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 Mikhail Skopenkov "Convergence of discrete period matrices"
We develop linear discretization of complex analysis on triangulations,
originally introduced by R. Isaacs, R. Duffin, and C. Mercat. We prove
existence and uniqueness of discrete Abelian integrals and convergence
of discrete period matrices to their continuous counterparts. The proofs
use energy estimates inspired by electrical
networks.
 Alexander Its "Painlevé Transcendents in Physics. The RiemannHilbert Point of View."
The classical Painlevé equations have been playing
increasingly important role in physics since 1970s1980s works of
Barouch, McCoy, Tracy, and Wu, and of Jimbo, Miwa,
Mori and Sato on the quantum correlation functions.
Since the early nineties, the Painlevé transcendents have become
a major player in the theory and applications of Random Matrices
as well (the pioneering works of Brézin and Kazakov, Duglas and Shenker,
Gross and Migdal, Mehta and Mahoux, and Tracy and Widom ).
In this talk we will try to review these, and also some of the
more recent results concerning Painlevé transcendents and
their appearance in random matrices, statistical
mechanics and quantum field theory. We will present a unified
point view on the subject based on the RiemannHilbert method.
The emphasis will be made on the various double scaling limits related to
the universal properties of random matrices for which Painlevé
functions provide an adequate special function environment'.
 Igor Rivin "Hyperbolic polyhedra"
In this lecture series I will talk about (mostly hyperbolic)
polyhedral geometry, and its connections to such diverse
subjects as:
 Computational geometry
 Low dimensional topology
 Geometric group theory
 Number theory
 Theoretical physics
The lectures will assume no prerequisites.
 Johannes Wallner "Circular arc structures"
In the realization of freeform architectural designs, key issues
are the simplicity of supporting and connecting elements as well
as repetition of costly parts. We study socalled circular arc
structures as a means to achieve this goal  they are meshes
with circular arcs as edges which may be interpolated by smooth
surfaces and which exhibit repetition in the vertex
configuration. We show relations to problem of approximation of
surfaces by cyclide patches, and also to discrete differential
geometry.
 Mikhail Skopenkov "On the boundary value
problem for discrete analytic functions"
The talk is on further development of discrete complex analysis
introduced by R. Isaacs, R. Duffin, and C. Mercat. We consider a
graph lying in the complex plane and having quadrilateral faces.
A function on the vertices is called discrete analytic, if for
each face the difference quotients along the two diagonals are
equal. We prove that the Dirichlet boundary value problem for
the real part of a discrete analytic function has a unique
solution. In the case when each face has orthogonal diagonals we
prove that this solution converges to a harmonic function in the
scaling limit (under certain regularity assumptions). This
solves a problem of S. Smirnov from 2010. This was proved
earlier by R. CourantK. FriedrichsH. Lewy for square lattices,
by P. G. CiarletP.A. Raviart for special kite lattices, and by
D. ChelkakS. Smirnov for rhombic lattices. In particular, this
proves the convergence for the Dirichlet problem on intrinsic
Delauney triangulations studied by A. BobenkoB. Springborn.
This also provides a new approximation algorithm for numerical
solution of the Dirichlet problem and has probabilistic
corollaries. The proof is based on energy estimates inspired by
alternatingcurrent networks theory.
 George Shapiro "Discrete Differential
Geometry in Twistor Space"
The complexified Lie quadric is the classical Pluecker quadric
of complex projective lines: the twistor space of Penrose. The
twistor construction, while important to physicists, also
describes the novel contactgeometry of 2spheres in the
4sphere, generalizing Lie sphere geometry. Starting from the
Pluecker quadric, the Lie quadric and the 4sphere are
identified by real structures as real subquadrics. I will
define a discrete integrable system by Steiner crossratio in
the Pluecker quadric that generalizes the quaternionic
crossratio system in the 4sphere. Then, I will show how the
complex crossratio system in CP^{1} is naturally
embedded into this larger construction, solutions being thus
given by quadrilateral nets. Finally, I will indicate how the
principal contact element nets of Bobenko and Suris are
generalized by "halftouching" contact elements in the 4sphere.
 Boris Hasselblatt "Legendrian knots and
nonalgebraic contact Anosov flows on 3manifolds"
We describe a surgery construction in a neighborhood of a
transverse Legendrian knot that gives rise to new contact
structures preserved by Anosov flows. In particular, this
includes examples on many hyperbolic 3manifolds.
