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Discrete Differential Geometry
Organized by: Alexander I. Bobenko (1), Richard Kenyon (2), John M. Sullivan (3) and Günter M. Ziegler (4)
(1) Fachbereich Mathematik, MA 8-3, Technische Universität Berlin, Str. des 17. Juni 136, 10623, BERLIN, GERMANY(2) Dept. of Mathematics, University of British Columbia, 1984 Mathematics Road, BC V6T 1Z2, VANCOUVER, CANADA
(3) Fakultät II - Institut f. Mathematik, MA 3-2, Technische Universität Berlin, Strasse des 17. Juni 136, D-10623, BERLIN, GERMANY
(4) Institut für Mathematik, Freie Universität Berlin, Arnimallee 2, 14195, BERLIN, GERMANY
The workshop \emph{Discrete Differential Geometry}, organized
by Alexander~I. Bobenko (Berlin), Richard W. Kenyon (Vancouver),
John M. Sullivan (Berlin) and G\"{u}nter M. Ziegler (Berlin),
was held March 5th to March 11th, 2006. The meeting was
very well attended, with almost 50 participants, from as far
away as Australia and China.
Discrete differential geometry is a new and active mathematical
terrain where differential geometry (providing the classical theory
for smooth manifolds) and discrete geometry (concerned with
polytopes, simplicial complexes, etc.) meet and interact.
Problems of discrete differential geometry also naturally
appear in (and are relevant for) other areas of mathematics.
Moreover, the process of discretizing notions, problems
and methods from the smooth theory often brings out new
connections and interrelations between different areas.
The workshop at Oberwolfach brought together researchers with
a wide variety of backgrounds, including of course discrete
geometry and differential geometry, but also integrable systems,
combinatorics, mathematical physics and geometry processing.
The exchange of ideas among different subfields helped to
build new bridges between these mathematical communities.
Discrete differential geometry can be said to have arisen
from the observation that when a notion from smooth geometry
(such as the notion of a minimal surface) is discretized ``properly'',
the discrete objects are not merely approximations of
the smooth ones, but have special properties of their own which
make them form in some sense a coherent entity by themselves.
The discrete theory would seem to be the more fundamental one:
The smooth theory can always be recovered as a limit, while
there seems to be no natural way to predict from the smooth
theory which discretizations will have the nicest properties.
One case where these ideas seem particularly well-developed
is for geometries described by integrable systems. The notion
of a discrete integrable system as given by consistency on
a cubic lattice has already shed new light on classical,
smooth integrable systems.
Another theme which arose repeatedly during the workshop
was that of circle patterns and sphere packings. These
can be used to discretize conformal maps, isothermic surfaces,
and elastic bending energy.
Since a computer works with discrete representations of
data, it is no surprise that many of the applications of
discrete differential geometry are found within computer
science, particularly in the areas of computational geometry,
graphics and geometry processing. The workshop brought
theoreticians together with people interested in these
and other applications.
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