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I will discuss some joint work with K. Uhlenbeck. There is a general method for constructing soliton hierarchies from a splitting of Lie algebras. We explain how formal scattering and inverse scattering, Hamiltonian structures, commuting conservation laws, Backlund transformations, tau functions, and Virasoro actions on tau functions can all be constructed in a uni ed way from such splittings.

When considering flat unitary bundles on a punctured Riemann surface, it is often convenient to have a space that includes all possible holonomies around the punctures; such a space is provided by the extended moduli space of Jeffrey. On the other hand, there are certain inconveniences, in particular no clear link to complex geometry via a Narasimhan-Seshadri type theorem. It turns out that the situation can be remedied quite nicely by considering bundles with framings taking values in a Grassmannian. Analogs for general structure groups, and in particular links with recent work of Martens and Thaddeus, will also be discussed. (This is joint work with U. Bhosle and I. Biswas.)

After introducing Killing-Yano tensors and their basic properties, I will concentrate on their applications to black hole physics. Namely, I will focus on two topics: i) symmetries of the Dirac operator in curved background and ii) generalized Killing-Yano tensors in the presence of skew-symmetric torsion and the classification of corresponding Euclidean metrics.

Generalized complex manifolds, like complex manifolds, admit a decomposition of the bundle of di
erential forms. When an analogue of the @ @ lemma holds there is a corresponding Hodge decomposition in twisted cohomology. We look at some aspects of this decomposition, in particular its behavior under deformations of generalized complex structure. We de ne period maps and show a Griths transversality result. We use Courant algebroids to develop the notion of a holomorphic family of generalized complex structures and show the period maps for such families are holomorphic.

In classical terms, an SKT structure is a Hermitian structure for which the Hermitian 2-form is closed with respect to the second order operator ddc. These structures arise naturally in the study of sigma models with (2; 0) or (2; 1)-supersymmetries, much like generalized Kahler structures arise in the (2; 2)-supersymmetric sigma model. While the introduction of generalized complex geometry has provided the correct framework to study generalized Kahler structures and great progress has been made in this area in the last few years, SKT structures laid forgotten. We will take a look at what the generalized complex framework can do for SKT structures and in the process dispel some misconceptions that have arisen over the years.

Spin liquid phases in frustrated magnets may arise in a variety of forms. Here we discuss the possibility of topological insulators of spinons or the fractionalized excitations in spin liquids. These phases should be characterized by "both" of the two popular and different definitions of topological orders, namely the long-range entanglement and the symmetry-protected topological order. We show an explicit construction of such a state in frustrated magnets on the pyrochlore lattice and discuss novel properties such as the finite surface thermal conductivity.

We report on the ab-plane optical properties of the magnetic field inducednormal state of underdoped La1.94Sr0.06CuO4  (Tc=5.5 K), the first such study. We apply strong magnetic fields (4 T and 16 T) along the c-axis. We find that a 4 T field is strong enough to destroy the superconducting condensate.  However at higher fields we observed a gap-like depression in the optical conductivity at low frequency along with parallel growth of a broad absorption peak at higher frequency just above the 5 meV gap. The loss of low frequency conductivity in the gap region is in good agreement with dc magneto resistance measurements on samples from the same batch.  The spectral weight loss in the depression at low frequency is recovered by the spectral weight in the broad peak. Significantly, this spectral weight equals the spectral weight of the superconducting condensate.  The broad peak tracks the SDW order seen by neutron scattering[1] and we suggest offers an optical signature of magnetism.

Chiral superconducting states have attracted an enormous amount of interest in recent years, due both to their intrinsic novelty as well as their potential for quantum information processing. They break both parity and time-reversal sym-metries and have been predicted to harbour Majorana fermions in vortex cores and along their edges.  A crucial challenge in the quest to find such states is identifying robust experimental probes of chirality.  In this talk, I will discuss an intrinsic, anomalous Hall effect that arises in multiband chiral superconductors. This effect arises from interband transitions involving time-reversal symmetry breaking chiral Cooper pairs. I will discuss the implications of this effect for the putative chiral p-wave superconductor, Sr2RuO4, and show that it can contribute significantly to Kerr rotation experiments. Since the magnitude of the effect depends on the structure of the order parameter across the bands, this result may also be used to distinguish between different models proposed for the superconducting state of Sr2RuO4.

