2008 CUNY Geometric Analysis Conference
Celebrating Józef Dodziuk's 60th Birthday
February 23-24, 2008
Abstracts
Saturday Schedule:
Sunday Schedule:
"Discretization and Differential Forms"
Abstract:
Even though the integral defines a duality in the algebra of forms on a
closed oriented manifold, there is no dual coproduct because of functional
analysis difficulties. Morally there should be a coproduct with a
compatibility with the product. This combined structure is called a
Frobenius Algebra and provides links with other discussions on manifolds
related to string theory and Chern-Simons Quantum Theory. Using ideas from
homotopical algebra one can produce discrete analogs of a Frobenius Structure
derived from the imperfect infinite dimensional one coming from differential
forms. I would like to discuss these ideas in my talk.
"Can one hear bi-invariant metrics on compact Lie groups?"
Abstract:
We show that bi-invariant metrics are spectrally isolated among all
left-invariant metrics on compact simple Lie groups. In fact, finitely many
eigenvalues suffice to locally distinguish any given bi-invariant metric
within
this class. In the case of simple Lie groups, the first two non-zero
eigenvalues suffice. (This is joint work with D. Schueth and C. Sutton.)
preprint
"Colorful Eigenfunctions"
Abstract: The lecture is about joint work with Marc
Maintrot, Alexandre Masserey, Loris Renggli and Klaus-Dieter Semmler.
The aim is to compute numericlly the eigenfunctions of the Laplacian on
compact Riemann surfaces and to
visualize the behaviour of the so-called small eigenfunctions, i.e. those
with eigenvalues less than 1/4. This is
related with Jozef Dodziuk's work. In the lecture, presented jointly with
Marc Maintrot, many examples are shown.
"Homotopy Algebras in Geometric Analysis"
Abstract:
Dodziuk's early work on cochain convergence and spectral results
lead naturally to analogous questions about non-linear structures, for
example the algebra of differential forms. I'll describe how a more general
notion, called a homotopy algebra, appears in the context of cochains,
currents and Lp forms, and discuss the analogous convergence issues.
"Atiyah's Question about Integrality of L2 Betti Numbers"
Abstract: L2-Betti numbers are defined as Murray-von Neumann dimension of
certain cohomology modules, defined over the von Neumann algebra of the
fundamental group of the space in question. Early on, Atiyah asked which
are the possible values of these numbers. If
the group is torsion-free, they are conjecturally always integers. A
famous consequence of this conjecture is the non-existence of zero
divisors in the group ring of such a fundamental group - as it then
embeds in a canonically defined skew field.
We present positive and negative results concerning this conjecture and
its generalisations. In particular, we draw light on the more algebraic
side of this question.
preprint
preprint
``Differentiability of Lipschitz maps from metric measure spaces to
Banach spaces."
Abstract: We will survey joint work with Bruce Kleiner on the almost
everywhere differentiability of Lipschitz maps from metric measure spaces to
Banach spaces and discuss applications to bi-Lipschitz nonembedding
theorems. We will also
indicate some related joint work with Kleiner and Naor on quantitative
nom-embedding theorems for the target L1.
"Amenable Groups"
Definitions: Given a locally compact group, G, a mean is a linear
functional on L infty of G which takes that constant function 1 to the
real number 1 and nonnegative functions to nonnegative numbers. We say
a mean is left invariant if it takes the same value on a function f
as it does on the left shift of f, fLg, for any value
g in G. The locally compact group G is called amenable
if it has a left invariant mean defined on it. When G is discrete
it is amenable iff there is a finitely additive left invariant
probability measure. The speaker will discuss Amenable Groups.
"A Survey of the Topology of Manifolds with Nonnegative Ricci Curvature"
Abstract:
While there have been many advances in the understanding of the topology
of complete noncompact manifolds with nonnegative Ricci curvature, this
area is wide open for further study. In particular, Milnor's famous 1969
conjecture that such a manifold has a finitely generated fundamental group
is still open even in dimension three. The speaker will survey a
selection of theorems and examples emphasizing the results with the
most geometrically intuitive proofs. A
survey article with open problems was written with Zhongmin Shen. A more
recent result of Michael Munn will also be mentioned
preprint.