The results are separate, and apparently paradoxical, and have implications for physics. First, when two exponentials compete, their interference can be dominated by the contribution with smaller exponent. Second, repeated differentiation of almost all functions in a wide class generates trigonometric oscillations ('almost all functions tend to cosx'). Third, it is possible to find band-limited functions that oscillate arbitrarily faster than their fastest Fourier component ('superoscillations').

In this talk I will describe proposals for quantum simulations in Ion traps and BECs. It will be shown that the radial degree of freedom of strings of trapped ions in the quantum regime may be prepared and controlled accurately through the variation of the external trapping potential while at the same time its properties are measurable with high spatial and temporal resolution. This provides a new test-bed giving access to static and dynamical properties of the physics of quantum-many-body systems and quantum phase transitions that are hard to simulate on classical computers. Furthermore, it allows for the creation of double well potentials with experimentally accessible tunnelling rates and with applications in testing the foundations of quantum physics and precision sensing. A scheme for the study of methods for detecting Unruh-like acceleration radiation effects in a Bose-Einstein condensate in a 1+1 dimensional setup will be descried. In particular, the dispersive effects of the Bogoliubov spectrum on the ideal case of exact thermalization will be explained.

In this talk we will review the recent developments in the field of integrability of the AdS/CFT correspondence. In particular, we will discuss the novel dynamical tests of the duality and summarize the breakdown of the asymptotic spin-chain picture at the wrapping order. We will also discuss the latest attempts to incorporate the non-local wrapping interactions.

Two different types of inverse problems will be discussed: (a) The inversion of certain integral transforms arising in medical imaging. (b) The generalized Dirichlet to Neumann map for certain physically significant boundary value problems, including the biharmonic equation in a semi strip and the Helmholtz equation in the interior of a canonical polygon.

The talk is based on a 1990 Phys. Rev. milestone paper Scientists attempt to understand physical phenomena by constructing models. A model serves as a link between scientists and nature, and one basic goal is to develop models whose solutions accurately reflect the nature of the physical process. But a dynamical model uses simplifying assumptions and approximations. The question of whether a model accurately reflects nature is one constantly faced by scientists. I will argue that there exist levels of mathematical difficulty, brought from the theory of dynamical systems, which can limit our ability to represent chaotic processes in nature using deterministic models. Specifically, in such cases, no model of such a system produces solutions of reasonable length that are realized by nature. Furthermore, for these processes, the numerical solutions of the models do not approximate any actual model solutions.

We give a brief introduction to recent advances in the AdS/CFT correspondence with particular emphasis on the role integrable structures enter this duality.

The Large Hadron Collider is just beginning it's work as the highest energy proton proton collider ever made. After a brief introduction to the machine I will concentrate on it's main goal - the search for the Higgs boson. The Higgs is the missing particle from the Standard Model of particle physics which makes the model well behaved at high energy and generates particle mass. Something has to be there to do the job of the Higgs but there are hints of flaws in the simplest model that have led to many suggestions for new physics from particle compositeness to extra space-time dimensions.

We will introduce deformations of partial derivatives (Dunkl operators) and discuss their actions on polynomial functions. This leads to various flavours of Hecke algebras and relates to the geometry of families of points in the plane. Eventually, this can be studied through microlocal analysis.

TBA

Several recent papers have investigated the extention of classical PT-Symmetric Hamiltonians to the classical domain. This approach to classical mechanics will be introduced as a generalisation of the familiar classical mechanics learnt at high school. The properties of these systems are often surprising and specific graphical studies can be quite beautiful. Familiar classical systems will be studied and brand new results will be presented including an exploration of chaotic systems.

The seminar had to be cancelled, but will be held next term.

There exists a family of deformations of Minkowski spacetime, and its Poincare symmetries, parameterized by a mass-scale kappa. I will discuss some recent work in which ideas from the theory of quantum groups are applied to the problem of formulating kappa-deformed quantum field theory.

In this talk I shall discuss special polynomials associated with rational solutions for the Painleve equations. The Painleve equations (PI-PVI) are six nonlinear ordinary differential equations that have been the subject of much interest in the past thirty years, which have arisen in a variety of physical applications. Further they may be thought of as nonlinear special functions and arise as symmetry reductions of soliton equations which are solvable by the inverse scattering method, such as the Korteweg-de Vries, Boussinesq and nonlinear Schroedinger equations. Rational solutions of the Painleve equations are expressible in terms of the logarithmic derivative of certain special polynomials. For PII these polynomials are known as the Yablonskii-Vorobev polynomials, first derived in the 1960's by Yablonskii and Vorob'ev. The locations of the roots of these polynomials is shown to have a highly regular triangular structure in the complex plane. The analogous special polynomials associated with rational solutions of PIII, PIV and PV are described and it is shown that their roots also have a highly regular structure.

We study an N-particle, two-mode Bose-Hubbard system, modelling a Bose-Einstein condensate in a double-well potential. By introducing effective complex energies to the modes we describe a coupling to a continuum. The eigenvalues of the resulting non-Hermitian matrix model are in general complex where the imaginary parts (resonance widths) describe the decay rate into the continuum. In dependence on the parameters the spectrum shows an intricate structure of exceptional points, at which two or more eigenvalues as well as the corresponding eigenstates are degenerate. In the present talk the effect of the interplay between the particle interaction and the non-Hermiticity on the spectrum is analysed, drawing special attention to the occurrence and unfolding of exceptional points. Some peculiarities are clarified using perturbational methods.

TBA

Modules for associative or Lie algebras are just vector spaces acted on by the algebra. This means that we can take the product of an algebra element with a module element in a sensible way. Typically, one cannot take the product of two module elements, though. However, in certain circumstances, it is possible to make a module into an associative or Lie algebra, in a way that is compatible with the action. I will start with some examples for finite-dimensional Lie algebras, then infinite-dimensional Kac-Moody Lie algebras and finally quantized enveloping algebras. The products we get will turn out to involve braidings, giving us braided Lie algebras and braided enveloping algebras. I will also demonstrate one use for this extra structure, namely gluing together the module and the original algebra to get bigger algebras.

I discuss how systems with non-abelian anyons can be used to build a topological quantum computer. Operations are performed by braiding the anyons, because the outcome of braiding is a purely topological property, such quantum computers should be robust against local errors. I will give several examples of how such anyons arise in fractional quantum Hall systems and in quantum loop models. Mathematical byproducts of this work are algebraic proofs and extensions of Tutte's identities for the chromatic polynomial (the zero-temperature Potts-model partition function).

We discuss the possibility of applying modern techniques of homological algebra to the composition and analysis of fugues.

This seminar deals with integrable models on graphs. After some motivations for studying such systems, I give the proof of the integrability of a Calogero model on a star graph (see J.Phys. A 40 (2007) 5491 and math-ph/0609057). I also discuss its symmetry. In particular, I emphasise the links with the complex reflection groups and twisted algebra. I finish with open problems in this quite recent domain of integrable models.

Reflection equation or boundary Yang-Baxter equation is an important object in the theory of quantum groups and integrable models, along with the famous RTT relation. In my talk I focus on the role of the related algebraic structures in the deformation quantization. We provide a survey of their basic properties from a geometric point of view.