Conference in Berlin, July 16  20, 2007
Overview
Daily Schedule
Talks
Discrete Nonlinear Hyperbolic Equations. Classification of Integrable Cases
We consider discrete nonlinear hyperbolic equations on Z^{2} of
the form
Q(
x_{m, n},
x_{m+1, n},
x_{m, n+1},
x_{m+1, n+1}) = 0.
Integrability is understood as 3Dconsistency, that is, as the possibility to
consistently impose equations of the same type on all the faces of a
threedimensional cube. This allows one to set these equations also
on multidimensional lattices Z^{N}. We classify integrable
equations with complex fields x and Q affinelinear with
respect to each argument. The method is based on the analysis of
singular solutions.
This is joint work with Alexander Bobenko and Yuri Suris.
YangBaxter Equation and Discrete Conformal Symmetry
We consider the FaddeevVolkov solution of the YangBaxter equation
connected with the modular double of the quantum group
U_{q}(sl_{2}). It defines an Isingtype lattice model with positive
Boltzmann weights where the spin variables take continuous values
on the real line. The free energy of the model is exactly
calculated in the thermodynamic limit. The model describes
quantum fluctuations of circle patterns and the associated
discrete conformal transformations connected
with the Thurston's discrete analogue of the Riemann mappings
theorem. In particular, in the quasiclassical limit the model precisely
describe the geometry of integrable circle patterns with prescribed
intersection angles.
A Universality Theorem for Realization Spaces of Polyhedral Maps
A universality theorem for maps in R^{3} is shown, stating
essentially that every semialgebraic set can occur as realization space of some
map (with a distinguished set of vertices); more precisely:
Theorem. (Universality Theorem for Maps) Let n≥0 and
G be a graph with vertex set
{v_{1}, ..., v_{5}, w_{1}, ...,
w_{n}}. Let P ⊆ R^{3n} be a
semialgebraic set defined over Q. Then there is a map M (on some
orientable manifold) containing G as an induced subgraph such that for
each subfield K ⊆ R a straight standard embedding f of
G in K^{3} can be extended to a polyhedral embedding
(i.e. the facets are strictly convex) of M if and only if
(f(w_{1}), ...,
f(w_{n})) ∈ P.
Here, "standard embedding"
means that v_{1}, ..., v_{5} are mapped onto some
fixed given projective base.
Corollaries. (1) For each strict subfield L of the field of
real algebraic numbers there is a map M which can be polyhedrally
embedded in R^{3} but not in L^{3}.
(2) The realizability problem for maps in R^{3} is polynomial
time equivalent to the "existential theory of the reals" and thus NPhard.
Dimers on Graphs and Discrete Spin Structures on Surfaces
A dimer configuration (or perfect matching) on a graph G is a choice of
edges of G such that each vertex of G is adjacent to exactly one of these
edges. It is known since the work of Kasteleyn that the number of dimer
configurations on G can be written as a linear combination of 4^{g}
Pfaffians, where g is the genus of a surface S where G can be embedded.
Each of these 4^{g} terms corresponds to a certain orientation of the edges
of G.
In this talk, I shall explain a correspondance between these orientations
of G and the spin structures on S. As a result, we obtain a very simple
and geometric proof of Kasteleyn's formula. Also, it allows to understand
the coefficients in this formula as the Arf invariant of the corresponding
spin structures.
This is joint work with N. Reshetikhin.
Stability of Boundary Measures
The differential geometry of piecewiselinear objects has been much studied
over the years. This lead to discrete notions of curvature that are consistent
with their smooth analogs, and possess convergence properties when the
piecewiselinear object tends to a smooth one. In this talk, I will discuss an
attempt to deal with the curvature of point cloud data. The main technical
result is an error bound showing convergence to the classical notions when the
sampling density increases.
This is joint work with F. Chazal and Q. Merigot
Plane foliations of the regular skew polyhedra {6,44} & {4,64}
Plane foliations of triply periodic surfaces have a rich topological
structure. The problem, introduced by S. P. Novikov in 1982,
lead to the discovery of a topological invariant whose dependence on the
direction of the bundles of cutting planes is of fractal nature. After
extending the main theorems to polyhedra we studied numerically in detail the
case of the infinite skew polyhedra {6,44} & {4,64} and ultimately discovered
a simple algorithm that allowed us to solve completely analytically the {4,64}
case.