We present large-scale quantum Monte Carlo simulations on a sign-problem free Bose-Hubbard model on the kagome lattice. This model supports a quantum Z2 spin liquid phase with fractional excitations and topological order, which can be characterized definitively through calculation of the topological entanglement entropy.  I will outline how the entanglement entropy can be measured in general using a direct implementation of the "replica trick", which allows for the study of entanglement scaling in a variety of other models amenable to study by QMC.  Finally, I will examine the kagome model's superfluid/spin-liquid transition, which is an example of an exotic deconfined quantum critical point called XY*, mediated by the fractional charges.  This fact is demonstrated in several universal quantities that we measure, and may also be reflected the scaling of entanglement entropy at the critical point.

Experiments where impurities were incorporated into helium nanodroplets have shown that the impurity freely rotates, and this has been attributed to the superfluidity of the nanodroplet [1]. Results from experiments with smaller helium clusters suggest that the onset of superfluidity is linked to system size and bosonic exchange effects [2]. We have used path integral techniques to investigate these systems and predict their spectroscopic behaviour in the microwave and the infrared regions of the spectrum. We are particularly interested in observing the superfluid response in clusters where the helium atoms have been substituted with parahydrogen molecules. Molecular hydrogen has been suggested as a potential candidate for the observation of superfluid response but this substance crystallizes before reaching a temperature low enough for superfluidity to appear. We will show theoretical and experimental results of a molecular superfluid response at the nanoscale via the formation of doped hydrogen clusters with a carbon dioxide probe molecule [3]. Properties such as density distributions, spectroscopic features, and effective rotational inertia can be extracted from the simulations. We will show new results for the case of asymmetric top molecules embedded in superfluid para-hydrogen clusters. A perspective on the current challenges of the field will be presented. [1] Grebenev, Toennies, and Vilesov, Science 279, 2083 (1998); Toennies and Vilesov, Angew. Chem.-Int. Edit. 43, 2622-2648 (2004). [2] Tang, Xu, McKellar and Jäger, Science 297, 2030 (2002) [3] Li, Le Roy, Roy, and McKellar, Phys. Rev. Lett. 105, 133401 (2010)

Charged colloidal particles present a controllable system for study a host of condensed matter/many body problems such as crystallization. 2D crystals are invariably hexagonal. Hexagons perfectly tile a flat plane but a soccer ball requires exactly 12 pentagons dispersed among the hexagons on its curved surface. Pentagons and hexagons are positive and negative topological charges, disclinations, sources for positive and negative curvature. But we have discovered that “Pleats”, grain boundaries which vanish on the surface (and play a similar role to fabric pleats) can provide a finer control of curvature. We experimentally investigate the generation of topological charge as flat surfaces are curved. For positive curvature, domes and barrels, there is one pentagon added for every 1/12 of a sphere. Negative curvature is different! For capillary bridges forming catenoids, pleats relieve the stress before heptagons appear on the surface. Pleats are important for controlling curvature from crystals on surfaces, to the shape of the spiked crown of the Chrysler building. Adding a particle to a flat surface produces an interstitial - usually an innocuous point defect. On a curved surface interstitials are remarkable, forming pairs or triplets of dislocations which can fission dividing the added particles into fractions which migrate to disclinations.    Work done with William Irvine, e.g. Nature 468, 947 (2010).

I will discuss string cosmology and the dynamics of multiple scalar fields in potentials that can become negative, and their features as (Early) Dark Energy models. The point of departure is the ``String Axiverse'', a scenario that motivates the existence of cosmologically light axion fields as a generic consequence of string theory. These fields can constitute part of the Dark Matter, suppressing structure formation in a manner similar to massive neutrinos. Future observations will constrain their existence to percent level accuracy. I couple such an axion to its corresponding modulus and give a detailed presentation of the rich cosmology of such a model, ranging from the setting of initial conditions on the fields during inflation, to the asymptotic future. A dynamical systems analysis reveals the existence of many fixed point attractors, repellers and saddle points, which I analyse in detail, and provide a geometric interpretation of. These fixed points can be used to bound the couplings in the model. A systematic scan of certain regions of parameter space reveals that the future evolution of the universe in this model can be rich, containing multiple epochs of accelerated expansion.

(Given time) I will also discuss the relevance of isocurvature perturbations in these models, and the motivation to study negative (terminal) vauca in the context of eternal inflation and the string landscape, as recently discussed by Susskind.

(based primarily on Marsh, Tarrant, Copeland and Ferreira (to appear) but also arXiv:1102.4851  arXiv:1110.0502)