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"Sasakian geometry is often described as an odd dimensional counterparts of K\""ahler geometry. There is a natural Riemannian metric on the space of Sasakian metrics, which in turn gives a geodesic equation on this space. It can be viewed as parallel case of a well-known geodesic equation for the space of K\""ahler metrics. The equation is connected to some interesting geometric properties of Sasakian manifolds. It is a complicated complex Monge-Amp\`ere type involving gradient terms. We discuss the problem of existence and regularity of solutions of this type of equations. This is a joint work with Xi Zhang."

I will describe how the geometry of supersymmetric AdS solutions of type IIB string theory may be rephrased in terms of the geometry of generalized (in the sense of Hitchin) Calabi-Yau cones. Calabi-Yau cones, and hence Sasaki-Einstein manifolds, are a special case, and thus the geometrical structure described may be considered a form of generalized Sasaki-Einstein geometry. Generalized complex geometry naturally describes many features of the AdS/CFT correspondence. For example, a certain type changing locus is identified naturally with the moduli space of the dual CFT. There is also a generalized Reeb vector field, which defines a foliation with a transverse generalized Hermitian structure. For solutions with non-zero D3-brane charge, the generalized Calabi-Yau cone is also equipped with a canonical symplectic structure, and this captures many quantities of physical interest, such as the central charge and conformal dimensions of certain operators, in the form of Duistermaat-Heckman type integrals.

I will discuss the existence problem of extremal Kahler metrics (in the sense of Calabi) on the total space of a holomorphic projective bundle P(E) over a compact complex curve. The problem is not solved in full generality even in the case of a projective plane bundle over CP^1. However, I will show that sufficiently ``small'' Kahler classes admit extremal Kahler metrics if and only if the underlying vector bundle E can be decomposed as a sum of stable factors. This result can be viewed as a ``Hitchin-Kobayashi correspondence'' for projective bundles over a curve, but in the context of the search for extremal Kahler metrics. The talk will be based on a recent work with D. Calderbak, P. Gauduchon and C. Tonnesen-Friedman.

In this talk we will discuss the (local) construction of a calibrated G_2 structure on the 7-dimensional quotient of an 8-dimensional quaternion-Kahler (QK) manifold M under the action of a group S^1 of isometries. The idea is to construct explicitly a 3-form of type G_2, using the data associated to the S^1 action and to the QK structure on M. In the same spirit, we can consider the level sets of the QK moment-map square-norm function on M, and again take the S^1 quotient: we will discuss in this case the construction of half-flat metrics in dimension 6, under suitable circumstances. This talk is based on a joint work with F. Lonegro, Y. Nagatomo and S. Salamon, still in progress.

It is well known that new physics at the electroweak scale could solve important puzzles in cosmology, such as the nature of dark matter and the origin of the cosmic baryon asymmetry. In this talk, I discuss some of the simplest, non-supersymmetric possibilities, their collider signatures, and the prospects for their discovery and identification at the LHC.

This is joint work with Francois Lalonde. Using an analogue of Seidel's homomorphism in Lagrangian Floer homology for one Lagrangian, we give a condition for a diffeomorphism on a Lagrangian to extend to a Hamiltonian diffeomorphism on the whole symplectic manifold.

This talk is a report on joint work with Mohammed Abouzaid and Ludmil Katzarkov about mirror symmetry for blowups, from the perspective of the Strominger-Yau-Zaslow conjecture. Namely, we first describe how to construct a Lagrangian torus fibration on the blowup of a toric variety X along a codimension 2 subvariety S contained in a toric hypersurface. Then we discuss the SYZ mirror and its instanton corrections, to provide an explicit description of the mirror Landau-Ginzburg model (possibly up to higher order corrections to the superpotential). This construction allows one to recover geometrically the predicted mirrors in various interesting settings: pairs of pants, curves of arbitrary genus, etc.

Quantum random walks have received much interest due to their non- intuitive dynamics, which may hold a key to radically new quantum algorithms. What remains a major challenge is a physical realization that is experimentally viable, readily scalable, and not limited to specific connectivity criteria. In this seminar, I will present an implementation scheme for quantum walking on arbitrarily complex graphs. This scheme is particularly elegant since the walker is not required to physically step between the nodes; only flipping coins is sufficient. In addition, by taking advantage of the inherent structure of the CS decomposition of unitary matrices, we are able to implement all coin operations necessary for each step of the walk simultaneously. This scheme can be physically realized using a variety of quantum systems, such as cold atoms trapped inside an optical lattice or electrons inside coupled quantum dots.

Recently methods of integrability were shown to be useful for solving gauge theories in various dimensions. I will make an introduction into integrability in two dimensions and demonstrate how the integrability works also for some three and four dimensional gauge theories.