1st seminar:

Title: Diffusion maps and geometric harmonics

Speaker: Stephane Lafon, Applied Mathematics, Yale University

When/where: Tuesday, May 18th, 2:00PM, Room 200 AKW

Abstract::

We investigate the problem of finding meaningful geometric descriptions
of data sets. The approach that we propose is based upon diffusion
processes. We show that by designing a local geometry that reflects some
quantities of interest, it is possible to construct a diffusion operator
whose eigendecomposition produces an embedding of the data into $\mathbb
R^n$ via a \emph{diffusion map}. In this space, the data points are
reorganized in such a way that the geometry combines all the local
information captured by the diffusion process, and the Euclidean
distance defines a diffusion metric that measures the proximity of
points in terms of their connectivity. The case of submanifolds of
$\\mathbb R^n$ is the object of greater attention, and we show how to
define different kinds of diffusions on these structures in order to
recover their Riemannian geometry. General types of anisotropic
diffusions are also addressed, and we explain their interest in the
study of differential and dynamical systems.

Secondly, we introduce a special set of functions that we term
\emph{geometric harmonics}. These functions allow to perform
out-of-sample extensions of empirical functions defined on the data set.
We show that the geometric harmonics, and the corresponding restriction
and extension operators are a valuable tool for the study
of the relation between the intrinsic and extrinsic geometries of a set.
In particular, they allow to define a multiscale extension scheme, in
which empirical functions are decomposed into frequency bands, and each
band is extended to a certain distance so that it satisfies some version
of the Heisenberg principle.

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2nd seminar:

Title: Interpolation with Prolates

Speaker: Mark Tygert, Applied Mathematics, Yale Unviersity

When/where: Tuesday, May 18th, 4:15PM, Room 200 AKW

Abstract: This talk will introduce the Prolate Spheroidal Wave Functions
of zeroth order, their connection with time-frequency localization (as
discovered by David Slepian, Henry Landau, and Henry Pollack), and their
application to numerical quadratures and interpolation schemes (as
introduced by my thesis advisor, Vladimir Rokhlin, and by his doctoral
student Hong Xiao). The talk will present a proof that certain
interpolation schemes work correctly. Yoel Shkolinsky, Vladimir Rokhlin,
and the speaker put together this proof, with the help of R. R. Coifman.
THE TALK WILL NOT BE APPROPRIATE FOR ANYONE WHO IS FAMILIAR WITH THE
NUMERICAL ASPECTS OF PROLATES, AS PRESENTED IN THE RECENT ARTICLE BY
XIAO, ROKHLIN, AND YARVIN.