Black holes that are in equilibrium with their surroundings are necessarily bounded by isolated horizons: non-expanding null hypersurfaces whose cross-sections are compact spacelike surfaces. Over the last decade this formalism has become a standard tool used in the study of classical and quantum gravity. In this talk I will discuss a recent application of isolated horizons to the study of supersymmetric black holes. Results include the fact that supersymmetric isolated horizons are necessarily extremal and have severely restricted geometry, electromagnetic charges and angular momentum.

We consider random Schrödinger operators on tree graphs and prove absolutely continuous spectrum at small disorder for two models.

The first model is the usual binary tree with certain strongly correlated random potentials. These potentials are of interest since for complete correlation they exhibit localization at all disorders. In the second model we change the tree graph by adding all possible edges to the graph inside each sphere, with weights proportional to the number of points in the sphere.

This is joint work with David Hasler and Wolfgang Spitzer.

Quantum information science, an interface area of mathematics, physics and computing science, is concerned with the manipulation, computation and communication of information, where the information is encoded in two (or more) level quantum systems called "qubits", unlike classical information, which is encoded in Boolean "bits". The devices used in this science are governed by the principles of quantum mechanics, which opens the possibility for a large range of applications. In this talk I will discuss the long standing additivity conjecture that the minimum entropy output of a completely positive trace preserving linear map, as measured using the von Neumann entropy, is additive under taking tensor products. After enormous efforts by the most experts in the field during the last 12 years, this conjecture has been recently proven to be false. Here I will present a slight modification to this (false) conjecture and discuss some recent progress and future directions.

This talk is based on a joint work with Shmuel Friedland, Aidan Roy, and Jon Yard.

The ordinary differential equations governing the radial parts of the perturbations of Kerr-Newman black holes admit Liouvillian (closed-form) solutions for a discrete set of explicit frequencies. For configurations of the black hole charge and angular momentum in which the angular parts of the perturbations approach the spherical harmonics we find Liouvillian solutions satisfying the boundary conditions for quasinormal modes.

Quantum mechanics prescribes the probabilities of possible outcomes when a measurement is made at a particular time. The times when events like decay occur are also random, but the theory provides no direct prescription for their probabilities, even though these times are often observed in experiments. We study a model where a system with a decaying state is coupled with a detector. If by itself the initial state decays exponentially, the detected time of decay also has an exponential distribution. If the initial state is in the domain of its Hamiltonian, precisely exponential decay is impossible, and a detector which is too precise retards the decay, i.e., the quantum Zeno effect occurs. But if the decay is approximately exponential, and the detector is not overly precise, the detected time also is approximately exponential.

The Moyal equation of a quantum observable corresponds to a phase space representation of its Heisenberg equation of motion. Because the latter has a close relation to the corresponding classical dynamics, the Moyal method is ideally suited to study the transition from classical to quantum behaviour in a system. The Moyal representation is related to the Wigner function like the Heisenberg picture is related to the Schroedinger picture. Unfortunately the Moyal equation is difficult to solve so that only few exact solutions are known.

We have studied nonlinear optical self-phase modulation of Kerr type using the Moyal equation for a single optical field mode. An exact solution for the annihilation operator is found. The phase space representation of this operator is related to the classical field amplitude by a complex factor that shows characteristic singularities in time. We show that these singularities disappear in the classical limit and demonstrate how the uncertainty relation prevents that observable quantities are affected by it.

We consider a system S = S₁ + S₂ of two spins 1/2 (qubits) interacting with several thermal reservoirs. Each spin j=1,2 is coupled to an individual reservoir R_j and the two spins interact collectively with a third reservoir R. All reservoirs are at the same temperature and do not interact directly. Each interaction between a spin and a reservoir has two channels, an energy-conserving and an energy-exchange one.

We analyze decoherence, thermalization and disentanglement of the system S. We show that due to the energy-exchange interactions, the system has a finite disentanglement time, we estimate that time and compare it to decoherence times.

This is joint work with Kassu Gebresellasie.

Asymptotic stability of small bound states in one dimension is proved in the framework of a discrete nonlinear Schrödinger equation with septic and higher power-law nonlinearities and an external potential supporting a simple isolated eigenvalue. The analysis relies on the dispersive decay estimates from Pelinovsky and Stefanov (2008) and the arguments of Mizumachi (2008) for a continuous nonlinear Schrödinger equation in one dimension. Numerical simulations suggest that the actual decay rate of perturbations near the asymptotically stable solitons is higher than the one used in the analysis.

This is a joint work with A. Stefanov and P. Kevrekidis.

We consider the nonlinear Schrödinger equations i¶_t y = H₀ y+ l|y|² y in R³ where H₀ = -D+ V and l is a given constant which can be positive or negative. Assume that the potential V is radial and decays sufficiently fast at infinity. Assume also that the linear Halmitonian H₀ has only two discrete eigenvalues e₀ < e₁ < 0 where e₀ is simple and e₁ has multiplicity 3. We show that there exist three branches of excited states and for certain finite codimesion subset in the space of initial data, we construct solutions y converging to each of these excited states in both non-resonant and resonant cases.

