minimum spanning tree of the complete graph
by Christina Goldschmidt (Oxford)

Consider the complete graph on n vertices with independent and identically distributed edge-weights having some absolutely continuous distribution. The minimum spanning tree (MST) is simply the spanning subtree of smallest weight. It is straightforward to construct the MST using one of several natural algorithms. Kruskal's algorithm builds the tree edge by edge starting from the globally lowest-weight edge and then adding other edges one by one in increasing order of weight, as long as they do not create any cycles. At each step of this process, the algorithm has generated a forest, which becomes connected on the final step. In this talk, I will explain how it is possible to
exploit a connection between the forest generated by Kruskal's algorithm and the Erdös-Rényi random graph in order to prove that M_n, the MST of the complete graph, possesses a scaling limit as n tends to infinity. In particular, if we think of M_n as a metric space (using the graph distance), rescale edge-lengths by n^{-1/3}, and endow the vertices with the uniform measure, then M_n converges in distribution a certain random real tree.

This is joint work with Louigi Addario-Berry (McGill), Nicolas Broutin (INRIA Paris-Rocquencourt) and Grégory Miermont (ENS Lyon).

15 October 2013

Speeding up convergence to equilibrium for diffusion processes
by Greg Pavliotis (Imperial College)

In many applications it is necessary to sample from a probability distribution that is known up to a constant. A standard technique for doing this is by simulating a stochastic differential equation whose invariant measure is the probability measure from which we want to sample. There are (infinitely) many different diffusion processes that have the same invariant distribution. It is natural to choose a diffusion process that converges as quickly as possible to equilibrium, since this reduces the computational cost. In this talk I will present some recent results on optimizing the rate of convergence to equilibrium by adding an appropriate non-reversible perturbation to the standard reversible overdamped Langevin dynamics. This is joint work with T. Leliever and F. Nier.

08 October 2013

Finite element methods for nonvariational PDEs
by Tristan Pryer (Reading)

Nonvariational PDEs are those not given in divergence form. Given their variational nature, finite element methods at first glance appear unnatural for this class of problem. We will examine how to design schemes that lend themselves to these problems with the ultimate goal of designing schemes for fully nonlinear PDEs (e.g. the Monge Ampere problem) and generic quasilinear PDEs (e.g. the infinity Laplacian). 

10 June 2013

Seeing sounds, hearing shapes and beyond
by Iosif Polterovich (Montréal)

Geometric spectral theory has a long and fascinating history. It goes back to the experiments of Chladni with vibrating plates and to the ground-breaking work of Rayleigh on the theory of sound, to Weyl's law for the asymptotic distribution of eigenvalues and to Kac's celebrated question "Can one hear the shape of a drum?''. In my talk, I will discuss some of the old problems and related recent developments in the field.

14 May 2013

Glass transitions and large deviations
by Rob Jack (Bath)

Understanding the behaviour of glassy materials is a long-standing problem in physics. In particular, it is not clear whether there is any well-defined phase transition that is connected to experimentally-observed glass transitions. By considering large deviations associated with trajectories (histories) of glassy systems, we show that novel dynamical phase transitions can occur in glassy systems, even if thermodynamic transitions are absent. We show how these can be investigated, both numerically and through analysis of Markov chains.

23 April 2013

2.30 pm:

Geometry and asymptotics of the Painlevé equations
by Nalini Joshi (Sydney)

Critical solutions of classical ODEs called the Painlevé equations arise as universal limits in many nonlinear systems. This talk focusses on my geometric approach to describing their asymptotic properties, which was initiated in collaboration with Duistermaat. I will also describe extensions of this approach to discrete Painlevé equations, which has been an area of great activity in recent times.

Much of the activity in this field has been concentrated on deducing the
correct discrete versions of the Painlevé equations, finding transformations and other algebraic properties and describing solutions that can be expressed in terms of earlier known functions, such as q-hypergeometric functions. In contrast, in this talk, I focus on finding properties of solutions that cannot be expressed in terms of earlier known functions.

4.00 pm:

Quadratic forms and Siegel's generalised theta series
by Lynne Walling (Bristol)

Quadratic forms are ubiquitous in maths and science, as they capture some geometric notions such as distance and orthogonality. Number theorists are interested in questions about integers; a classical number theory question is: Given a positive definite quadratic form Q, and given a natural number n, for how many integer vectors x do we have Q(x)=n? Generalisations of the classical theta series can help us study this question, as they have some remarkable transformation properties that help us understand them.