 Francesco Calogero "Isochronous dynamical
systems, the arrow of time and the definitions of "chaotic"
versus "integrable" behaviors"
Given any (autonomous) dynamical system, other (also
autonomous) dynamical systems can be invented which behave
essentially, or even exactly, in the same way for an arbitrarily
long time T' but are isochronous (completely periodic
with an arbitrarily assigned period T>>T'). This
finding is also applicable to the most general (overall
translationinvariant, nonrelativistic, classical or quantal)
Hamiltonian manybody model that subtends much of macroscopic
physics and cosmology. It raises the issue of the ``arrow of
time'' and of the distinction among the ``chaotic'' and
``integrable'' behaviors of a dynamical system, suggesting the
need to invent new definitions of these behaviors that refer to
a finite time interval (all current definitions refer to
the behavior over infinite time). These findings have
been obtained in collaboration with F. Leyvraz.
 Sebastian Heller "A spectral curve for
Lawson's genus 2 surface"
A minimal surface in the 3sphere can be described by its
associated family of flat connections. This leads to an
algebrogeometric description of minimal tori based on the fact
that the first fundamental group of a torus is abelian. For a
minimal surface of higher genus the associated family of flat
connections has nonabelian holonomy. In order to get a better
understanding of these connections we use the abelianization
program due to Hitchin and others, which yields a natural notion
of a spectral curve for Lawson's minimal surface of genus 2.
 Sergey Agafonov "Singular hexagonal 3webs
with holomorphic Chern connection and infinitesimal
symmetries"
A finite collection of foliations form a web. Blaschke
discovered that already for a 3web in the plane, there is a
nontrivial local invariant, namely the curvature form. Thus any
local classification of 3webs necessarily has functional moduli
if no restriction on the class of webs is imposed. The most
symmetric is a hexagonal 3web when the curvature is supposed to
vanish identically. In a regular point a hexagonal 3web is
locally diffeomorphic to 3 families of parallel lines. For
singular points, where at least two foliations are not
transverse, two hexagonal 3webs are not necessarily locally
diffeomorphic. We provide a complete classification of hexagonal
singular 3web germs in the complex plane, satisfying the
following two conditions:
1) the Chern connection form remains holomorphic at the singular
point,
2) the web admits at least one infinitesimal symmetry at this
point.
As a byproduct, classification of hexagonal weighted
homogeneous 3webs is obtained.
 Dmitry Korotkin "Higher genus
generalization of Weierstrass sigmafunctions"
There exists two main different pictures in the theory of
elliptic functions  the Jacobi picture and the Weierstrass
picture. The significant difference between them is the modular
invariance of the Weierstrass picture. The Jacobi picture allows
a welldeveloped generalization to higher genus realized by
Riemann thetafunctions and related objects. On the other hand,
the notion of higher genus sigmafunctions remained essentially
undeveloped until recently (except the hyperelliptic case
considered by Klein and the very special class of socalled
(n,s)curves considered more recently by Buchstaber and his
coworkers). In our talk we introduce a natural notion of higher
genus sigmafunction for generic Riemann surface. The basic
feature which is required from this object is its modular
invariance, in analogy to the elliptic case. Technically, our
definition is based on an appropriate generalization of the
genus one formula expressing sigmafunction via Jacobi's
thetafunction θ_{1}. The talk is based on a joint work
with V.Shramchenko.
 Ian Marshall "Poisson structure associated
to differential and difference operators  with the Toda
lattice and KdV as examples"
I will describe a Poisson structure on the space of curves or
the space of polygons in $\mathbb{R^n}$ from which a series of
Poisson structures on several associated spaces may be obtained
by Poisson symmetry arguments. Amongst these associated spaces
one may find differential operators and difference operators and
their respective reductions. I will present in detail the
concrete cases $n = 2, 3$ for which the natural examples are the
KdV and the Toda lattice. The construction is a simple
application of the theory of Poisson Lie groups and it is
intended that it will serve as an illustration of that subject,
accessible to nonspecialists. It is an extension of a result of
Frenkel, Reshetikhin and SemenovTianShansky and therefore is
expected to have ramifications in the study of lattice or
quantum Walgebras. It is also expected to be a natural setting
for discrete integrable systems analogous to those of socalled
KdVtype.
