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Plenary Speakers / Conférenciers principaux
- INGRID DAUBECHIES, PACM, Department of Mathematics, Princeton University,
New Jersey 08544-1000, USA
An iterative algorithm for linear inverse problems with
a sparsity constraint
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There exist many different algorithms to compute the (approximate)
inverse of an operator K to recover an approximation of a function
f from data that represent Kf corrupted by noise. Recently,
several approaches have been proposed that are adapted to the case when
f has a sparse expansion in e.g. a wavelet basis. It turns out
that one can find such an approximation to f by means of an iterative
algorithm that uses repeatedly the simple thresholding operator that
solves the problem when K = Id. The successive approximations
converge in norm, and provide a stable regularization of the problem
when the inverse problem is ill-conditioned. (This is joint work with
Christine De Mol and Michel Defrise.)
- ROLAND GLOWINSKI, University of Houston
Operator splitting methods for initial Value problems:
application to the direct numerical simulation of
particulate flow and to computational differential
geometry
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The goals of this lecture are three-fold:
To discuss several operator-splitting methods (OSM) for the
time-discretization of initial value problems.
To combine OSM with finite element and fictitious domain methods in
order to simulate particulate flow for various types of incompressible
viscous fluids.
To combine OSM with mixed finite element methods in order to solve
highly nonlinear partial differential equations originating from
differential geometry (and other fields) such as Eikonal, Monge-Ampè
re's, Pucci's, etc. (see figure below).
This presentation will include the results of numerical experiments,
validating the methodology under consideration. They show, in
particular, the robustness, flexibility, and versatility of operator
splitting methods.
- GERHARD HUISKEN, Tuebingen/Albert Einstein Institute
Surgery for geometric evolution equations
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Geometric evolution equations like the Ricciflow of metrics and the
mean curvature flow of hypersurfaces can be used to smoothen and to
uniformize the geometry of manifolds just like a heat equation can be
used to approximate equilibrium states. In recent years and months
possible singularities of Ricciflow and mean curvature flow have been
understood in such detail that an extension past singularities has
become possible leading to a classification of large classes of
manifolds. The lecture describes joint work with C. Sinestrari on
surgery procedures and longtime existence results for mean curvature
flow of hypersurfaces. The methods in particular provide a complete
classification of 3-dimensional hypersurfaces of positive scalar
curvature in Euclidean 4-space. The lecture will illustrate major
techniques and explain the relation of these results to the work of
Hamilton and Perelmann on Riemannian 3-manifolds.
- JAMES LEPOWSKY, Rutgers University, Piscataway, New Jersey, 08854-8019, USA
An introduction to vertex operator algebra theory
and some of its problems
-
Vertex operator algebra theory is an inherently "non-classical"
subject deeply related to "monstrous moonshine" and many other themes
in mathematics, and to string theory in physics. I will motivate,
introduce and sketch a selection of the main themes and problems,
including some compelling current ones, in this exciting area.
- DENNIS SHASHA, Courant Institute, NYU, New York, New York 10012
Upstart puzzles
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The writer of puzzles often invents puzzles to illustrate a principle.
The puzzles, however, sometimes have other ideas. Sometimes, they
speak up and say that they would be so much prettier as slight variants
of their original selves. The dilemma is that the puzzle-writer
sometimes can't solve those variants. Sometimes he finds out that his
colleagues can't solve them either, because there is no existing theory
for solving them. We discuss a few such upstarts inspired originally
from architecture, zero-knowledge proofs, diplomacy, prime numbers, and
computational geometry. They have given a good deal of trouble to a
certain mathematical detective whom I know well.
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