I will introduce these generalised theta series and discuss how number theorists have approached the question posed above, as well as generalisations of this question.

 

12 March 2013

Analysis of some nonlinear PDEs from multi-scale geophysical applications
by Bin Cheng (Surrey)

This talk is regarding PDE systems from geophysical applications with multiple time scales, in which linear skew-self-adjoint operators of size 1/epsilon gives rise to highly oscillatory solutions. Analysis is performed in justifying the limiting dynamics as epsilon goes to zero; furthermore, the analysis yields estimates on the difference between the multiscale solution and the limiting solution. We will introduce a simple yet effective time-averaging technique which is especially useful in general domains where Fourier analysis is not applicable.

26 February 2013

Reappraisal of 2+ myths in the theory of nonlinear waves
by Thomas J. Bridges (Surrey)

There are 2+ parts to the talk. The first part considers the role of horizontal vorticity in shallow water. With the traditional scaling, the Euler equations reduce to the shallow water equations (SWEs) if the initial values of the horizontal vorticity field are taken to be zero. This property is exact. However, it is unstable in that small perturbations of the initial data may grow exponentially [1]. This observation suggests that the SWEs, although a perfectly valid mathematical model may be invalid as a model for realistic shallow water oceanographic flows.

In the second part, a new perspective on the emergence of the KdV equation is presented. The conventional view is that the KdV equation arises as a model when the dispersion relation of the linearization of some system of partial differential equations has the appropriate form, and the nonlinearity is quadratic. A new mechanism is presented which shows that the KdV equation always takes a universal form where the coefficients in the KdV equation are completely determined from the modulation properties of the background state [3] { even when the background state is the trivial solution! Well known examples such as the KdV equation in shallow water hydrodynamics and in the emergence of dark solitary waves [2] are predicted by the new theory.

The geometry of modulation is the backbone of the above KdV emergence. Modulation becomes even more interesting in higher dimension. The + part of the talk is to present a new mechanism for rolls in pattern formation bifurcating to planforms. A modulation equation is derived which has a sequence of multi-pulse planforms on periodic background. An example where this bifurcation occurs is the Swift-Hohenberg equation.

References:
1. TJB & D.J. Needham. Breakdown of the shallow water equations due to growth of the horizontal vorticity, J. Fluid Mech. 679, 655-666 (2011).

2. TJB. Emergence of unsteady dark solitary waves from coalescing spatially periodic patterns, Proc. Royal Soc. London A 468, 3784{3803 (2012).

3. TJB. A universal form for the emergence of the KdV equation, Proc. Royal Soc. London A (in press, 2013).

19 February 2013

Quantum normal forms and transition state theory: Microlocal analysis of chemical reactions
by Roman Schubert (Bristol)

Transition state theory provides a framework to describe a large class of chemical reactions. In this framework a chemical reaction is described by the passage over a barrier in the potential energy landscape of a molecule. We applied methods from microlocal analysis to study this situation, in particular we use quantum normal form theory to approximate the Hamilton operator by a simpler one whose dynamics we can solve explicitly. This allows to compute a local S-matrix, resonances and reaction probabilities. We developed explicit algorithms to compute these quantities and we illustrate the results with some examples.

12 February 2013

The large-time dynamics of atoms
by Mathieu Lewin (Paris)

We consider an atom containing N electrons and one nucleus of charge Z, and we ask the question of how many of the electrons can stay in the vicinity of the nucleus for an arbitrary long time. For both the linear many-body Schrödinger equation and its Hartree nonlinear approximation, we prove a bound on this number of electrons, which is the time-dependent equivalent of a famous result of Lieb in 1984. Work in collaboration with Enno Lenzmann (Basel, Switzerland).

05 February 2013

Science from a sheet of paper
by Tadashi Tokieda (Cambridge)

Most colloquia/public lectures take a highly developed piece of mathematics and attempt to make it accessible.

This talk will do the opposite: we take a daily object that does not yet look like mathematics at all - a sheet of paper - and, by cutting, twisting, folding, crumpling, we will develop mathematics out of it before our own eyes.

Everything is elementary but new even to experts. The talk is interwoven with many demos, which you will be able to repeat yourselves afterwards and to investigate further.