 Elena Klimenko "Discrete groups of
isometries in hyperbolic 3space"
The full group of orientationpreserving isometries of
hyperbolic 3space H^{3} is isomorphic to the
matrix group PSL(2,C). We discuss the following question:
When does a pair of such matrices generate a discrete group Γ,
that is, when is the quotient space H^{3}/Γ a
nice geometric object  a hyperbolic manifold or orbifold? The
goal of this talk is to explain some work on the structure of
the parameter space of twogenerator discrete subgroups of
PSL(2,C) and to show the corresponding manifolds and
orbifolds.
 Vladimir Dragovic "From Poncelet porisms
to quadgraphs: discrete differentialgeometric structures
approach"
We present discrete differentialgeometric structures and
configurations that appeared in our study of Poncelet porisms,
like the PonceletDarboux grids and the Weyr chains. We give
their natural higherdimensional and highergenera
generalizations, we introduce a notion of discrete billiard
geodesics, and we get generalizations of the Darboux theorem,
the GriffithsHarris space theorem, and the Poncelet theorem.
This leads to progress in a thirtyyearold programme of
Griffiths and Harris to understand higherdimensional analogues
of Poncelet porisms and a synthetic approach to higher genera
addition theorems. Among several applications, a new view on the
Kowalevski top is derived. It is based on the classical
differentialgeometric notion of Darboux coordinates, the modern
concept of $n$valued BuchstaberNovikov groups, and a new
notion of discriminant separability. An unexpected relationship
with the Great Poncelet Theorem for a triangle is established.
We provide a classification of strongly discriminantly separable
polynomials of degree two in each of three variables. We discuss
a strong relationship with the theory of quad graphs, developed
by Adler, Bobenko and Suris, and with the YangBaxter equation.
 Vasilisa Shramchenko "Hurwitz Frobenius
manifolds and cluster algebras"
My talk will consist of two parts. In the first part, I will
define Frobenius manifolds and Hurwitz spaces (moduli spaces of
functions over Riemann surfaces) and explain the main idea for
constructing various Frobenius manifold structures on Hurwitz
spaces. Frobenius manifolds were introduced by Dubrovin to give
a geometric reformulation of the WDVV
(WittenDijkgraafVerlindeVerlinde) system of differential
equations, which describes deformations of topological field
theories. In the second part of the talk, I will give a brief
introduction to cluster algebras, show how they arise from ideal
triangulations of surfaces and speak about joint work with
Ibrahim Assem and Ralf Schiffler in which we study symmetries of
such algebras.
 Grigory Mikhalkin "Complex algebraic
geometry in the tropical limit"
Tropical geometry can be thought of as complex algebraic
geometry after passing to a certain limit, ultimately ignoring
the phase of complex numbers. This limit is analogous to the
quasiclassical limit of quantum mechanics. Resulting notions are
also often simpler, more intuitive and more combinatorial in
their nature. As in quantum mechanics there is correspondence
principle allowing us to connect classical and tropical
geometries.
 Hartmut Weiß "Starrheit und
Flexibilität von hyperbolischen Kegelmannigfaltigkeiten
und Polyedern"
Eine hyperbolische Kegelmannigfaltigkeit ist eine glatte
dreidimensionale Mannigfaltigkeit versehen mit einer
singulären geometrischen Struktur. Die Singularitäten
dieser Struktur sind von polyedrischer Natur. So bilden etwa
Verdoppelungen hyperbolischer Polyeder eine natürliche
Beispielklasse. Wir diskutieren die Deformationstheorie
hyperbolischer Kegelmannigfaltigkeiten sowie ihre Anwendung auf
eine Frage von Stoker über die Starrheit konvexer Polyeder.
