I will give a brief account of some examples of non-Hermitian Hamiltonian systems with real spectra, which have appeared in the literature over the last four decades. I will discuss the spectral properties of these type of Hamiltonians and explain how their reality results from PT-symmetry, quasi-Hermiticity, pseudo-Hermiticity or supersymmetry. Subsequently I review the general technicalities needed to formulate a consistent quantum mechanical system in this context by constructing an appropriate metric and domain. I will provide some Lie algebraic examples. Taking PT-symmetry as a guiding principle one may construct deformations of integrable models, such as Calogero-Moser-Sutherland models, the Korteweg-deVries or Burgers equation. It turns out that some of these deformations are supersymmetry preserving. Others even leave the integrabilty in tact and therefore lead to new types of integrable system.

I will discuss the expansion of a Bose-Einstein condensate, released from a harmonic trap, in the presence of a random potential. The potential causes scattering of the condensate, preventing its free propagation. Under appropriate conditions some part of the condensate will get stuck (Anderson localization), whereas the rest will diffuse away. Expansion of a cold Fermi gas will be also discussed.

We present a model to study decoherence on non--interacting qubits. Single qubit decoherence (as measured by purity) is obtained in linear response. Two qubit decoherence is solved using the spectator configuration. Entanglement evolution is considered. The n-qubit case is studied using weaker assumptions. Various results are exemplified using a kicked spin chain as a toy model.

In analogy to Beck and Cohen's superstatistics (1), we connect the canonical Gaussian ensembles of the random-matrix theory (RMT) to their superstatistical generalizations through the fluctuation of an intensive parameter, the local density of states (2). On one hand, the superstatistical RMT, seen from the present perspective, may bear interest per se because of the additional nontrivial fluctuations introduced in a simple model. On the other hand, it may constitute a useful statistical paradigm for the analysis of the spectral fluctuations of systems with mixed regular-chaotic dynamics. In contrast to other proposals for applying RMT to mixed dynamics, the superstatistical approach yields ensemble of matrices, which are invariant with respect to base transformation. The formalism has been checked by the analysis of experimental resonance spectra of mixed microwave billiards (3). The spectra for each billiard are represented as time series in which the level order plays the role of time. Each series is shown to have two relaxation times as required by superstatistics, which involves the folding of two distribution functions. Analysis of the time series suggests that the superstatistical parameter has an inverse-chi-square distribution. The experimental distribution nearest-neighbor level spacings and strength functions agree with the corresponding predicted distributions. (1) C. Beck and E.G.D. Cohen, Physica A 322, 267 (2003). (2) A.Y. Abul-Magd, Phys.Rev. E 71, 066207 (2005). (3) A.Y. Abul-Magd, B. Dietz, T. Friedrich, and A. Richter, Phys. R

After a short introduction to the Painleve equations, I'll study the Hamiltonian structure of the second Painleve hierarchy, an infinite sequence of nonlinear ordinary differential equations containing PII as its simplest equation. The n-th element of the hierarchy is a non linear ODE of order 2n in the independent variable z depending on n parameters denoted by t(1),...,t(n-1) and alpha(n). I'll introduce new canonical coordinates and obtain Hamiltonians for the z and t(1),...,t(n-1) evolutions. I'll give explicit formulae for these Hamiltonians showing that they are polynomials in our canonical coordinates.

In this talk I will first briefly review the extreme value statistics of independent random variables and show how three limiting distributions (Gumbel, Frechet and Weibull distributions) emerge there. Next I'll discuss the celebrated Integer Partition problem of Hardy and Ramanujan and its connection to the ideal Bose gas. We will see how the same three limiting distributions of extreme value statistics appear in the Integer Partion/Bose gas problem. The connection between these three different problems respectively in probability theory, number theory and statistical physics is intriguing and fascinating.

The cylindrical Kadomtsev-Petviashvily equation (CKP) also known as Johnson equation was introduced by R.S. Johnson in 1978 in the context of describing surface waves in a shallow imcopressible fluid. Later it was derived also as describing the internal waves in a stratified medium by V.D. Lipowskij in 1995. The same equation was obtained in 2000 for description of nonlinear acoustic waves by S. Leble and A. Sukhov. We present a large families of explicit solutions to the CKP-I equation and the CKP -II equation obtained by means of the algebrogeometrical approach and by use of the Darboux transformation method. Some plots of these solutions will be also presented as well as the hamiltonian formulation of the CKP model. Particular case of the obtained solutions with crossed parabolic profiles correspond well enough to the real waves observed in thin films of shallow water being cooled along the inclined plane. This talk is based on a recent joint work with A.O Smirnov and Ch. Klein.