29 January 2013

Rigorous results for the minimal speed of Kolmogorov-Petrovskii-Piscounov monotonic fronts with a cutoff
by Rafael Benguria (P. Universidad Catolica de Chile, Santiago)

In this talk I will present the effect of a cut-off on the speed of pulled fronts of the one dimensional reaction diffusion equation. To accomplish this, we first use variational techniques to prove the existence of a heteroclinic orbit in phase space for traveling wave solutions of the corresponding reaction diffusion equation under conditions that include discontinuous reaction profiles. This existence result allows us to prove rigorous upper and lower bounds on the minimal speed of monotonic fronts in terms of the cut-off parameter $\varepsilon$. From these bounds we estimate the range of validity of the Brunet-Derrida formula for a general class of reaction terms.

This is joint work with M. C. Depassier (Santiago) and M. Loss (Atlanta).

22 January 2013

Rogue waves, vortices and polynomials
by Peter Clarkson (Kent)

In this talk I shall discuss special polynomials associated with rational solutions of the Painlevé equations and of the soliton equations which are solvable by the inverse scattering method, including the Korteweg-de Vries, Boussinesq and nonlinear Schrödinger equations. I shall illustrate applications of these polynomials to rogue waves and vortex dynamics.

15 January 2013

Steady Water Waves
by Walter Strauss (Brown University, USA)

The mathematical study of water waves began with the derivation of the basic mathematical equations of fluids by Euler in 1752. Later, water waves played a central role in the work of Poisson, Cauchy, Stokes, Levi-Civita and many others. It remains a very active area to the present day.I will consider a classical 2D traveling water wave with vorticity. By means of local and global bifurcation theory using topological degree, we now know that there exist many such waves (exact smooth solutions of the Euler equations with the physical boundary conditions) of large amplitude. This work is joint with Adrian Constantin. I will also exhibit some numerical computations of such waves, obtained jointly with Joy Ko. Many fundamental problems remain open.

20 November 2012

Noise and Parametrizations in complex systems
by Valerio Lucarini (Hamburg and Reading)

The investigation of forced and dissipative non-equilibrium system is one of the key fields of research in mathematical physics, statistical physics and complex systems science. Using rigorous methods of statistical mechanics, we derive novel explicit expressions for impact of rather general noise (additive or multiplicative, white or red) on the statistical properties of a chaotic dynamical systems. We prove that information on the system's response to noise informs on the systems'

response to deterministic perturbations. In the second part, we focus on the problem of providing a closure for the problem of parameterizing the fast dynamics multi-level systems. Through response theory, we derive explicit expressions for the deterministic and stochastic components of parametrization and find also a novel contribution describing a memory effect, usually neglected. Such a term vanishes in the limit of infinite time scale separation between the slow and fast dynamics. By establishing a connection with the Mori-Zwanzig theory, we also explain that these results have relevance for the general problem of coarse graining and for the issue of applicability of the fluctuation-dissipation theorem in chaotic systems. These findings could be of relevance for practically devising new parametrizations for geophysical and other complex fluid flows.

13 November 2012

Computational Room Acoustics - an overview
by Peter Svensson (Norwegian University of Science and Technology, Trondheim)

Computational room acoustics deals with solving the Helmholtz equation which is an innocent looking PDE. However, the size of realistic problems turns out to be huge enough to call for other methods than solving the PDE accurately. Three different categories of approaches are common:

1. The PDE is solved accurately for low frequencies, in small rooms.

2. For a wide range of cases, high-frequency asymptotic solution methods are used, the geometrical acoustics assumption, and usually one hopes for the best when it comes to accuracy for lower frequencies.

3. Quite a different approach, which is surprisingly successful, is based on so-called diffuse-field assumptions. Then the locally-averaged intensity/energy of the sound field is studied. Very simple relationships come out of this, dating back to the person who is considered to have started room acoustics as a science: W. C. Sabine, physics professor at Harvard in the end of the 1800s.

The presentation will give examples from all of these categories, and try to tie them to the fundamental equations.

30 October 2012

Meteorology and analytical methods: a recent application to mountain waves
by Miguel Teixeira (Dept of Meteorology, Reading)

While historically analytical mathematical methods have been closely linked with developments in meteorology, in recent years the massive use of numerical simulations has pushed them to a somewhat more secondary position. However, such methods, which may be used to devise simple idealized models, are still quite valuable, since they allow us to understand physical processes at work in meteorological problems, explore the parameter space much more effectively, and interpret the results of numerical simulations and plan new ones in a more rational way.

