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Next Seminar
| 2 December |
Speaker: Richard Montgomery (UCSC)
Title: Compactifying the fibers of jet spaces, Nash blow-up, and Contact
Geometry.
Abstract:
The space $J^1 (M, R)$ of 1-jets of real-valued functions on an (n-1)-manifold
M provides one of the standard examples of a contact manifold,
its dimension being $2n +1$. The space $J^k = J^k (M, R)$ of k-jets of functions also has a
canonical distribution (subbundle of the tangent bundle)
and the k-jet of any smooth function on $M$ is an integral submanifold for this distribution.
The contact group, which contains the diffeomorphism group of $M \times R$
acts on $J^k$ in such a way as to preserve this distribution. And this action is transitive.
The fibers of $J^k \to M$ are not compact, rather they are affine spaces.
We show how the Cartan prolongation procedure provides for a minimal compactification
of these fibers in such a way that the contact group still acts on the whole space by symmetries.
However the action is no longer transitive when $k > 2$, and the fibers are no longer
smooth spaces when $k > 3$, and $n>2$. The central problem is to classify
the orbits of this action. The key to understanding the orbits is
to realize points of the compactified space as the iterated prolongations of singular hypersurface
in $M \times R$. The process of prolonging hypersurfaces is identical to the algebraic geometer's
`Nash blow-up'. This process leads to a complete resolution of the classification problem
when $n=2$ -- the case of curves in the plane, and is mostly wide open when $n = 3$ -- the case of surfaces in space
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Fall Quarter
| 30 September |
Speaker: Brian Clarke (Stanford)
Title: Metric Structures on the Manifold of Riemannian Metrics
Abstract: The manifold $\mathcal{M}$ of all smooth Riemannian metrics over
a closed, finite-dimensional base manifold carries itself several natural Riemannian metrics.
We discuss the metric geometry of a particular one of these, the $L^2$ metric, chosen for its
applications in Teichm\"uller theory and in previous investigations of the geometry and
topology of $\mathcal{M}$. The main result is a description of the completion of
$\mathcal{M}$ with respect to the $L^2$ metric.
At the end of the talk, we discuss some directions for further study, including applications
to Teichm\"uller theory and the moduli space of Riemannian metrics. We also discuss how the
metric geometry of the $L^2$ metric relates to that of a newly discovered metric structure on
$\mathcal{M}$.
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| 7 October |
Speaker: Boris Vertman (Stanford)
Title: The regular-singular Sturm-Liouville operators and their zeta-
determinants.
Abstract: Recent advances in the computation of zeta-determinants for
Laplace-type operators with specific regular-singular potentials of model
type and general boundary conditions at the singularity have been made by
Klaus Kirsten, Paul Loya and Jinsung Park. A formula for zeta-determinants
for a general class of regular-singular potentials, however only for
specific boundary conditions at the singular end, is due to Matthias
Lesch.
This poses the question whether appropriate results can also be achieved for
Sturm-Liouville operators with general regular-singular potentials and
general boundary conditions. We answer this question affirmatively and
provide a formula for the zeta-determinant in terms of the
Wronski-determinant of the boundary value problem, generalizing the
earlier results of Lesch and Kirsten-Loya-Park.
This is a joint project with Matthias Lesch.
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| 14 October |
Speaker: Kai Cieliebak (Munich)
Title: Stable Hamiltonian structures
Abstract: Stable Hamiltonian structures are natural geometric structures
on odd-dimensional manifolds, a special case of which are contact
structures. They first appeared in Hamiltonian
dynamics as a stability condition on energy levels. Recently, they
have gained importance as the geometric structures underlying symplectic
field theory. The goal of this talk is to give an introduction to stable
Hamiltonian structures and illustrate their geometric, dynamical and
topological aspects with many examples.
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| 21 October |
Speaker: No Speaker
Title: N/A
Abstract: N/A
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| 28 October |
Speaker: Andrea Malchiodi (SISSA)
Title: Variational theory for a class of singular Liouville equations
Abstract: We study an elliptic equation on compact surfaces with exponential nonlinearities and singular data, motivated by Chern-Simons theory or from the Gaussian curvature prescription problem.
We prove new existence results using a new improved Moser-Trudinger inequality, which is scaling invariant, combined with variational arguments.
