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Connections in Geometry and Physics 2009

Conference Date: 
Vendredi, Mai 8, 2009 (All day) to Dimanche, Mai 10, 2009 (All day)

 

The main objective of this workshop is to gather together researchers, from Canada and around the world, who work at the interface between geometry and physics. A secondary objective is to further increase Canada's presence and visibility in geometry within the wider international mathematical community, and to serve as an opportunity for geometers in the region with common interests to meet. This in particular influenced the choice to focus the workshop on three main research areas: elliptic and parabolic equations in geometry, geometry and topology of moduli spaces, and structures in symplectic geometry.

 

 

Vestislav Apostolov, Université du Québec à Montréal

Denis Auroux, MIT

Chris Brav, University of Toronto

Albert Chau, University of British Columbia

Andrew Dancer, Oxford University

Andrea Gambioli, Université du Québec à Montréal

Veronique Godin, University of Calgary

Pengfei Guan, McGill University

Sergei Gukov, University of California Santa Barbara

John Harnad, Concordia University

Shengda Hu, University of Waterloo

Colin Ingalls, University of New Brunswick

Niky Kamran, McGill University

Jeffrey Morton, University of Western Ontario

James Sparks, Oxford University

Ben Weinkove, University of California San Diego

Benjamin Young, McGill University

Fabien Ziltener, University of Toronto

 

Vestislav Apostolov, Université du Québec à Montréal

Extremal Kahler metrics on projective bundles over a curve

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.


Denis Auroux, MIT

Mirror symmetry for blowups

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.


Chris Brav, University of Toronto

Faithfulness of braid group actions on derived categories

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.


Albert Chau, Univeristy of British Columbia

Lagrangian mean curvature flow for entire Lipschitz graphs

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.


Andrew Dancer, Oxford University

Ricci Solitons with Large Symmetry Group

We produce new examples of Ricci solitons, including many of non-Kahler type, by looking for solutions with symmetries, thus reducing the equations to dynamical systems


Andrea Gambioli, Université du Québec à Montréal

Special geometries associated to quaternion-Kahler 8-manifolds

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.


Veronique Godin, University of Calgary

Relative string topology

I'll discuss how to get an interesting invariant of submanifolds by using the ideas of string topology


Pengfei Guan, McGill University

A Geodesic Equation for the Space of Sasakian metrics

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.


Sergei Gukov, University of California Santa Barbara

Branes and Quantization


John Harnad, Concordia University

Hamiltonian structure of isomonodromic deformations of rational connections on the Riemann sphere

The classical "split" rational R-matrix Poisson bracket structure on the space of rational connections over the Riemann sphere provides a natural setting for studying deformations.  It can be shown that a natural set of Poisson commuting spectral invariant Hamiltonians, which are dual to the Casimir invariants of the Poisson structure, generate all deformations which, when viewed as nonautonomous Hamiltonian systems, preserve the generalized monodromy of the connections, in the sense of Birkhoff  (i.e., the monodromy representation, the Stokes parameters and connection matrices).  These spectral invariants may be expressed as residues of the trace invariants of the connection over the spectral curve.  Applications include the deformation equations for orthogonal polynomials having "semi-classical" measures.  The $\tau$ function for such isomonodromic deformations coincides with the Hankel determinant formed from the moments, and is interpretable as a generalized matrix model integral.  They are also related to Seiberg-Witten invariants. (This talk is based in part on joint work with: Marco Bertola, Gabor Pusztai and Jacques Hurtubise).


Shengda Hu, University of Waterloo

Lagrangian Seidel homomorphism and an application

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.


Colin Ingalls, University of New Brunswick

Spaces of Linear Modules on Regular Graded Clifford algebras

The space of regular noncommutative algebras includes regular graded Clifford algebras, which correspond to base point free linear systems of quadrics in dimension n in P^n.  The schemes of linear modules for these algebras can be described in terms of this linear system. We show that the space of line modules on a 4 dimensional algebra is an Enriques surface called the Reye congruence, and we extend this result to higher dimensions.


Niky Kamran, McGill University

Wave equations in Kerr geometry

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.


Jeffrey Morton, University of Western Ontario

Groupoids of Connections and Higher-Algebraic QFT

This talk will discuss, illustrated by a toy example, how to construct "higher-algebraic" quantum field theories using groupoids.  In particular, the groupoids describe configuration spaces of connections, together with their gauge symmetries, on spacetime, space, and boundaries of regions in space.  The talk will describe a higher-algebraic "sum over histories", and how this construction is related to usual QFT's, and particularly the relation to the case of the Chern-Simons theory.


James Sparks, Oxford University

The geometry of the AdS/CFT correspondence

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.


Ben Weinkove, UC San Diego

The Kahler-Ricci flow on Hirzebruch surfaces

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.


Benjamin Young, McGill University

Combinatorics inspired by Donaldson-Thomas theory

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.


Fabien Ziltener, University of Toronto

Symplectic Vortices and a Quantum Kirwan map

A Hamiltonian action of a Lie group on a symplectic manifold $(M,\omega)$ gives rise to a gauge theoretic deformation of the Cauchy-Riemann equations, called the symplectic vortex equations. Counting solutions of these equations over the complex plane leads to a quantum version of the Kirwan map. In joint work with Christopher Woodward, we interpret this map as a weak morphism of cohomological field theories.

 

Funding provided in part by:

Perimeter Institute

University of Waterloo

Fields Institute

NSERC-CRSNG