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I will discuss the metric behavior of the Kahler-Ricci flow on Hirzebruch surfaces assuming that the initial metric is invariant under a maximal compact subgroup of the automorphism group. I will describe how, in the sense of Gromov-Hausdorff, the flow either shrinks to a point, collapses to P^1 or contracts an exceptional divisor. This confirms a conjecture of Feldman-Ilmanen-Knopf. This is a joint work with Jian Song.

"In this joint work with Jingyi Chen and Weiyong He, we prove existence of longtime smooth solutions to mean curvature flow of entire Lipschitz Lagrangian graphs. A Bernstein type result for translating solitons is also obtained."

I will describe some combinatorial problems which arise when computing various types of partition functions for the Donaldson-Thomas theory of a space with a torus action. The problems are all variants of the following: give a generating function which enumerates the number of ways to pile n cubical boxes in the corner of a room. Often the resulting generating functions are nice product formulae, as predicted by the recent wall-crossing formulae of Kontsevich-Soibelman. There are now a variety of techniques, both geometric and combinatorial, to compute these formula. My work uses the entirely combinatorial techniques, namely vertex operators and the planar dimer model; these techniques can be applied essentially "bare-handed" and rely very little upon the underlying algebraic geometry.

Inspired by homological mirror symmetry, Seidel and Thomas constructed braid group actions on derived categories of coherent sheaves of various varieties and proved faithfulness of such actions for braid groups of type A. I will discuss joint work with Hugh Thomas giving some faithfulness results for derived braid group actions of types D and E

Quite a bit of progress has been achieved over the past seven years in understanding from a rigorous mathematical perspective the long time dynamics of waves in the Kerr geometry of a rotating black hole in equilibrium. A proof of the Penrose process for scalar waves has notably been given in this context. I will review some of these results, obtained in collaboration with Felix Finster, Joel Smoller and Shing-Tung Yau. I will also indicate a number of open problems.

"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.

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.

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.