 Franz Schuster "Integralgeometrie und
isoperimetrische Ungleichungen"
In diesem Vortrag werden einige der faszinierenden Entwicklungen
im Bereich der Integralgeometrie der letzten Jahre und die
dadurch entstandenen Beziehungen etwa zur Differentialgeometrie
und Funktionalanalysis vorgestellt. Im Zentrum dieser
Entwicklungen steht hierbei der fundamentale Begriff der
\emph{Bewertung}. Dabei handelt es sich um Funktionen $\phi$ auf
konvexen Körpern (oder allgemeineren Teilmengen des
$\mathbb{R}^n$) mit Werten in einer Abelschen Halbgruppe und der
Eigenschaft, dass $\phi(K) + \phi(L) = \phi(K \cup L) + \phi(K
\cap L)$ wann immer $K, L$ und $K \cup L$ konvex sind. Als
Verallgemeinerung des Maßbegriffes haben Bewertungen schon
lange eine wichtige Rolle in der geometrischen Analysis
gespielt. Die Ursprünge des Begriffes gehen insbesondere
auf Dehns Lösung des 3. Problems von Hilbert, sowie die
dadurch entstandene Zerlegungstheorie von Polytopen zurück.
Die starken Beziehungen zwischen der affinen Geometrie und der
Theorie der Bewertungen haben sich erst in den letzten 5  10
Jahren manifestiert. So konnten etwa eine Reihe fundamentaler
Operatoren auf konvexen Körpern, wie
Projektionenkörper und SchnittkörperOperatoren,
durch ihre Bewertungseigenschaft und Kompatibilität mit
affinen Abbildungen klassifiziert werden. Diese Resultate haben
Anwendung in der Theorie affin isoperimetrischer Ungleichungen
gefunden, wodurch klassische Ungleichungen der euklidischen
Geometrie signifikant verschärft werden konnten.
 Max Wardetzky "Discrete Differential
Operators  structure preservation, convergence &
applications"
I will present an approach of studying polyhedral surfaces from
the perspective of function spaces and differential operators
acting between those. This includes a Sobolev theory on
polyhedral surfaces viewed as Euclidean cone manifolds. When
moving to the discrete setting, I will discuss both the
importance of maintaining core structural properties of the
smooth case and convergence in the limit of refinement. With
structure preservation being the main theme of my talk, I will
point out some unavoidable limits to this desire, for example
when studying discrete LaplaceBeltrami operators. In terms of
applications, I will also touch upon the importance of geometric
structure preservation for the efficient yet accurate simulation
of physical systems. Analyzing manifolds from the perspective of
functions over them naturally leads to Morse theory. In this
spirit, I will additionally present a recent combination of a
structure preserving version of Morse theory on simplicial
manifolds  Forman's discrete Morse theory  with persistent
homology. I will show that this combination leads to a proof of
tightness of the lower bound on the number of critical points
provided by the celebrated stability theorem of persistent
homology and an attendent efficient algorithm for optimal noise
removal for functions on simplicial 2manifolds.
 Olaf Post "Spektrale Eigenschaften
verzweigter Strukturen "
Unter einer verzweigten Struktur sei hier ein Raum verstanden,
der gemäß eines diskreten Graphen in einfache
Bausteine zerlegt werden kann. Typische Beispiele sind diskrete
Graphen selber, metrische Graphen (topologische Graphen, bei
denen jeder Kante eine Länge zugeordnet wird), sowie eine
tubenartige Umgebung eines eingebetteten metrischen Graphen. Auf
jedem dieser Beispiele kann ein natürlicher
LaplaceOperator definiert werden. Im Vortrag werden einige
interessante Beziehungen zwischen den Spektren dieser Räume
vorgestellt sowie ihre Anwendungen diskutiert. Beispielsweise
hat Colin de Verdière verwandte Methoden benutzt, um zu
zeigen, dass die Vielfachheit des ersten nichtverschwindenden
Eigenwertes auf einer mindestens dreidimensionalen
Mannigfaltigkeit beliebig hoch sein kann.
 Raphael Boll "Classification of 3D
consistent quadequations"
We consider 3D consistent systems of six possibly different
quadequations assigned to the faces of a cube. The wellknown
classification of 3D consistent quadequations, the socalled
ABSlist, when all six equations are the same up to parameters,
is included in this situation. The extension of these equations
to the whole lattice Z^{3} is possible by reflecting the
cubes. This construction allows to show the integrability of all
quadequations by deriving Bäcklund transformations and
zerocurvature representations for all of them.
In this talk we will present certain tools for the
classification of these systems, some structural attributes of
the classified systems, the embedding of these systems in the
lattice Z^{3} and the procedure of deriving
Bäcklund transformations and zerocurvature representations
for quadequations.
