An integrable shock, or discontinuity, is a type of 'defect' allowed, in the sense of not destroying the property of integrability, within (at least some) integrable field theories. A main example is the sine-Gordon model, or, more generally, the a-series of affine Toda field theories. Such 'defects' have interesting properties in the classical and quantum domains especially with regard to their interactions with solitons - or indeed with each other, if there is more than one of them and they are allowed to move. This talk will present a survey of recent results and describe some open problems.

Motivated by a desire to find a useful 2d Lorentz-invariant reformulation of the AdS_5 x S 5 superstring world-sheet theory in terms of physical degrees of freedom we construct a Pohlmeyer-reduced version of the corresponding sigma model. Starting with a coset space string sigma model in the conformal gauge and writing the classical equations in terms of currents one can fix the residual conformal diffeomorphism symmetry and kappa-symmetry and introduce a new set of variables (related locally to currents but non-locally to the original string coordinate fields) so that the Virasoro constraints are automatically satisfied. The resulting gauge-fixed equations can be obtained from a Lagrangian of a non-abelian Toda type: a gauged WZW-model with an integrable potential coupled also to a set of 2d fermionic fields. We show that in the special case of the AdS_2 x S2 superstring model the reduced theory is equivalent to the N=2 supersymmetric extension of the sine-Gordon model.

I'll introduce a combinatorial problem on tilings of the plane by rhombi and explain how the solution is related to the representation theory of groups.

Random Matrix Theory (RMT) has many applications in Physics and other Sciences. The starting point is given by the global symmetries of the underlying Hamiltonian or other relevant operator of the theory to be described. In this talk I will focus on the so-called chiral ensembles and their complex extensions, generalising the Wigner-Dyson classes that were first introduced. The corresponding Gaussian RMT with NxN matrices can be solved exactly using Laguerre Polynomials, both for matrices with real and complex eigenvalues. An example where these find applications are the spectra of Dirac operators in Lattice Gauge Theory.

Solution of the Heisenberg spin chain s=1/2 (spectrum of the energy operator and its eigenvectors) is related with three algebras: the Lie algebra sl(2), group algebra of symmetric group S_N and an infinite dimentional Hopf algebra, Yangian Y(sl(2)). Using representaions of these algebras there are generalization of this model to higher spins s=1, 3/2, 2, ...However there is also possibility to geberalize these algebras preserving structure of their representations. In particular the group algebra of S_N will be substituted by the Temperle -- Lieb algebra.

I shall present a review of the vertex operator approach to solvable lattice models. I shall briefly describe how this technique is being applied to the computation of dynamical structure functions - the latter being directly assessable via neutron scattering experiments.

D-branes in Calabi-Yau manifolds are interesting objects from the point of view of string phenomenology and of pure mathematics, and they have been studied using a conformal field theory as well as geometric methods. In the past few years, a new tool has emerged in the form of matrix factorisations of Landau-Ginzburg potentials, which gives a rather efficient computational handle on the properties of topological Calabi-Yau branes. I will try to give an overview of some of these developments.

A not entirely formal discussion of some aspects of the so called PT-symmetric quantum mechanics will involve the following four more or less open problems: 1. What are chances of a closed-form description of boundaries of stability (i.e., of the allowed domain D of physical parameters) in a generic PT-symmetric model? (hint: we shall demonstrate that they are better than expected) 2. What happens if the coordinates are allowed to be topologically nontrivial? (hint: one gets the so called quantum toboggans) 3. Should and could a realistic PT-symmetric model be also based on a non-Hermitian form of the parity? (we intend to recommend it) 4. Could the use of a realistic PT-symmetric model be made compatible with a time-dependent metric? (we shall show how).

TBA

We show that the full three-dimensional architecture of simple viruses is encoded by affine symmetries. In particular, the locations and structures of all material boundaries can be predicted with our method, including the organisation of the capsid proteins and the genomic material. This approach departs radically from the 2-dimensional schematic representations of viral capsids in Caspar-Klug Theory and its recent generalisations, and opens up new possibilities to study the immuno-dominant epitopes and viral evolution. Applications to the modelling of virus assembly are also discussed, and we show that the counting of assembly pathways can be cast into a Hamiltonian paths problem

We discuss a bijection between the Z_k parafermions CFT quasi-particle basis and the RSOS lattice paths. This bijection also implies a bijection between Bressoud lattice paths and RSOS lattice paths. We generalize this result for the graded parafermionic models and use it to construct a new fermionic character formula for the M(k+1,2k+3) minimal models, which are dual to the graded parafermions.

In this talk I will discuss how the formalism of master equations may be applied to study the stochastic dynamics of a number of problems in population biology and biochemistry. When they contain a large number of constituents, the behaviour of these systems may be analysed using an expansion in the system size. To leading order the deterministic analogues of the models can be compared to the equations which are normally written down on phenomenological grounds. At next to leading order a simplified stochastic description is obtained. Attention will focus on systems for which the deterministic description fails to predict cycles, but where large cycles are found at next-to-leading order through a resonant amplification of demographic fluctuations. The generality and applicability of these results will be discussed.

After introducing very generally the idea of defect and why they are important, I will review an approach to incorporate them in classical integrable field theories. It is essentially a lagrangian approach in which a defect is implemented as internal boundary conditions on the fields. The various models treated in this way so far (sine-Gordon, nonlinear Schrodinger, Korteweg-de Vries and its modified version) share common features which suggested an underlying general structure. I will propose another approach, directly based on the inverse scattering method, which exhibits these common features in a unified way. The main advantage of this approach is that it allows to discuss integrability in the presence of a defect systematically. It may also simplify the quantization of the models.

This is joint work with R. Marsh (Leeds). I will describe m-cluster categories of type A using a category of diagonals of a regular polygon. This generalises a result of Caldero, Chapoton and Schiffler for m=1.

An atom exposed to an intense laser field may loose one or several of its electrons by photoionization. The field effectively turns all bound states into resonances of finite lifetime. For fields of constant intensity, this process is amenable to a time-independent description based on the method of complex scaling and on the Floquet theory of differential equations with periodic coefficients. The ionization rate of the state and the position of multiphoton resonances between Stark-shifted states can then be inferred from the spectrum of complex quasienergies of the Hamiltonian. After a general introduction to this approach, the talk will concentrate on a recent study of the origin of the sharp enhancements of emission of fast photoelectrons at certain intensities found a few years ago in the ionization of rare gases.

Given an initial quantum state and a final quantum state in a Hilbert space, there exist Hamiltonians H that transform one into the other. Subject to the constraint that the difference between the largest and smallest eigenvalues of H is held fixed, which H achieves this transformation in the least time? For Hermitian Hamiltonians this time has a nonzero lower bound. However, among complex PT-symmetric Hamiltonians satisfying the same energy constraint, this time can be made arbitrarily small without violating the time-energy uncertainty principle. The talk will discuss the possible implications of this result.

Please check the local City seminar website for abstract.

A recent paper by Kapustin and Witten synthesises ideas from pure mathematics and quantum field theory that have been developing independently for more than thirty years. Thus the Langlands programme, itself a unification scheme in mathematics, relates to electromagnetic duality and the topological twisting of supersymmetry. An attempt will be made to explain these ideas.