This is a joint work with Stephen Gustafson.

We study the critical behavior of statistical lattice models in 2d using the homotopy of their Fortuin-Kasteleyn (FK) cluster. A FK cluster on a torus is said to be of type {a,b}, a,b Î Z, if it possible to draw a curve belonging to the cluster that winds a times around the first cycle of the torus as it winds -b times around the second. Even though the Q-Potts models make sense only for Q integers, they can be included into a family of models parametrized by b = ÖQ for which the FK clusters can be defined for any real b Î (0,2]. For this family, we study the probability p({a,b}) of a given type of clusters as a function of the torus modular parameter t = t_r+it_i. We compute the asymptotic behavior of some of these probabilities as the torus becomes infinitely thin. Exponents describing these behaviors are defined and related to weights h_r,s of the extended Kac table for r,s integers, but also half-integers. Numerical simulations are also presented.

Joint work with Alexi Morin-Duchesne.

A complete solution to the problem of orthogonal separation of variables of the Hamilton-Jacobi equation in three-dimensional Minkowski space is obtained. The solution is based on the underlying ideas of Cartan geometry and ultimately developed into a general new algorithm that can be employed in the study of Hamiltonian systems defined by natural Hamiltonians within the framework of Hamilton-Jacobi theory. To demonstrate its effectiveness, we investigate, from this viewpoint, the Morosi-Tondo integrable system derived as a stationary reduction of the seventh-order Korteweg-de Vries flow to show explicitly that the system in question is an orthogonally separable Hamiltonian system. The latter result is a new characterization of the Morosi-Tondo system.

This is joint work with Joshua Horwood and Raymond McLenaghan.

The relativistic Vlasov-Maxwell system (RVM in short) is a kinetic model that arises in plasma physics and describes the time evolution of a collisionless plasma whose particles interact through the self-induced electromagnetic field. The plasma is assumed to be at high temperatures, thus the particles may travel at speeds comparable to the speed of light. The main open problem concerning this system is to prove whether or not classical solutions develop singularities in finite time. Glassey and Strauss established the existence and uniqueness of local in time classical solutions for smooth and compactly supported initial data. They showed that such solutions can be continued globally in time provided the momenta of the particles are controlled. Subsequently, they proved that such control is achieved if their kinetic energy density remains bounded for bounded times. In this talk, we show that the latter assumption can be weaken to the boundedness of the spatial density.

This talk represents joint work with Brigitta Vermesi. Using Stein's method, Chatterjee and Crawford proved a CLT for the energy density in the high temperature region of the Sherrington-Kirkpatrick mean-field spin glass model. Using Chen's extension of Stein's method, we proved CLT's for the dilute mean-field spin glass model known as the Viana-Bray model, where only a fraction of edges are present, leaving a random graph. The fluctuations are larger and some new features appear.

In this talk we will review some recent results on the (local exact) controllability of (certain) nonlinear Schrödinger equations with interior, boundary, and bilinear controls. Some applications to quantum control systems will be discussed.

We study the existence of multi-vortex solutions to the Ginzburg-Landau equations external potential in R². We show that if the external potential is strong enough (in some sense) and has widely spaced critical points, then there exists multi-vortex solutions centered near the critical points of the external potential.

This is joint work with Aaron Pakylak.

A procedure is developed to average tensor fields over simply connected differentiable manifolds. Parallel transport in metric affine spaces is reviewed and arguments are made to ensure that an averaging procedure be well defined. We argue that in a Riemannian space (V_n) only parallel transport along geodesics provides for a well-defined averaging procedure. Alternatively, if one wishes to develop a well-defined averaging procedure that is independent of the path used for parallel transport, then one must use a Weitzenbock connection on a flat manifold (a space of distant parallelism or W_n). With these preliminaries, we define of the average of a tensor field over a finite region in both V_n and W_n spaces, and conclude with comparisons to some of the other averaging procedures.

A Monte Carlo method for sampling self-avoiding walks and polygons is presented. The method (GAS, for Generalised Atmospheric Sampling), samples polygons or walks along weighted sequences by implementing elementary moves generated by the positive, negative and neutral atmospheric statistics of the polygons or walks. GAS is a self-tuning algorithm which samples from uniform distributions over lengths of polygons and walks in an interval [0,n_max] - this implementation is called "flatGAS" (flat histogram GAS).

In the case of polygons, states along a sequence realised by GAS are weighted such that the average weight of states of length n edges is proportional to p_n (the number of polygons of length n). Hence, GAS is an approximate enumeration method, and I shall present data estimating the numbers of cubic lattice polygons of specified knot type.

We obtain solvability conditions for some elliptic equations involving non-Fredholm operators with the methods of spectral theory and scattering theory for Schroedinger-type operators.