I will explain how a novel use of dimer models in string theory sheds light on the non-commutative geometry of toric Calabi-Yau 3-fold singularities (and also explain what this means).

One dimensional spin chains systems represent an ideal place to study a wide range of non perturbative phenomena such as exotic phases, fractional excitations and statistics. A method to study these systems is the so called Matrix Product States, which provides an ansatz for the ground state and excitations. The MPS method exploits the entanglement properties of spin chains but it is limited to describe systems with short range entanglement. For this reason it cannot properly describe critical systems where the entanglement entropy grows with the size. In this talk we shall present an extension of the MPS which overcomes this difficulty using Conformal Field Theory. The ansatzs so obtained have strong resemblences with the Laughlin wave function of the Fractional Hall effect.

The average quantum physicist on the street believes that a quantum-mechanical Hamiltonian must be Dirac Hermitian (invariant under combined matrix transposition and complex conjugation) in order to guarantee that the energy eigenvalues are real and that time evolution is unitary. However, the Hamiltonian H=p2+ix3, which is obviously not Dirac Hermitian, has a real positive discrete spectrum and generates unitary time evolution, and thus it defines a fully consistent and physical quantum theory. Evidently, the axiom of Dirac Hermiticity is too restrictive. While H=p2+ix3 isnot Dirac Hermitian, it is PT symmetric, that is, invariant under combined space reflection P and time reversal T. The quantum mechanics defined by a PT-symmetric Hamiltonian is a complex generalization of ordinary quantum mechanics. When quantum mechanics is extended into the complex domain, new kinds of theories having strange and remarkable properties emerge. Some of these properties have recently been verified in laboratory experiments. If one generalizes classical mechanics into the complex domain, the resulting theories have equally remarkable properties.

We want to know all reductive subgroups of the five exceptional algebraic groups. In low characteristic some unusual subgroups appear which are contained inside parabolic subgroups. I will show how to find these by an archaeological dig.

After an overview of some of the achievements of the AdS/CFT correspondence when applied to realistic field theories, I will describe recent progress in the area.

A pedagogical introduction into the ideas, propects and successes of spacetime quantization will be presented, with emphasis on what it tells us about hte problems of quantizing gravity and renormalizing quantum field theory.

A new framework within which an integrable defect may be described is introduced in the classical context (type II defect). It relays on the introduction of a new degree of freedom located on the defect and coupled to the discontinuity of the fields across the defect. Its features will be analyzed and few concrete examples will be provided. In particular, this new scheme allows to provide a description of the Tzitzeica model with a defect, which was not possible in the previous approach (type I defect).

Motivated by recent work on frustrated spin chains and anyonic fusion chains, we define a 1D quantum Hamiltonian (based on the Temperley-Lieb algebra) which contains these models as special cases. This Hamiltonian contains a new integrable point with Z_2-symmetry, which we have solved exactly. In this talk, I will first explain the physical motivations and recall basic facts on the Temperley-Lieb algebra, a central object in 2D Statistical Mechanics. Then I will present analytical and numerical results for the phase diagram, and comment on perspectives in the fields of Condensed Matter Theory and Integrable Models.

The classic Faraday wave experiment consists of a horizontal layer of fluid that spontaneously develops a pattern of standing waves on its surface as it is driven by vertical oscillation with amplitude exceeding a critical value. Faraday wave experiments have consistently produced patterns with remarkably high degrees of symmetry. Quasipatterns, which are quasiperiodic in any spatial direction, are particularly interesting since there is, as yet, no satisfactory theoretical understanding of their formation. We use multi-frequency parametric forcing to investigate the formation of patterns and approximate quasipatterns in a model partial differential equation, which plays the same role for the Faraday wave experiment that the Swift--Hohenberg equation plays for convection. We exploit three-wave resonant interactions to design forcing functions that ought to produce complex patterns, and make quantitative comparisons between weakly nonlinear predictions and the solutions of the PDE. This comparison reveals the limitations of the theory, and we explore ways in which these limitations can be addressed. Based on: Design of parametrically forced patterns and quasipatterns, by A.M. Rucklidge and M. Silber. SIAM J. Applied Dynamical Systems 8 (2009) 298-347.

In this seminar, I will give an introduction to a series of ideas which suggest that many aspects of quantum fied theories, including the celebrated N=4 super Yang-Mills, may be most simply understood in terms of a dual theory in twistor space. No previous knowledge of string theory or twistor theory will be assumed.