In this talk we will introduce the main theorems of the theory and then
present the main numerical and analytical results about the two polyhedra
above.
Measuring Periodicity in Gene Expression with Persistence
The work presented in this talk is motivated by microarray
experiments aimed at illuminating gene regulation in
embryonic somite development. This development is
approximately periodic, generating one somite at a time.
The microarray experiment yields a onedimensional function
representing expression per gene. We measure the extent
to which a function follows the same periodic pattern and
is a candidate in participating in the process. To this
end we simplify functions, integrate numbers of critical
points, and prove the stability of the resulting measure.
We evaluate the resulting rankings based on genes for
which there is biological evidence of their direct
involvement in somite development.
The microarray work is due to Olivier Pourquie and MaryLee
Dequeant. The assessment of periodicity of the expressions
of few thousand gene is done in collaboriation with
Yuriy Mileyko.
Tutte's Embedding Theorem Reproved
Tutte's celebrated "spring embedding theorem" of 1963
states that a 3connected planar graph can be drawn in the plane as a
nondegenerate straightline drawing with convex faces by pinning down
its boundary vertices to form a convex shape, and positioning each
interior vertex at the centroid of its neighbors. This theorem is the
basis of many geometry parameterization techniques.
Over the years a number of quite lengthy proofs of Tutte's theorem
have been proposed. In this talk we give a short proof based on simple
properties of harmonic oneforms and a discrete index theorem. Using
this machinery, we also show how to generalize the theorem to the case
of a nonconvex boundary.
Mixed Volumes, Rigidity, and Colin de Verdière Matrices
One of the definitions of mixed volumes comes from parallel displacements of
the faces of a convex polyhedron. The Hessian of the volume function with
respect to
such displacements has a constant signature, given by AlexandrovFenchel
inequalities. This observation allows us to reprove or reinterpret some known
results. We are going to mention the infinitesimal rigidity of convex
polytopes, the Lovász construction of Colin de Verdière matrices for
planar graphs, and the hyperbolic polyhedra of shapes of polytopes due to
Thurston.
Neighborly Cubical Polytopes and Spheres
We prove that the neighborly cubical polytopes (previously studied
jointly with Günter M. Ziegler) arise as a special case of the
neighborly cubical spheres constructed by Babson, Billera, and Chan.
By relating the two constructions we obtain an explicit description
of a nonpolytopal neighborly cubical sphere and,
further, a new proof of the fact that the cubical equivelar surfaces of
McMullen, Schulz, and Wills can be embedded into R^{3}.
This is joint work with Thilo Rörig.
Lie Group and Homogeneous Variational Integrators:
Towards a Geometrically Exact Model of Elasticity
I will survey recent work on the synthesis of Lie group techniques and
variational integrators to construct symplecticmomentum methods which
automatically stay on Lie groups and homogeneous spaces without the need for
constraints, local coordinates, or reprojection.
These provide the basis for constructing geometrically exact numerical schemes
for representing flexible structures and surfaces arising in modern engineering
applications. We also anticipate that a combination of such techniques with
notions of discrete curvature arising in discrete differential geometry will
provide a more rational approach towards the numerical simulation of
elasticity.
Rigidity of Polyhedral Surfaces
We study rigidity of polyhedral surfaces and the moduli space of polyhedral
surfaces using variational principles. Curvaturelike quantities for polyhedral
surfaces are introduced. Many of them are shown to determine the polyhedral
metric up to isometry. The action functionals in the variational approaches are
derived from the cosine law and the Lengendre transformation of them. These
include energies used by Colin de Verdière, Brägger, Rivin,
Cohen & Kenyon & Propp, Leibon and Bobenko & Springborn for variational principles on
triangulated surfaces.
Discrete Riemann Surfaces
Generalized Curvatures
The aim of this lecture is to present a coherent
framework for defining
suitable curvature measures associated to a huge class of subsets of
E^{N}. These measures appear as local geometric invariants,
involving 1 or 2 differentials in the smooth case, (basically 1 for the length,
the area and the volume and 2 for the curvatures). These general geometric
invariants coincide with the standard ones in the smooth case, but are also
adapted to triangulations, meshes, algebraic and subanalytic sets, and
"almost" any compact subsets of E^{N}. Moreover, the continuity
for a suitable topology and the inclusionexclusion principle will be
systematically satisfied.
Constructing Polyhedral Surfaces
I will discuss two problems: isometric realization of
polyhedral surfaces in R^{3} and volumeincreasing
isometric deformation of polyhedral surfaces (inflation).
