The 3rd DIFFERENTIAL GEOMETRY DAY
Wednesday the 11th of May 2005
Peter Røgen (Lyngby)
Geometry in Structural Biology
Because (macro)molecular shape frequently defines the function of
bio-molecules, it seems evident that geometric methods should be an essential
component of any attempt to understand and simulate biological systems.
Existing techniques developed for much smaller molecular systems, however,
rely primarily on sequence and, in some cases, structure information and use
statistical and/or energy based methods to analyse the relationship between
biological structure and function. Although there have been significant
advancements in the field, a systematic solution of many of the most important
biological problems is still elusive, including ab initio protein structure
prediction, the protein folding process, and ligand to protein docking.
In the talk I will give an introduction to Structural Biology as seen by a
geometer and treat automatic classification of protein structures by use of
geometric shape descriptors in more detail.
A protein is a long chain molecule with smaller side chains. The large scale
shape of a protein is given by its backbone, which may be considered as an
open polygonal space curve. I will present a fast way to tell the difference,
not just between a tied and an untied shoelace, but between 400 different
types of tied shoelaces when allowing deformations of the individual types.
Mikael Bengtsson (Kalmar)
On the classification of nonnegatively curved metrics on trivial R^2-bundles over S^3
"Nonnegatively curved metrics on S^2 x R^2", Detlef Gromoll and Kristopher Tapp
ask what nonnegatively curved metrics on S^n x R^k may look like. They remark
that e.g. S^2 x R^2 admits very large families of such metrics when the product
is nontrivial, and continues to give the classification of the trivial product,
which turns out to be more rigid. A similar classification is underway in the
cases S^2 x R^3 (e.g. by Tapp: "Rigidity for nonnegatively curved metrics on
S^2 x R^3." and Marenitch: "Rigidity of non-negatively curved metrics on open
five-dimensional manifolds") as well as in S^3 x R^2 (Marenitch and Bengtsson:
"On non-negatively curved metrics on open five-dimensional manifolds.").
In the latter case the classification turns out to be dependent on whether a
certain vector field has zeroes or not. Further, the set of zeroes of this
vector field - if there are any at all - has a fairly rigid structure in itself.
Martin Svensson (Lund)
Harmonic morphisms from symmetric spaces
Harmonic morphisms are maps between Riemannian manifolds which pull back local
harmonic functions on the codomain to local harmonic functions on the domain.
Equivalently, they are characterized as solutions to an over-determined non-linear
system of partial differential equations. This makes the question of existence
very difficult to answer in general, and there are examples where no such maps,
not even locally defined, can exist. In my talk I will discuss the progress of
finding solutions to the existence problem for harmonic morphisms from Riemannian
symmetric spaces. I will describe a construction which produces harmonic morphisms
from symmetric spaces of type II, i.e. simple, compact and connected Lie groups.
By duality, these maps in turn induce harmonic morphisms from symmetric spaces
of type IV, which, in some cases, turn out to be even globally defined.
Karsten Grosse-Brauckmann (Darmstadt)
Constant mean curvature surfaces: from real world applications to recent results
Surface that minimize area under a volume constraint have constant mean curvature.
I will mention examples of such surfaces beyond the well known soap bubbles.
While for many decades mathematical existence results for complete surfaces
with constant mean curvature had been scarce, now, surprisingly, many examples
are known. I will describe an existence result for embedded genus 0 surfaces
with ends, obtained jointly with R. Kusner (Amherst) and J. Sullivan (Berlin).