Differential Geometry / Géométrie différentielle
(J. Chen, Organizer)
- C. AREZZO, Università di Parma
To be announced
- J. BRYAN, University of British Columbia, Vancouver
Gromov-Witten theory of local Calabi-Yau's and TQFT
We study the local Gromov-Witten theory of a curve in a Calabi-Yau
3-fold. We show this theory can be exhibited as a 1+1 dimensional
topological quantum field theory. This theory is semi-simple and leads
to nice closed formulas for the local Gromov-Witten invariants.
- MIHAIL COCOS, University of British Columbia
On the heat-equation method of M. Gaffney
The study of L2 harmonic forms on a complete Riemanian manifold is a
very interesting and important subject. It has also found numerous
applications in the field of Mathematical Physics and other related
areas of Mathematics. As early as 1956 M. Gaffney in his PhD thesis at
the University of Chicago proved that the Heat Equation for forms of
arbitrary degree has solution for all time t > 0 and the solution is
smooth (similar results were also obtained by F. Browder). He also
showed that the ``heat flow'' generated by solving this equation
preserves the De Rham cohomology class of the initial data, thus giving
a topologycal method of constructing nontrivial L2 harmonic forms on
certain manifolds with ``rich'' topology. Using Gaffney`s ``heat
flow'' the author of the present paper gives a nice geometrical
interpretation of the vanishing of the L2 cohomology space and also
proves that a conformal change of metric in the case of the hyperbolic
space has no effect on the L2 cohomology.
- A. FRASER, Brown University
Fundamental groups of manifolds with positive isotropic curvature
Positive isotropic curvature (PIC) is a natural and much studied
curvature condition which includes manifolds with pointwise
quarter-pinched sectional curvatures and manifolds with positive
curvature operator. By results of Micallef and Moore there is only one
topological type of compact simply connected manifold with PIC; namely
any such manifold must be homeomorphic to a sphere. On the other hand,
there is a large class of non-simply connected manifolds with PIC. An
important open problem has been to understand the fundamental group of
manifolds with PIC. In this talk we describe a new result in this
direction. The techniques used involve minimal surfaces.
- JACQUES HURTUBISE, CRM
G-bundles over an elliptic curve and dynamical r-matrices
The r-matrix formalism of integrable systems ties in very nicely with
the geometry of rigid bundles over a curve. Recently, through work of
Felder, Etingopf and Varchenko, this formalism was extended to a
particular non-rigid case, that of G-bundles over an elliptic curve.
In joint work with Markman, the geometry of these systems is examined,
and a description is given of the Poisson geometry, as well as a proof
- X. LIU, Notre Dame
Solving universal equations in Gromov-Witten invariants
Universal equations for Gromov-Witten invariants are obtained from
relations in homology classes of moduli spaces of stable curves. When
the underlying manifolds have semisimple quantum cohomology, the
generating functions of genus-2 Gromov-Witten invariants can be
explicitly expressed in terms of genus-0 and genus-1 invariants by
solving already existing universal equations. These equations can also
be used to study the Virasoro conjecture.
- P. LU, Deparment of Mathematics, McMaster University, Hamilton,
Ontario L8S 4K1
Cheeger-Gromov type compactness theorem on oribifolds
We will discuss the generalization of Cheeger-Gromov type compactness
theorem to orbifolds. We will also discuss a compactness theorem on
orbifolds in Ricci flow.
- D. MATESSI, Centre de Recherche Mathematiques, Universite de Montreal,
Families of special Lagrangian tori
We will show some ways to construct isometric embeddings of
one or two parameter families of special Lagrangian submanifolds.
As an application we show that in dimension 3, the moduli space
of special Lagrangian tori is not in general ``special Lagrangian''
in the sense of Hitchin, as it is instead in dimension 2.
- M. WANG, McMaster University, Hamilton, Ontario L8S 4K1
Painleve analysis, Diophantine conditions, and special holonomy
I will report on joint work with A. Dancer of Jesus College, Oxford.
We apply the method of S. Kowalevskaya as further developed by
Painlevé, and modern authors such as Ablowitz, Ramani, and Segur to a
quadratic system associated to the cohomogeneity 1 Ricci-flat
equations. The existence of ``large'' families of convergent
Painlevé expansions in the cases under study leads to restrictions on
the principal orbit type, including diophantine conditions on its
dimension (e.g. existence of integral points on elliptic
curves). Some of the diophantine conditions are related to metrics
with special holonomy.
- R. WENTWORTH, Johns Hopkins University
The Yang-Mills flow revisited
This talk will outline work in progress with G. Daskalopoulos on the
convergence properties of the Yang-Mills flow on holomorphic vector
bundles over higher dimensional Kaehler manifolds.
- BAOZHONG YANG, Stanford University, Stanford, California 94305, USA
Analytic aspects of gauge theories in higher dimensions
I'll talk about some recent results on the analytical aspects for
Yang-Mills theories in dimensions greater than 4. These include
singular-compactness theorems, removable singularity theorems, as well
structure of singularities.