I will survey both geometric and computational aspects
of these problems.
Weierstrass Representation of Discrete Surfaces
The Weierstrass representation for minimal surfaces in terms of a pair
ψ = (ψ_{1}, ψ_{2})
of holomorphic functions is well known. More generally,
Taimanov and Konopelchenko have shown that a very similar representation is
possible for any surface in R^{3}
in terms of a ψ that satisfies
Dψ = 0
for a certain Dirac operator
D. This can be viewed as a "quaternionic holomorphicity condition" for
ψ. Any two surfaces f, g obtained from the same D are
conformal to each other and in fact
H_{f} df = H_{g} dg,
where H denotes the mean curvature. We present a discrete version of
this theory.
This is joint work with Christoph Bohle and Franz Pedit.
Global Parameterization of Surface Meshes using Branched Coverings
We introduce a new algorithm for the automatic computation of a global parameterization
of arbitrary simplicial 2manifolds. The parameter lines are guided by a given
frame field, for example, by principal curvature frames. The parametrization is
globally continuous and allows a remeshing of the surface into quadrilaterals.
The algorithm converts a given frame field into a single vector field on a branched
covering of the 2manifold, and then generates an integrable vector field by a
Hodge decomposition on the covering space. Except an optional smoothing and alignment
of the initial frame field, the algorithm is fully automatic and generates
high quality quadrilateral meshes.
This is joint work with Felix Kälberer and Matthias Nieser.
Cyclic Structures and Incidence Theorems
The talk investigates the the interplay of incidence theorems on points lines,
circles and conics and their underlying combinatorial structure.
In particular we will focus on the aspect how proofs of incidence theorems
can be considered as cyclic structures on oriented manifolds in various ways.
There we will investigate several different approaches, that lead to various
elegant proving methods. Furthermore we will consider the interplay of
incidence theorems on circles embedded in the complex plane and the role of
arguments of cross ratios of their points.
Bertrand Curves, Geodesics of Constant Torsion and their Discretization
A little known result due to Razzaboni states that there exists a class of
integrable parallel surfaces on which classical Bertrand curves and their
offset mates form geodesics. In the particular case of curves of constant
torsion,
the Betrand mates and therefore the corresponding Razzaboni surfaces coincide
and it turns out that the associated Bäcklund transformation algebraically
resembles the classical Bäcklund transformation for pseudospherical surfaces.
Even though it appears that this connection does not go beyond a 'resemblance'
in the continuous case, it emerges that its precise geometric origin may be
revealed at the discrete level. In addition, as a byproduct, it becomes
transparent why these particular Razzaboni surfaces come in pairs and are related
by an analogue of the classical Bianchi transformation for pseudospherical
surfaces.
On the Rigidity of Weakly Convex Polyhedra
Cauchy proved that convex polyhedra are rigid. Dehn later
proved that they are also infinitesimally rigid.
We are interested in a conjectured
generalization: a polyhedron P which is weakly convex (its
vertices are on the boundary of a convex domain) and
decomposable (it can be cut into convex pieces without
any interior vertex) is infinitesimally rigid. This is
true if P is starshaped with respect to one of its
vertices.
Partly in collaboration with Bob Connelly and Ivan
Izmestiev.
Closed Polyhedral 3Manifolds
of Nonnegative Curvature
We show that a closed polyhedral 3manifold of nonnegative
Alexandrov curvature admits a Riemannian metric of nonnegative
sectional curvature, and, therefore, can be covered by S^{3},
S^{2} × S^{1}, or S^{1} ×
S^{1} × S^{1}.
Random Triangulations and Emergent Conformal Structure
Discrete conformal geometry via circle packing began with a 1985
conjecture of Bill Thurston, now a theorem of Rodin and Sullivan,
on the approximation of conformal mappings of plane regions. This talk
will discribe an extension of Thurston's conjecture to circle packings of
random triangulations. Experimental evidence will be given suggesting that
conformal structure is an "emergent" phenomenon  that the discrete
conformal structures provided by circle packings of
increasingly fine random triangulations will converge in probability to
classical conformal structure.
Two Connections Between Combinatorial and Differential Geometry
There is a rich interplay between combinatorial and differential geometry.