Using methods of integrability, enormous progress has been done in calculating the spectrum of operators in N=4 Super Yang-Mills theory as well as of energies of strings. I will discuss advances in finding the spectrum of the Konishi operator, the simplest nontrivial operator of finite size, which is already a hard problem. A further mathematical problem is to prove integrability. I will discuss the algebraic objects appearing in the AdS/CFT correspondence, namely Yangians, and also show how they are related to the spectral equations.

The AdS/CFT correspondence establishes an equivalence of a four-dimensional supersymmetric gauge theory and string theory in Anti-de-Sitter space. Remarkably, the AdS/CFT system can be solved exactly with the help of integrability methods typical for two-dimensional models. I will review the general framework of the AdS/CFT duality, describe how integrability arises in AdS/CFT and how it can be used to compute the exact non-perturbative spectrum of supersymmetric Yang-Mills theory.

The 11 year solar magnetic cycle is driven by a hydromagnetic dynamo. However, the exact nature of this dynamo mechanism is still not fully understood, and there are several scenarios proposed to explain the observed behaviour. Here I will briefly overview observations of the solar magnetic field before moving on to discuss why magnetic buoyancy is a key ingredient in the models proposed to explain the observations of the large-scale magnetic field. I will go on to discuss the recent progress in our understanding of magnetic buoyancy with emphasis on double-diffusive magnetic buoyancy.

I will review the gauge/string correspondence, with particular emphasis on recent exact results based on integrable techniques.

Recently the mathematical structure of the static correlation functions of the spin-1/2 Heisenberg chain has been completely understood. It has been proved by Boos, Jimbo, Miwa and Smirnov that, after appropriate regularizations, all static correlation functions become polynomials in a one-point and a special neighbour-two-point correlation function. We review this important result and show that those two functions allow for an efficient description in terms of the solutions of linear and non-linear integral equations. This opens the way for applications in quantum field theory and condensed matter physics.

We consider quivers/skew-symmetric matrices under the action of mutation (in the cluster algebra sense). We classify those which are isomorphic to their own mutation via a cycle permuting all the vertices, and give families of quivers which have higher periodicity. The periodicity means that sequences given by recurrence relations arise in a natural way from the associated cluster algebras. We present a number of interesting, new nonlinear recurrences, necessarily with the Laurent property, of both the real line and the plane. In particular, we show that some of these recurrences can be linearised and, with certain initial conditions, give integer sequences which contain all solutions of some particular Pell equations.

Many integrable field theories are known to have topological excitations, so-called solitons in their spectra. Although their existence is not directly tied to integrability, the existence of infinitely many local conserved quantities plays an important role in their dynamics. Recently we investigated whether such excitations can survive a nonintegrable perturbation. We found a very simple and comprehensive picture for models that are formulated as perturbations of conformal field theories. It turns out that in many cases solitons are confined by the nonintegrable perturbations, but there are large classes of models for which nontrivial topological excitations do survive.

When a salt-stratified fluid is cooled from the side, a two-dimensional convection pattern of cells is formed along the vertical side-wall and develops into horizontal intrusions which grow away from the wall. Such conditions exist in the oceans along melting icebergs, and the growth of these intrusions prevents melt-water from rising to the surface. As the intrusions grow, a series of layers is created in the salt water, with stepped temperature and salinity profiles. In this talk, I will describe an experimental and theoretical study carried out while at the ITG in Cambridge on the formation and growth of these double-diffusive intrusions. Observations were made of growth rates of the intrusions, of internal velocities, and of temperature and salinity distributions. The rate of growth of the intrusions was found, surprisingly, to depend on the length of the experimental tank, with the end-wall playing a role in their evolution right from the beginning of the experiment (Malki-Epshtein, Phillips and Huppert, JFM 2004). Internal waves are visualised using particle tracing methods and are shown to propagate throughout the experimental tank, maintaining hydrostatic equilibrium and having a strong role in setting up the layers. Future application of the results of this study to the large scale could have significant implications for the modelling of oceanic double-diffusive processes, which are believed to drive large vertical and lateral fluxes of heat and salt.

Topological orders are non-symmetry breaking phases of matter, which provide a new paradigm in condensed-matter physics. Their characterization is a major open problem, although considerable progress has been made since the proposal of the topological entanglement entropy. In this talk I will review how topological orders have been proposed to serve as robust quantum memories. Then the dynamical stability of topologically ordered quantum states will be examined. Conditions will be derived on the type of dynamical evolution that allows quantum recurrence and preservation of the initial topological order.

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