24th
Annual
Geometry Festival
in memory of
Detlef Gromoll
SUNY Stony
Brook, Stony Brook, NY
April 1719, 2009

Detlef Gromoll
(19382008) 


Jeff Cheeger, New York University
Quantitative Behavior
of Maps from the Heisenberg Group to L^{1}
Let (X,d)
denote a metric space and let C
denote a collection of metrics on X such that d'∈C implies c·d'∈C for
any real number c>0.
Put ρ(d,C)=
inf_{d'∈C}
inf{c: d' ≤ d ≤ c·d'}.
Let L denote the
collection of metrics on X of the form d'(x_{1},x_{2})=f(x_{1})f(x_{2})_{L1},
for some map f: X→L^{1}, and let N denote the collection of
metrics
d on X such that (X,d^{1/2})
embeds
isometrically in L^{2}.
It is easy to verify that L⊂N, and so ρ(d,N)≤ ρ(d,L). If X is finite with
cardinality n, it was shown by Bourgain that ρ(d,L)=O(log n)
for any metric d.
Although the problem of computing ρ(d,L)
exactly is equivalent to other fundamental problems for which there is
believed to be no polynomial time algorithm, there
is a quadratic time algorithm for computing ρ(d,N). Goemans and Linial
conjectured that for some universal constant C>0, independent of
n,
ρ(d,N)≤
C· ρ(d,C).
Their conjecture was refuted by KhotVishnoi (2005) who gave a sequence
of
examples for which the best C grows at least like a constant times
log(log n). We will discuss a very difference sequence based on the
Heisenberg group, for which C grows at least like (log n)^a for some
explicit a. This is joint with with Kleiner and Naor. It is an
outgrowth of ealier work of LeeNaor and CheegerKleiner.

Wolfgang Meyer,
Westfälische WilhelmsUniversität Münster
The Contributions of
Detlef Gromoll to Riemannian Geometry  Part 1
(presented by Blaine Lawson, Stony Brook)
The lecture
will include D. Gromoll's contributions to the
Differentable Sphere Theorem, the Diameter Rigidity Theorem, metric
fibrations, the structure theory for complete
manifolds of nonnegative sectional curvature, results for nonnegative
Ricci
curvature, and closed geodesics.

Wolfgang Meyer,
Westfälische WilhelmsUniversität Münster
The Contributions of
Detlef Gromoll to Riemannian Geometry  Part 2
(presented by Michael Anderson, Stony Brook)
The lecture
will include D. Gromoll's contributions to the
Differentable Sphere Theorem, the Diameter Rigidity Theorem, metric
fibrations, the structure theory for complete
manifolds of nonnegative sectional curvature, results for nonnegative
Ricci
curvature, and closed geodesics.

Marcos
Dajczer,
Instituto de Matematica Pura e Aplicada
Conformal Killing
graphs with prescribed mean curvature
I will discuss
the existence and uniqueness of graphs with prescribed mean curvature
function over a bounded domain in Riemannian manifolds endowed with a
conformal Killing vector field. The domain is contained in a
hypersurface in the integrable orthogonal distribution and the graph is
a
hypersurface transversal to the flow lines of the field.
This is joint work with Jorge H. Lira.

Guofang Wei,
University of
California Santa Barbara
Smooth Metric Measure
Spaces
Smooth metric
measure spaces are Riemannian manifolds with
a conformal change of the Riemannian measure and occur naturally as
measured GromovHausdorff limit of Riemannian manifolds. The important
curvature quantity here is the BakryEmery Ricci tensor,
which corresponds to the (synthetic) Ricci curvature lower bound for
(nonsmooth) metric measure spaces. What geometric and topological
results for Ricci curvature can be extended to BakryEmery Ricci
tensor? Recently there are many developments. We will discuss
comparison geometry and rigidity in this direction.

Karsten
Grove,
University
of Notre Dame
Positive Curvature:
the Quest for Examples
We will discuss
recent progress including the discovery and construction of new
examples.

Christina
Sormani, Lehman
College and CUNY Graduate Center
The Intrinsic Flat
Distance between Riemannian Manifolds
We define a new
distance between oriented Riemannian manifolds that we call the intrinsic
flat distance based upon AmbrosioKirchheim's theory of
integral currents on metric spaces. Limits of sequence of manifolds,
with a uniform upper bound on their volumes, the volumes of their
boundaries and diameters are countably H^{m}
rectifiable metric spaces with an orientation and multiplicity that we
call integral current spaces.
In general the GromovHausdorff and intrinsic flat limits do not agree.
However, we show that they do agree when the sequence of manifolds has
nonnegative Ricci curvature and a uniform lower bound on volume and
also when the sequence of manifolds has a uniform linear local
geometric contractibility function. These results are proven using work
of GreenePetersen, Gromov, CheegerColding and Perelman.
We present an example of three manifolds with positive scalar curvature
constructed using GromovLawson connected sums attaching two standard 3
spheres with increasingly many tiny wormholes which converge in the
Gromov Hausdorff sense to the standard three sphere but in the
intrinsic flat sense to the 0 space due to the cancelling orientation
of the two spheres. We conjecture this cannot occur if we exclude
spaces with interior minimal surfaces. This is joint work with S.
Wenger;
transparencies and related preprints are available
here

Gabriel
Paternain,
University of Cambridge
Transparent
Connections over Negatively Curved Surfaces
Let M be a
closed orientable surface of negative curvature.
A unitary connection on a Hermitian vector bundle is said to be
transparent
if its parallel transport along closed geodesics of g is the identity.
In this talk I will try to show that SU(2)transparent connections can
be
understood in terms of B\"acklund transformations and
that the trivial connection is locally unique.

Dedication of Detlef Gromoll Common Room