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| 4 November |
Speaker: Rod Gover (Auckland)
Title: The Poincare-Einstein programme and overdetermined PDE
Abstract:
A compact manifold with boundary is said to have a Poincare-Einstein
structure if its interior is equipped with a negative curvature
Einstein metric, in terms of which the boundary is suitably ``at
infinity''. A central problem is to relate the conformal geometry of
this boundary to the Riemannian structure of the interior, and this is
linked to the ideas behind Maldacena's AdS/CFT correspondence in
String theory. There is a natural approach to aspects of this problem
via conformal geometry, a certain overdetermined PDE and its
prolongations. This approach also leads to a natural way to extend
the programme, and new problems in geometric analysis.
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| 11 November |
Speaker: Daniel Grieser (Oldenburg) (3pm in PDE seminar)
Title: The exponential map on a singular surface
Abstract:
Understanding the geodesics on a singular space is of interest
both from the point of view of studying the inner geometry of such
spaces and also from a PDE point of view, since the geodesics are the
expected trajectories of singularities of solutions of the wave
equation. In the case of conical singularities, it was proved by Melrose
and Wunsch that the geodesics hitting the conical point foliate a
neighborhood of that point smoothly. In other words, there is a smooth
exponential map based at the singular point. We consider a different
class of singularities and prove that while the exponential map based at
the singularity is not smooth, its precise asymptotic behavior (to any
order) near the singularity can be described completely in terms of
certain blow-ups of the space and of its cotangent bundle. Joint work
with V. Grandjean.
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| 11 November |
Speaker: Erez Lapid (Hebrew)
Title: Analytic aspects of the trace formula
Abstract:
In the 1950's Selberg developed the trace formula and used it
among other things to show the existence in abundance of Maass forms
for quotients of the upper half plane by congruence subgroups.
In the late 70's and early 80's Arthur widely extended the
trace formula to the context of arithmetic quotients G(Q)\G(A)
for any reductive group G defined over Q.
I will discuss the structure of the trace formula (especially on the
spectral side) and what is needed in order to extend classical results
about spectral asymptotics to the non-compact case.
The new results are joint work with Tobias Finis and Werner Muller
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| 18 November |
Speaker: Pierre Albin (Courant)
Title: Equivariant cohomology and resolution
Abstract:
The equivariant cohomology of a manifold
with a group action is, in some sense, the
cohomology of the space of orbits. I will describe
joint work with Richard Melrose where we make this
precise.
In fact our method of lifting the group action and
the equivariant cohomology to a manifold with
corners and smooth orbit space also allows us to
extend the `delocalized' equivariant cohomology of
Baum, Brylinski, and MacPherson from actions of
Abelian Lie groups to actions of arbitrary compact
Lie groups.
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| 25 November |
No Seminar due to Thanksgiving
|
| 2 December |
Speaker: Richard Montgomery (UCSC)
Title: Compactifying the fibers of jet spaces, Nash blow-up, and Contact
Geometry.
Abstract:
The space $J^1 (M, R)$ of 1-jets of real-valued functions on an (n-1)-manifold
M provides one of the standard examples of a contact manifold,
its dimension being $2n +1$. The space $J^k = J^k (M, R)$ of k-jets of functions also has a
canonical distribution (subbundle of the tangent bundle)
and the k-jet of any smooth function on $M$ is an integral submanifold for this distribution.
The contact group, which contains the diffeomorphism group of $M \times R$
acts on $J^k$ in such a way as to preserve this distribution. And this action is transitive.
The fibers of $J^k \to M$ are not compact, rather they are affine spaces.
We show how the Cartan prolongation procedure provides for a minimal compactification
of these fibers in such a way that the contact group still acts on the whole space by symmetries.
However the action is no longer transitive when $k > 2$, and the fibers are no longer
smooth spaces when $k > 3$, and $n>2$. The central problem is to classify
the orbits of this action. The key to understanding the orbits is
to realize points of the compactified space as the iterated prolongations of singular hypersurface
in $M \times R$. The process of prolonging hypersurfaces is identical to the algebraic geometer's
`Nash blow-up'. This process leads to a complete resolution of the classification problem
when $n=2$ -- the case of curves in the plane, and is mostly wide open when $n = 3$ -- the case of surfaces in space
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