We will give first a geometric proof of a combinatorial result, and then
a combinatorial analysis of a geometric moduli space. The first is joint
work with Ivan Izmestiev, Rob Kusner, Günter Rote, and Boris Springborn;
the second with Karsten GrosseBrauckmann, Nick Korevaar and Rob Kusner.
In any triangulation of the torus, the average vertex valence is 6.
Can there be a triangulation where all vertices are regular (of valence 6)
except for one of valence 5 and one of valence 7? The answer is no.
To prove this, we give the torus the metric where each triangle is
equilateral and then explicitly analyze its holonomy. Indeed, techniques
from Riemann surfaces can characterize exactly which euclidean cone
metrics have full holonomy group no bigger than their restricted holonomy
group (at least when the latter is finite).
Next we consider the moduli space M_{k} of Alexandrovembedded surfaces of
constant mean curvature which have k ends and genus 0 and are contained in
a slab. We showed earlier that M_{k} is homeomorphic to an open manifold
D_{k} of dimension 2k3, defined as the moduli space of spherical metrics
on an open disk with exactly k completion points. In fact, D_{k} is the
ball B^{2k3}; to show this we use the Voronoi diagram or Delaunay
triangulation of the k completion points to get a tree, labeled by
logarithms of crossratios. The combinatorics of the tree are tracked
by the associahedron, and the labels give us a complexification of the
cone over its dual. We note similarities to the spaces of labeled trees
used in phylogenetic analysis.
Discrete Koenigs Nets and Discrete Isothermic Surfaces
Koenigs nets allow a great number of seemingly unrelated but actually
equivalent characterizations, for instance, as conjugate nets
f: R^{2} ⊃ U → R^{N}
admitting a dual net
f^{*}: R^{2} ⊃ U → R^{N}
such that
f^{*}_{x} = α f_{x}
and
f^{*}_{y} =  α f_{y}
with a scalar function α.
This definition belongs to affine geometry, but the resulting
class of nets turns out to be projectively invariant: Koenigs nets can be
equivalently characterized, e.g., by the equality of both their Laplace invariants,
or by the existence of a representative
F
in the space of homogeneous
coordinates which satisfies the so called Moutard equation
F_{xy} = β F.
Isothermic surfaces are those carrying an orthogonal Koenigs net.
I will present a systematic discretization of all these notions and
interrelations. In particular, a new characterization of discrete isothermic
surfaces as circular Koenigs nets will be discussed, along with its further
generalizations.
Geometry of MultiLayer Freeform Structures for Architecture
The geometric challenges in the architectural design of freeform
shapes come mainly from the physical realization of beams and
nodes. We approach them via the concept of parallel meshes, and
present methods of computation and optimization. We discuss planar
faces, beams of controlled height, node geometry, and multilayer
constructions. Beams of constant height are achieved with
the new type of edge offset meshes. Mesh parallelism is also the
main ingredient in a novel discrete theory of curvatures. These
methods are applied to the construction of quadrilateral, pentagonal
and hexagonal meshes, discrete minimal surfaces, discrete constant
mean curvature surfaces, and their geometric transforms. We
show how to design geometrically optimal shapes, and how to find
a meaningful meshing and beam layout for existing shapes.
This is joint work with H. Pottmann, Y. Liu, A. Bobenko, and W. Wang.
Classification of 3D Integrable Scalar Discrete Equations
We classify all integrable 3dimensional scalar discrete quasilinear
equations Q_{3}=0 on an elementary cubic cell of the lattice Z^{3}. An
equation Q_{3}=0 is called integrable if it may be consistently imposed
on all 3dimensional elementary faces of the lattice Z^{4}. Under the
natural requirement of invariance of the equation under the action of
the complete group of symmetries of the cube we prove that the only
nontrivial (nonlinearizable) integrable equation from this class is
the wellknown dBKPsystem.
This is joint work with Sergey Tsarev.
Discrete Laplace Operators: No Free Lunch
Discrete Laplace operators are ubiquitous in applications spanning geometric
modeling to simulation. For robustness and efficiency, many applications
require discrete operators that retain key structural properties inherent to
the continuous setting. Building on the smooth setting, we present a set of
natural properties for discrete Laplace operators for triangular surface
meshes. We prove an important theoretical limitation: discrete Laplacians
cannot satisfy all natural properties; retroactively, this explains the
diversity of existing discrete Laplace operators. Finally, we present a family
of operators that includes and extends wellknown and widelyused ones.
This is joint work with Saurabh Mathur, Felix Kälberer, and Eitan Grinspun.