 Felix Günther "Integrable discrete
Laplace equations with convex variational principles "
In this talk, we investigate the Lagrangian structure of 2D
discrete integrable systems. We describe the systems with convex
action functionals. The relation to circle patterns is
discussed.
 Olga Holtz "From box splines to discrete
geometry: zonotopal algebra, analysis, and combinatorics"
I will discuss the main ideas behind the socalled zonotopal
algebra, motivated initially by the development of box splines
in approximation theory, which leads, rather unexpectedly, to
many new results in commutative algebra, geometry and
combinatorics. This is joint work with Amos Ron and, partly,
with Zhiqiang Xu. I will also try to explain some related
results of de Concini, Procesi, Vergne, Sturmfels, Postnikov,
Ardila, and others.
 Udo HertrichJeromin "Special
Lieapplicable surfaces"
I will aim to describe how linear Weingarten surfaces in space
forms appear as special Lieapplicable surfaces.
 Ulrich Pinkall "Notes on Discrete Exterior
Calculus"
There are some serious inconsistencies in the common
presentation of Discrete Exterior Calculus. Here I present the
results of my recent efforts to make mathematical sense out of
this theory.
 Vladimir Bazhanov "A master solution to
the quantum YangBaxter equation and classical discrete
integrable equations"
We obtain a new solution of the startriangle relation with
positive Boltzmann weights which contains as special cases all
continuous and discrete spin solutions of this relation, that
were previously known. This new master solution defines an
exactly solvable 2D lattice model of statistical mechanics,
which involves continuous spin variables, living on a circle,
and contains two temperaturelike parameters. If one of the
these parameters approaches a root of unity (corresponds to zero
temperature), the spin variables freezes into discrete
positions, equidistantly spaced on the circle. An absolute
orientation of these positions on the circle slowly changes
between lattice sites by overall rotations. Allowed
configurations of these rotations are described by classical
discrete integrable equations, closely related to the famous Q_{4}equations
by Adler, Bobenko and Suris. Fluctuations between degenerate
ground states in the vicinity of zero temperature are described
by a rather general integrable lattice model with discrete spin
variables. In some simple special cases the latter reduces to
the KashiwaraMiwa and chiral Potts models.
 Dmitry Sinitsyn "Geodesics on deformed
spheres studied using an asymptotic Hamiltonian reduction"
We address the problem of describing geodesics on surfaces which
are perturbations of the standard 2D or 3Dsphere. Considering
the geodesics as the trajectories of a freely moving particle on
the surface and using perturbation theory, we derive an averaged
Hamiltonian system that asymptotically describes the slow
evolution of the angular momentum of the particle. The averaged
Hamiltonian is obtained by an integral transformation of the
deformation function defining the surface. A procedure of this
type was first employed by Poincare for finding closed
geodesics. In the case of deformations of the 2Dsphere, the
averaged system has one degree of freedom and is therefore
integrable and admits a complete description in terms of phase
portraits. In the 3Dsphere case it has two degrees of freedom
and is integrable only for a certain class of deformations. We
study second and fourth order polynomial surfaces using the
technique, obtaining a classification of the averaged system
dynamics types.
 Ivan Izmestiev "Infinitesimal rigidity of
convex surfaces via the HilbertEinstein functional"
An infinitesimal isometry of a surface in R^{3} is a
deformation that preserves lengths of curves on the surface in
the first order. It is known that convex surfaces without flat
pieces are infinitesimally rigid. For convex polyhedra this was
proved by Legendre, Cauchy, and Dehn; for smooth convex surfaces
by Liebmann and Blaschke; and for general convex surfaces by
Pogorelov. We present a unified approach to the discrete and
smooth cases based on variations of the HilbertEinstein
functional. Instead of deforming an embedding of the surface, we
reduce the problem to deformations of the metric in the domain
bounded by the surface. (It suffices to consider deformations in
the class of ``warped product'' metrics.) We show that every
deformation that preserves the metric on the boundary and leaves
the metric in the interior Euclidean in the first order is
trivial. This implies infinitesimal rigidity of the surface.
 Alexander Its "Asymptotics of Toeplitz,
Hankel and Toeplitz + Hankel determinants with
FisherHartwig singularities. The RiemannHilbert Approach."
