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Faithfulness of braid group actions on derived categories
Chris Brav University of Toronto
PIRSA:09050034Extremal Kahler metrics on projective bundles over a curve
Vestislav Apostolov University of Quebec
PIRSA:09050031Minimal Electroweak Scale Cosmology at the LHC
Michael Ramsey-Musolf University of Wisconsin–Madison
Lagrangian Seidel homomorphism and an application
Shengda Hu Wilfrid Laurier University
PIRSA:09050029
Combinatorics inspired by Donaldson-Thomas theory
Benjamin Young McGill University
PIRSA:09050035I 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.Faithfulness of braid group actions on derived categories
Chris Brav University of Toronto
PIRSA:09050034Inspired 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 EWave equations in Kerr geometry
Niky Kamran McGill University
PIRSA:09050033Quite 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.Ricci Solitons with Large Symmetry Group
Andrew Dancer University of Oxford
PIRSA:09050049We produce new examples of Ricci solitons, including many of non-Kahler type, by looking for solutions with symmetries, thus reducing the equations to dynamical systemsA Geodesic Equation for the Space of Sasakian metrics
Pengfei Guan McGill University
PIRSA:09050036"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."The geometry of the AdS/CFT correspondence
PIRSA:08050063I 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.Relative string topology
PIRSA:09050032I'll discuss how to get an interesting invariant of submanifolds by using the ideas of string topology.Extremal Kahler metrics on projective bundles over a curve
Vestislav Apostolov University of Quebec
PIRSA:09050031I 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.Special geometries associated to quaternion-Kahler 8-manifolds
PIRSA:09050030In 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.Minimal Electroweak Scale Cosmology at the LHC
Michael Ramsey-Musolf University of Wisconsin–Madison
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.Lagrangian Seidel homomorphism and an application
Shengda Hu Wilfrid Laurier University
PIRSA:09050029This 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.Mirror Symmetry for Blow Ups
Denis Auroux University of California, Berkeley
PIRSA:09050028This 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.