It is shown that non-intersecting Brownian particles, leaving from one point on the real line and forced to return to one or two points lead to some infinite-dimensional diffusions, which are described by non-linear PDE's. These equations have their origin in the theory of multicomponent KP equations.

Quantum graphs have become a paradigm for quantum chaos. It is indeed known that universal interference effects can be observed in the statistics of their spectra. Here, we investigate the possibility for the modulus of the high energy eigenfunctions to become uniformly spread over the graph - a property known as quantum ergodicity. A field theoretical method enables to identify the class of quantum ergodic graphs, and to predict the rate with which quantum ergodicity is obtained.

Standard nonperturbative covariant gauge fixing procedure leaves the theory with Gribov copies and on lattice even Neuberger zero-zero problem. Due to this Neuberger problem, BRST and SUSY on lattice are still open and urgent questions to be addressed. I will introduce the problems using Landau gauge for a simple toy model, compact QED on a one dimensional lattice, and propose a modification which completely resolves Gribov-Neuberger problems on this simple toy model and even the higher dimensional generalization. This gauge-fixing term for compact QED on lattice is the classical XY-model Hamiltonian, and in condensed matter terms the problem is to get ALL extrema of this Hamiltonian exactly. To give a full analytical proof for the higher dimensional generalization, I will need to introduce a tailor-made terminology of Algebraic Geometry. I will also go on proposing two algorithms for gauge-fixing on lattice that use sophisticated applied mathematics and give efficient results derived from Numerical Algebraic Geometry.

I will report on some joint work with Arno Kujlaars on the two-matrix model in random matrix theory. In this model one is interested in the eigenvalue statistics for two NxN hermitian Matrices, M1 and M2, taken random from exp(-NTrV(M1)-NTrW(M2)+tauTrM1M2)dM1dM2. Here V and W are two polynomials of even degree and tau is called coupling contant. The eigenvalues statistics of M1 and M2 can be expressed in terms of certain biorthogonal polynomials. Therefore, if one can find asymptoticss for the biorthogonal polynomials then one also knowns the asymptotics for the eigenvalue statistics. Asymptotic results for the biorthogonal polynomials are known in the physics literature, however, they are without rigurous proofs. I will discuss a rigorous approach for a special case. The key element is a Deift-Zhou steepest descent analysis for a 4x4 Riemann-Hilbert problem.

I will discuss a recent work which concerns the critical behavior of eigenvalues in ensembles 1/Z(n,N) det M(2 alpha) exp(-N Tr V(M)) dM with alpha greater than -1/2, where the factor det M(2alpha) induces critical eigenvalue behavior near the origin. Supposing that the limiting mean eigenvalue density associated with V is regular, and that the origin is a right endpoint of its support, one can compute (using the Deift-Zhou steepest-descent method) the limiting eigenvalue correlation kernel in the double scaling limit as n, N to infinity such that n(2/3) (n/N-1) = O(1). It turns out that the limiting kernel can be described through a distinguished solution of the thirty fourth Painleve equation. This solution is related to a particular solution of the Painleve II equation, which however is different from the usual Hastings-McLeod solution. The talk is based on joint work with Alexander Its and Arno Kuijlaars.

The eigenvalues of large unitary random matrices show universal local behavior, only depending on the so-called scaling regime. It is known that in the bulk of the spectrum, local correlations of the eigenvalues are given in terms of the sine kernel, while at the edge one obtains the Airy kernel. In certain critical random matrix ensembles, other limiting correlation kernels can occur. Near singular interior points, the limiting kernel is related to the Hastings-McLeod solution of the Painleve II equation. Near singular edge points on the other hand, one obtains a kernel related to the second member of the Painleve I hierarchy. We describe how one can obtain these kernels rigorously in double scaling limits using the Riemann-Hilbert approach.

The distributions of eigenvalues of random matrices display universal behavior in two ways. On the one hand these distributions appear in many areas of mathematics and physics including such fields as number theory and combinatorics which have no obvious connection to random matrices. On the other hand these distributions are universal in the sense that there are large classes of ensembles displaying the same distributions in the limit of large matrix dimensions. In this talk both aspects of universality will be briefly surveyed. We will also present recent universality results (of the second type) for orthogonal and symplectic ensembles.

We consider the statistical mechanics of a single classical particle placed in an N dimensional gaussian random potential. This is a model system showing several features of glassy systems. The connection between replica symmetry breaking and the complexity of stationary points and minima is revealed in this model by random matrix methods.