We will discuss some new asymptotic results obtained via the
integrable systems techniques, notably the RiemannHilbert
method, in the area of Hankel and Toeplitz determinants whose
symbols possess FisherHartwig singularities on a smooth
background. Specifically, we will discuss the proof of the
BasorTracy conjecture concerning the Toeplitz determinants, the
asymptotics of Hankel and Toeplitz + Hankel determinants on a
finite interval, and the Painlevétype crossover formulae
describing a transition between the Szegö and the
FisherHartwig type of asymptotic behavior for Toeplitz
determinant. The talk is based on the joint works with P. Deift,
T. Claeys, and I. Krasovsky.
 Daniel Matthes "Some nonlinear equations
of fourth order, their gradient flow representation, and
ideas for discretization"
There are several parabolic PDEs of fourth order  like the
CahnHilliard equation and the HeleShaw flow  that can
formally be written as steepest descent of a free energy
functional with respect to a metric tensor. In this talk, some
situations are discussed in which this formal idea can be made
rigorous, i.e., when there exists a genuine metric on the
infinitedimensional space of densities that induces the desired
metric tensor and allows to define and analyse the gradient flow
of the free energy functional. A byproduct of this is a method
to construct weak solutions by means of timediscrete
"minimizing movements". In the last part of the talk, we discuss
a geometrically motivated spatial discretization of the latter
that gives rise to a stable numerical scheme.
 Esfandiar Navayazdani "Subdivision of
ManifoldValued Data"
There has been a growing interest in multiscale resolution of
nonlinear data in the recent years. A main issue therein is
smoothness. In the present work we consider subdivision of
manifoldvalued data determined by transformations which reflect
the nonlinearity and a corresponding linear scheme. Our approach
results in a full description of conditions on those
transformations and the underlying linear subdivision scheme in
order to achieve higher smoothness for the nonlinear scheme and
generalizes all previous results in this setting. Moreover, we
present applications to several areas, such as rigid motions,
diffusion tensors and array signal processing.
 Peter Schröder "A Simple Geometric
Model for Elastic Deformations"
We advocate a simple geometric model for elasticity: distance
between
the differential of a deformation and the rotation group.
It comes with rigorous differential geometric underpinnings,
both smooth and discrete, and is computationally almost as
simple and efficient as linear elasticity. Owing to its
geometric nonlinearity, though, it does not suffer from the
usual linearization artifacts. A material model with standard
elastic moduli (Lamé parameters) falls out naturally, and
a minimizer for static problems is easily augmented to construct
a fully variational 2nd order time integrator. It has excellent
conservation properties even for very coarse simulations, making
it very robust.
Our analysis was motivated by a number of heuristic,
physicslike algorithms from geometry processing (editing,
morphing, parameterization, and simulation). Starting with a
continuous energy formulation and taking the underlying geometry
into account, we simplify and accelerate these algorithms while
avoiding common pitfalls. Through the connection with the Biot
strain of mechanics, the intuition of previous work that these
ideas are ``like'' elasticity is shown to be spot on.
Joint work with Isaac Chao, Ulrich Pinkall, and Patrick Sanan.
 Tom Banchoff "SelfLinking, Inflections,
and the Normal Euler Class for Smooth and Polyhedral
Surfaces in FourSpace"
The normal Euler class for a simplicial surface in fourspace
can be defined by analogy with the geometric description of this
class for a smooth surface. We present a combinatorial formula
for the normal Euler class in terms of the selflinking numbers
of spherical polygons and inflection faces of polyhedra, related
to a construction of Gromov, Lawson, and Thurston. The talk will
feature computer graphics illustrations.
 Volker Mehrmann "Structured
differentialalgebraic systems and their structured
discretization"
We discuss a new class of structured system of differential
algebraic equations that arise from optimal control, in the
variation analysis of high speed trains and many other
applications.
These systems are higher order generalization of Hamiltonian
systems (which is not surprising since they arise from
variational problems). The corresponding flow of these systems
has a generalized symplectic property and therefore it is
important to generate discretization methods that preserve this
property. We will discuss classes of such discretizations and
some of their properties.
 Ulrike Bücking "Approximation of
conformal mappings by conformally equivalent triangle meshes
in case of an equilateral lattice"
We approximate a given conformal mapping f on a compact set K in
the plane, where f is defined on some open neighborhood of K.