In this talk I will present an example of an application of such methods to the problem of internal gravity waves generated by stratified flow over orography, known as mountain waves. The two basic techniques employed are the WKB approximation, which allows us to approximately calculate the drag force associated with these waves for a wind profile with a generic vertical variation, as long as this variation is relatively slow; and contour integration, which is used to evaluate the partial absorption of the wave momentum flux at so-called critical levels, where the horizontal wavenumber vector is perpendicular to the mean wind vector.

 

23 October 2012

Some statistical problems in probabilistic forecasting and
data assimilation
by Jochen Broecker (Reading)

In this talk, I intend to give an overview over my current research. Time permitting, I will be more specific about some problems. I will start talking about probabilistic forecasts which have become very popular in weather and climate. Such forecasts can be objectively evaluated using various statistical means. Although this is a useful and by now more or less complete formalism, I think that there are major issues remaining with the interpretation of the probabilities we obtain from our weather and climate models.

A second line of my research is variational data assimilation. In essence, data assimilation is an inverse problem which requires regularisation, and consequently, the question arises how to set the regularisation parameter. I will present a method which is akin to Mallows' Cp for simple linear regression. I believe, however, that other methods for model selection might be applicable as well, after some modification; this might be a future line of my research.

16 October 2012

Colloquium:

Consequences of Integrability to Linearity and to Medical Imaging
by Thanasis Fokas (Cambridge)

A new method for analyzing boundary value problems (BVPs) for linear and integrable nonlinear PDEs, extending ideas of the inverse scattering transform method, was introduced in [1] and further developed by several researchers. First,this method will be introduced by using linear evolution PDEs in the half-line as illustrative examples. Then, regarding the nonlinear analogues of these problems it will be emphasised that: (i)For linearizable BVPs, the new method is as effective as the usual inverse scattering transform method. (ii) For general BVPs, with either decaying, or t-periodic boundary conditions, it yields effective long time asymptotic formulae.

The related development of the emergence of a new analytical approach to inverting integrals with applications in medical imaging, will be mentioned.

References [1] A.S. Fokas, A Unified Transform Method for Solving Linear and Certain Nonlinear PDEs, Proc. R. Soc. Lond. A 453, 1411-1443 (1997).

 

09 October 2012

MAGIC Lecture:
Dynamics of Social perceptions, Attitudes Norms and Applications to Digital Media and Security
by Peter Grindrod (Reading)

The challenges from within applications often drive us to postulate new mathematical models and develop new methods of analysis. This is especially true for complex systems, whether discrete or continuous, where the dynamics at a microscopic level, together with specified coupling, results in emergent structures and dynamics at the meso or macro scales.

Modern digital platforms, for social networking, commerce, communications, provide new, virtual, places for individuals to work, rest and play; and also provide us with vast amounts of data. But without models we would be merely observing and commentating. If we want to draw inferences, to make forecasts, to spot aberrations; or to de-risk decisions to intervene, modify or control behaviour, then we surely require some models.

Here we consider complexity within social systems, which, at the micro level have inconsistent, partially rational, and often unreliable people. There are no conservation laws (energy, mass, momenta), yet systematic patterns of behaviour occur at a small group or population levels. We shall illustrate some problems and ideas from the work carried out within the Centre for the Mathematics of Human Behaviour, here at University of Reading.

17 May 2012

Social dynamics -what can we learn from silicon footprints?
Michael Macy (Cornell University) 

This talk will be about Prof Macy's recent and current work:

- Using the nearly complete record of all telephone calls made in the UK over a one-month period (with Rob Claxton and Nathan Eagle, Science 328: 1029) to investigate whether members of communities that differ socio-economically have different social network structures. There are strong theoretical reasons to believe that they do, with the causal direction running both ways. The surprising result was not that the theories were wrong but that the differences in network structure were so dramatic. People in advantaged communities have much more open network.
- Using worldwide Twitter messages to measure diurnal and seasonal mood variations across diverse cultures (with Scot Golder, Science 333: 1878). From Africa to the U.A.E, and from Asia to South America, we found that people are happiest in the morning and it is all down hill from there. Curiously, the work-week pattern is the opposite of the work-day pattern. The results indicate that periodic biological processes involving changes in cortisol levels are closely associated with sleep cycles which in turn are constrained by cultural patterns, particular the structure of the work week.
- Using Twitter content from the Middle East to track the spread of Arab Spring (with Jon Kleinberg, Noona Oh, Michael Siemon, Silvana Toska, and Shaomei Wu).

