All seminars (unless otherwise stated) will take place on Mondays at 3.00 pm in Room 505 which is located on the 5th floor of the Mathematics Department. See Where to Find Us for further details. There will be tea afterwards in room 606.

If you require any more information on the Applied seminars please contact Professor Yaroslav Kurylev e-mail: y.kurylev AT ucl.ac.uk or tel: 020-7679-7896.

## 17 January 2011

Prof. Ulf Leonhardt - University of St Andrews

##### Invisibility cloaking and perfect imaging

Abstract:

Invisibility has been a subject of fiction for millennia, from myths of the ancient Greeks and Germans to modern novels and films. In 2006 invisibility turned from fiction into science, primarily initiated by the publication of first ideas for cloaking devices and the subsequent demonstration of cloaking for microwaves. Perfect imaging is the ability to optically transfer images with a resolution not limited by the wave nature of light. Advances in imaging are of significant importance to modern electronics, because the structures of microchips are made by photolithography. In the seminar I explain our latest progress in cloaking and imaging where ideas from the geometry of curved space are used in electrical and optical engineering.

## 24 January 2011

Prof. Paul Glendinning - University of Manchester

##### Attractors and invariant measures for the border collision normal form

Abstract:

The border collision normal form is a map of the plane defined by two affine maps, one in the left hand half-plane and one in the right hand half-plane, and which is continuous across the y-axis. These arise naturally in the study of switched control systems, electric circuit theory and some mechanics problems. Indeed, they are so natural that mathematicians (both pure and applied) studied them before they were given a name in 1992! Despite a literature extending back to the 1970s much is still unknown about the dynamics of these maps. I will describe recent results proving the existence of two-dimensional attractors under some conditions ('known' to exist numerically since the 1980s), and also describe some conditions which allow us to show the existence of chaotic attractors with invariant measures.

## 31 January 2011

Dr. Oleg Derzho - Memorial University of Newfoundland, Canada

##### On variability of polar currents due to Rossby wave trains

Abstract:

A simple analytical model for low frequency and large-scale variability of the Antarctic Circumpolar Current (ACC) will be presented. Physical mechanism of the variability is related to temporal and spatial variations of the ACC mean flow due to circularly propagating nonlinear barotropic Rossby wave trains trapped between the major fronts of the ACC. The Rossby wave patterns are predicted to rotate with a specific angular velocity, which depends on the magnitude and width of the mean current. It is shown that the specific form of the stream function-vorticity relation defines spatial structure of the rotating pattern, including its zonal wave number. The similarity between the simulated patterns and the recently discovered Antarctic Circumpolar Wave (ACW) is highlighted.

## 07 February 2011

Dr. Patrizio Neff - University of Duisburg-Essen, Germany

##### On linear and nonlinear static Cosserat models

Abstract

I consider linear and nonlinear Cosserat and extended continuum models in the static case, which may describe non-classical deformation behaviour and size-dependent results, useful e.g. for nano-devices. Focus is on the variational formulation and the mathematical treatment. I will make connections to plasticity theory and models for shells and plates.

## 14 February 2011

READING WEEK - NO SEMINAR

## 21 February 2011

Prof. Wolfgang Wendland - University of Stuttgart, Germany

##### Boundary integral equations, Trefftz elements, Levi functions

Abstract:

The combination of FETI and BETI (Finite, respectively Boundary Element Tearing and Interconnecting Methods) using Trefftz elements have recently been developed for simulating electrical machines with complicated geometry and strongly varying material properties. For domain decomposition methods applied to elliptic problems, the use of Trefftz elements allows to achieve higher resolution accuracy - similar to the h-p methods. In two-dimensional problems where the differential equation coefficients are real analytic, Trefftz elements can be constructed via analytic extension and by the use of Bergman-Vekua operators. An alternative approach in two and three dimensions is based on local domain-boundary integral equations employing Levi function approximation of the spatial Green's function. In this lecture, these ideas are exemplified for the Dirichlet problem of a second order elliptic partial differential equations with variable coefficients in a bounded domain. Keywords: domain decomposition, Trefftz elements, Levi functions and local domain-boundary integral equations.

## 28 February 2011

Dr. Alberto Ruiz - University Autonoma de Madrid

##### The BornÂ´s approximation in potential inverse scattering

Abstract:

We will study what information about the actual potential can be recover and reconstructed from the Born aproximation in the Schroedinger equation. In particular we prove that the rough part of the potential can be recovered, from different sets of inverse scattering data.

## 07 March 2011

Prof. Peter Gritzmann - Munich University of Technology

##### On Clustering Bodies, Gravity Polytopes and Power Diagrams: Computational Convexity in Agriculture

Abstract:

In geometric clustering, m objects in some R^d have to be partitioned into k clusters according to certain balancing constraints so as to optimize some distance-based objective function. The most prominent example in our context is that of the consolidation of farmland.

In particular, we present new structural and algorithmic results for underlying convex sets and diagrams that are at the basis of surprisingly tight approximation algorithms for optimal clusterings.

## 14 March 2011

Prof Gregorii Seregin - Oxford and St-Petersburg

##### On the divergence free drifts

Abstract:

In the talk, we are going to discuss fine properties of solutions to the heat equation with the divergence free drift which is a bounded in time function taking values in the space of functions with bounded mean oscillations. The problem is motivated by Liouville type theorems for the Navier-Stokes equations. Our approach is based on higher integrability and Harnack's inequality. Some results are almost sharp.

## 21 March 2011

Dr Djoko Wirosoetisno - Durham

##### Navier-Stokes equations on the beta-plane

Abstract:

We show that the solution of the 2d Navier-Stokes equations on the beta-plane becomes more zonal as the rotation rate increases. Moreover, it is shown that for sufficiently large (but finite) rotation, the resulting flow is steady and stable, implying that the global attractor of the 2d NSE reduces to a point. We also discuss possible extension to the rotating sphere.