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Next Seminar
| 10 March |
Speaker: Andrew Hassell (ANU)
Title: Quasi-orthogonality of boundary values of eigenfunctions,
and applications to a numerical method for computing eigenfunctions.
Abstract:
Consider Dirichlet eigenfunctions for a smooth bounded plane domain.
The normal derivatives of these eigenfunctions are known, at least heuristically,
to be "quasi-orthogonal" when the eigenvalues are sufficiently close. I will discuss
a new result -- with a remarkably simple proof -- expressing this quasi-orthogonality,
and apply it to give sharp theoretical bounds on the accuracy of the "method
of particular solutions" for numerically computing such eigenfunctions and eigenvalues.
This is joint work with Alex Barnett (Dartmouth).
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Winter Quarter
| 6 January |
Speaker: Sergey Cherkis (Trinity)
Title: Yang-Mills Instantons in Curved Backgrounds
Abstract:
A Yang-Mills instanton is a connections with the
self-dual curvature on a vector bundle over a
four-manifold. Instantons play an important role in
differential geometry and physics. Since the
original construction of Atiyah, Drinfeld, Hitchin,
and Manin of instantons on flat space it was
generalized in a number of ways by Kronheimer and
Nakajima and by Nahm. All of these generalizations
are restricted to base spaces that are flat or have
a flat orbifold limit. We present a general
construction for instantons on essentially curved
ALF spaces. For the case of instanton number one,
it allows us to find the explicit metrics on their
moduli space as well as the explicit instanton
connection.
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| 13 January |
Speaker: Peter Perry (Kentucky)
Title: Spectral Geometry of Manifolds Hyperbolic Near Infinity
Abstract:
This talk concerns joint work with David Borthwick
and with David Borthwick, Tanya Christiansen, and
Peter Hislop. A conformally compact manifold is a
compact manifold with boundary, $X$, together with
a metric $g$ having the property that $x^2 g$ is a
smooth metric up to the boundary, where $x$ is a
defining function for the boundary of $X$. The
Riemannian manifold $(X,g)$ is \emph{hyperbolic near
infinity} if the sectional curvatures of $g$ are
identically $-1$ in a neighborhood of the boundary.
Since $X$ is non-compact, ``most'' of the spectral
data for the Laplacian are the scattering
resonances, defined as poles of the meromorphically
continued resolvent of the Laplacian. In this talk
we will review recent results on the geometric
content of the resonances, including their
distribution in the complex plane, and the extent to
which the resonances constrain the Riemannian metric
in a compact subset of $X$.
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| 20 January |
Speaker: No Seminar
Title:
Abstract:
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| 27 January |
Speaker: Richard Melrose (MIT)
Title: Resolution and Compactification of Moduli and configuration spaces
Abstract:
In this talk I will describe some of the significant
properties of three compact manfolds with corners obtained, respectively, as
the resolution of a group action (joint work with Pierre Albin), the
asymptotic configuration space for a vector space and the
compactified moduli space of magnetic monopoles (joint work with Michael
Singer). I will not have time to discuss these constructions in any
detail but intend instead to emphasize their common, particularly their
iterative features and how these can be expected to appear elsewhere.
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| 3 February |
Speaker: Francisco Martin (Granada)
Title: Results on the Calabi-Yau problem for minimal surfaces.
Abstract:
A natural question in the global theory of minimal surfaces, first
raised by Calabi and later revisited by Yau, asks whether or not there
exists a complete immersed minimal surface in a bounded domain D in
Euclidean space. Consider a domain D which is convex (possibly all R^3)
or which is smooth and bounded. Given any open surface M, we prove that
there exists a complete, proper minimal immersion f: M ---> D.
Moreover, if D is smooth and bounded, then we prove that the immersion f
can be chosen so that the limit sets of distinct ends of M are disjoint,
connected compact sets in the boundary of D.
Finally, we will prove that the results above are sharp, in the sense
that they fail to be true when D is neither convex or smooth and
bounded.
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| 10 February |
Speaker: Yng-Ing Lee (Taiwan)
Title: Eternal Solutions to Lagrangian Brakke Flow
Abstract:
In a joint work with M.T. Wang, we construct Hamiltonian stationary
Lagrangian shrinkers and expanders for mean curvature flow asymptotic to a
pair of Schoen-Wolfson cones, and show that they can be glued together to
yield eternal solutions to Lagrangian Brakke flow without mass loss. Here
Brakke flow is a weak formulation of mean curvature flow, and Schoen-Wolfson
cones are obstructions to the existence of special Lagrangian surfaces. It
is conjectured that among all Schoen-Wofson cones, only (2, 1) cone is area
minimizing. We can use Brakke flow to distinguish (2, 1) cone from other
cones, and prove an infinitesimal version of the conjecture.
We later generalize these results to higher dimensions and in particular
obtain higher dimensional analog of Schoen-Wolfson cones with various
topology. Other non-Hamiltonian stationary eternal solutions are also
obtained.
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| 17 February |
Speaker: Tom Banchoff (Brown)
Title: Geometry of Normal Euler Numbers for Smooth and Polyhedral Surfaces in Four-Space
Abstract:
The normal Euler number for an orientable or non-orientable
smooth surface immersed in Euclidean four-space is the algebraic number
of its intersections with the surface deformed along a generic normal
vector field, for example the mean curvature vector field. We compare
two definitions of the normal Euler number that are geometric enough to
work as well for polyhedral surfaces, involving inflection faces and the
self-linking numbers of spherical polygons in the three-sphere, related
to a construction of Gromov, Lawson, and Thurston as interpreted by
Kuiper. This presentation will be illustrated by interactive computer
graphics.
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| 24 February |
Speaker: Spyros Alexakis (Toronto)
Title: A black hole uniqueness theorem
Abstract:
I will discuss recent joint work with A. Ionescu and S.
Klainerman on the black hole uniqueness problem. A classical result of
Hawking (building on earlier work of Carter and Robinson) asserts that any
vacuum, stationary black hole exterior region must be isometric to the Kerr
exterior, under the restrictive assumption that the space-time metric should
be analytic in the entire exterior region. We prove that Hawking's theorem
remains valid without the assumption of analyticity, for black hole
exteriors which are apriori assumed to be "close" to the Kerr exterior
solution in a very precise sense. Our method of proof relies on certain
geometric Carleman-type estimates for the wave operator.
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| 3 March |
Speaker: Zhou Zhang (Michigan)
Title: Scalar curvature behavior for the Kahler-Ricci flow
Abstract:
The optimal existence result of the Kahler-Ricci
flow indicates the possibility of finite and infinite time
singularities by cohomology characterization. We focus on
the behavior of scalar curvature, which can be very different
in these two cases.
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| 10 March |
Speaker: Andrew Hassell (ANU)
Title: Quasi-orthogonality of boundary values of eigenfunctions,
and applications to a numerical method for computing eigenfunctions.
Abstract:
Consider Dirichlet eigenfunctions for a smooth bounded plane domain.
The normal derivatives of these eigenfunctions are known, at least heuristically,
to be "quasi-orthogonal" when the eigenvalues are sufficiently close. I will discuss
a new result -- with a remarkably simple proof -- expressing this quasi-orthogonality,
and apply it to give sharp theoretical bounds on the accuracy of the "method
of particular solutions" for numerically computing such eigenfunctions and eigenvalues.
This is joint work with Alex Barnett (Dartmouth).
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Fall Quarter
| For the Fall 2009 Schedule go
here |
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