Bio:
Michael W. Macy is Goldwin Smith Professor of Sociology, Professor of Information Science, and Director of the Social Dynamics Laboratory at Cornell. His research team has used computational models, online laboratory experiments, and digital traces of device-mediated interaction to explore familiar but enigmatic social patterns such as diurnal mood changes, the emergence and collapse of fads, the spread of self-destructive behaviors, the critical mass in collective action, the polarization of opinion, segregation of neighborhoods, and assimilation of minority cultures. Recent research uses 509 million Twitter messages to track diurnal and seasonal mood changes in 54 countries, and complete UK call logs to measure the economic consequences of network structure. His research has been published in leading journals, including Science, PNAS, American Journal of Sociology, American Sociological Review, and Annual Review of Sociology.

08 May 2012

The integrable cubic Szegö equation on the real line
Oana Pocovnicu (Imperial College)

In this talk we consider the cubic Szegö equation: i u_t = Pi_+ (|u|^2u) on the real line, where Pi_+ is the Szegö projector on non-negative frequencies. This equation was recently introduced as a model of a non-dispersive non-linear Hamiltonian PDE by P. Gérard (Orsay, France) and S. Grellier (Orleans, France). The most remarkable property of the Szegö equation is its integrability, in the sense that it possesses a Lax pair. Integrability allows us to find an explicit formula of solutions, that we will present in the first part of the talk. As an application, we will show that generic solutions decompose into a finite sum of solitons over the time (phenomenon called "soliton resolution"). Secondly, we will exhibit a non-generic solution for which the soliton resolution no longer holds. The high Sobolev norms of this non-generic solution grow to infinity over the time, which indicates an unstable behavior. Finally, we will consider a non-linear wave equation and prove that it can be approximated for a long time by the Szegö equation. As a consequence, we will obtain an instability result for the non-linear wave equation.

03 May 2012

1377 questions and counting - what can we learn from online math
Ursula Martin CBE (Queen Mary University of London)

Online blogs, question answering systems and distributed proofs provide a rich new resource for understanding what mathematicians really do, and hence devising better tools for supporting mathematical advance.

In this talk we discuss the first steps in such a research programme, looking at two examples through a sociological lens, to see what we can learn about mathematical practice, and whether the reality of mathematical practice supports the theories of philosophers of mathematics such as Polya and Lakatos, or the expectations of computer scientists devising software to do maths.

Polymath provides structured way for a number of people to work on a proof simultaneously: we analyse a polymath proof of a math olympiad problem to see what kinds of techniques the participants use. Mathoverflow supports asking and answering research level mathematical questions: we look at a sample of questions about algebra, and provide a typology of the kinds of questions asked, and consider the features of the discussions and answers they generate.

Finally we outline a programme of further work, and consider what our results tell us about opportunities for further computational support for proof and question answering.

Bio: Professor Ursula Martin is Professor of Computer Science and Director of the impactQM project at Queen Mary University of London, where she was until recently Vice-Principal for Science and Engineering. She works at the interface of mathematics and computer science, and current projects include providing a formal logic to capture control engineering; network analysis of resaerch impact; and crowdsourced mathematics. She is a member of numerous boards and committees, and active in activities to promote women in science.

24 April 2012

A New Non-Classical Class of Optimal Variational Problems
Alan Zinober (Sheffield)

The Calculus of Variations was initiated in the 17th Century and forms a basic foundation of modern optimal (maximising or minimising) variational problems, nowadays often called optimal control. An introduction to the Calculus of Variations with some sample examples will be presented. This will include the Euler-Lagrange and Hamiltonian formulation together with the associated final boundary value conditions. Solution of the necessary conditions (using Maple or by hand) or the numerical shooting method can be used to solve the resulting Two Point Boundary Value Problem (TPBVP), a set of differential equations.

A new non-classical class of variational problems has been motivated by recent research on the non-linear revenue problem in the field of economics. This class of problem can be set up as a maximising problem in the Calculus of Variations (CoV) or Optimal Control. However, the state value at the final fixed time, y (T ), is a priori unknown and the integrand to be maximised is also a function of the unknown y (T ). This is a non-standard CoV problem that has not been studied before. New final value costate boundary conditions will be presented for this CoV problem and some results will be presented.

24th January 2012

Homogenisation in finite elasticity for composites with a high contrast in the vicinity of rigid-body motions
Kirill Cherednichenko

I will describe a multiscale asymptotic framework for the analysis of the macroscopic behaviour of periodic two-material composites with high contrast in a finite-strain setting. I will start by introducing the nonlineardescription of a composite consisting of a stiff material matrix and soft, periodically distributed inclusions. I shall then focus on the loading regimes when the applied load is small or of order one in terms of the period of the composite structure. I will show that this corresponds to the situation when the displacements on the stiff component are situated in the vicinity of a rigid-body motion. This allows to replace, in the homogenisation limit, the nonlinear material law of the stiff component by its linearised version. As a main result, I derive (rigorously in the spirit of $\Gamma$-convergence) a limit functional that allows to establish a precise two-scale expansion for minimising sequences.

This is joint work with M. Cherdantsev and S. Neukamm.

10th January 2012

On the singularities of a free boundary through Fourier expansion
Georg Weiss (Düsseldorf)

Abstract (pdf)

6th December 2011

Progress towards loosely coupled parallelism in optimization
Raphael Hauser (Oxford)

Multicore processors and graphics cards are fast becoming standard features of personal computers. This additional computational power presents huge opportunities for scientific computing, but harnessing it is nontrivial, for it requires the development of new numerical linear algebra algorithms based on loosely coupled parallelism (low synchronicity, low communication overhead). In this talk we present such an algorithm for the computation of the leading part singular value decomposition, and we discuss the uses of this approach in continuous optimization.

Joint work with Daniel Goodman (University of Manchester) and Sheng Fang (University of Oxford).

8th November 2011

Mathematical Problems of the Q-tensor Theory of Liquid Crystals
Arghir Zarnescu (Sussex)

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 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 Sir 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.

11th October 2011

Freak waves and probabilistic forecasting
Peter Janssen (ECMWF)

I will start from the Hamiltonian formulation of surface gravity waves. For weak nonlinearity there is a natural distinction between free and bound gravity waves. This distinction allows the introduction of a relatively simple evolution equation, which is called the Zakharov equation.

From this evolution equation a number of interesting properties of weakly nonlinear water waves may be derived. In this talk I will concentrate on only one property, namely the instability of a uniform wave train which is an example of a four-wave interaction process. In the field of water waves this instability is called the Benjamin-Feir instability. Because it is a four-wave interaction, the instability, under different names, is also found in other fields. Examples are nonlinear Optics and Plasma Physics.

In its most simple form the Benjamin-Feir Instability leads to the generation of envelope solitons which provides a possible explanation for the generation of 'freak' waves. Because these extreme events are rare, forecasting of freak waves is difficult. Nevertheless, the probability of these extreme events can be estimated and at ECMWF probabilistic forecasting of 'freak' waves was introduced a number of years ago.

15th June 2011

Finite Sections of Random Jacobi Operators
Marko Lindner (TU Chemnitz)

To approximately solve an infinite linear system Ax=b one often resorts to literally truncating the matrix A and the right hand side b and to solve the remaining finite system instead.

In this talk we show recent results about good and bad ways of truncation and present a strategy for the case when A has three random diagonals.

21st March 2011

Geometric control of the Steklov spectrum of a domain
Alexandre Girouard (Neuchatel)

Abstract:  The Dirichlet-to-Neumann map is a first order elliptic pseudodifferential operator acting on the boundary of a bounded regular domain. Its spectrum is called the Steklov spectrum of that domain. In this talk I will discuss how the geometry of an ambient manifold can be used to control the Steklov spectrum of a bounded domain in terms of its isoperimetric ratio. On surfaces, it is possible to control the spectrum in terms of the genus. The proofs use classical complex analysis and some results in abstract metric geometry. This is joint work with Bruno Colbois and Ahmad El Soufi, and also with Iosif Polterovich.

24th January 2011

The numerical solution of partial differential equations on surfaces with the Closest Point Method

Colin Macdonald (Oxford)

Abstract:  Solving partial differential equations (PDEs) on curved surfaces is important in many areas of science. The Closest Point Method is a new technique for computing numerical solutions to PDEs on curves, surfaces and more general domains. For example, it can be used to solve a pattern-formation PDE on the surface of a rabbit. A benefit of the Closest Point Method is its simplicity: it is easy to understand and straightforward to implement on a wide variety of PDEs and surfaces. In this presentation, I will introduce the Closest Point Method andhighlight some of the research in this area. Example computations (including the in-surface heat equation, reaction-diffusion onsurfaces, level set equations, high-order interface motion, and Laplace-Beltrami eigenmodes) on a variety of surfaces will demonstrate the effectiveness of the method.


 

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