Take the part of an equilateral triangular lattice with edge
length \eps>0 contained in K. Using the values of logf' on
the boundary points, there is a conformally equivalent
triangular mesh. In this way we can define a discrete conformal
mapping. The goal of this talk is to prove the convergence of
this discrete conformal map to f for \eps to 0.
 Gero Friesecke "Energieminimierung und
Kristallisation"
Statische Probleme der atomaren Festk"orpermechanik laufen auf
die Minimierung einer potentiellen Energiefl"ache
$E(x_1,...,x_N)$ (englisch: potential energy surface, PES) "uber
Atompositionen $x_i\in\R^d$ hinaus. Prototypisch sind
Paarpotentialmodelle. In der Festk"orpermechanik liegt die Zahl
N der Atome zwichen $105$ (Cluster, KohlenstoffNanotuben) und
$10^{23}$ (makroskopische Materie), mathematisch idealisiert
durch den Limes $N\to\infty$.
Ich bespreche informell (und  if the audience is willing 
interaktiv) klassische und aktuelle Ans"atze und Fortschritte
bez"uglich der fundamentalen Frage, warum Minimierer sich
typischerweise (experimentell und numerisch) in kristalliner
Ordnung arrangieren.
 Vladimir Oliker "Designing lenses with help
from geometry and optimal transport"
Mirror and lens devices converting an incident plane wave of a
given cross section and intensity distribution into an output
plane wave irradiating at a given target set with prescribed
intensity are required in many applications. Most of the known
designs are restricted to rotationally symmetric mirrors/lenses.
In this talk I will discuss designs with freeform lenses, that
is, without a priori assumption of rotational symmetry. Assuming
the geometrical optics approximation, it can be shown that the
functions describing such freeform lenses satisfy
MongeAmp\`{e}re type partial differential equations. Because of
strong nonlinearities analysis of these PDE's is difficult.
Fortunately, many such problems can also be formulated
geometrically and lead to problems in calculus of variations in
which instead of solving the nonlinear PDE's one needs to find
extrema of some Fermatlike functionals.
Furthermore, discrete versions of such problems can be
formulated and are useful for numerics. In this talk I will
describe some of these results in the case of the lens design
problem.
 Raphael Boll "On NonSymmetric Discrete
Relativistic Toda Systems and Their Relation to QuadGraphs"
In this talk, we establish a relation of nonsymmetric discrete
relativistic Toda type equations to 3D consistent systems of
quadequations . Moreover, we will present a master equation
which includes both symmetric and nonsymmetric discrete
relativistic Toda type equations as particular or limiting
cases. In addition, we will give a survey at adequate
derivations of nonsymmetric discrete relativistic Toda type
equations from this master equation. Our construction allows for
an algorithmic derivation of the zero curvature representations
and yields analogous results for the continuous time case.
 Andrear Pfadler "HirotaKimura
Discretization Of Integrable Systems of Ordinary
Differential Equations"
A discretization scheme orginally proposed by W. Kahan yields 
when applied to several completely integrable systems 
explicit, birational maps, which are again integrable. At the
moment, it is, however, not clear whether this is true for all
algebraically completely integrable systems with a quadratic
right hand side. In this talk, a general intriduction to Kahan's
discretization scheme, which is called HirotaKimura
discretization in the "integrable" context, will be given and
the results of the application of this method to several
algebraically integrable systems will be presented. Also, there
will be a special emphasis on algorithmic aspects playing a role
in the study of integrable, birational mappings.
 Stefan Felsner "Triangle Contact
Representations"
We are interested in triangle contact representations of planar
graphs with homothetic triangles. That is, vertices are
represented by a set of disjoint triangles all identical up to
scaling and translation, two triangles touch exactly if there is
an edge between the corresponding vertices.
The big conjecture is that every 4connected plane triangulation
has such a representation. We outline a roadmap for a proof of
the conjecture and report on partial results and experimental
evidence. If time allows we draw a connection to squarings.
 Hannes Sommer "Energypreserving
discretization of incompressible fluids"
This is a joint talk with the students seminar in Differential
Geometry. In this talk I present results of the thesis of Dmitry
Pavlov (advisors: Jerrold Marsden and Mathieu Desbrun), in which
Pavlov geometrically derives discrete equations of motion for
fluid dynamics from first principles. In the case of an ideal
fluid (no viscosity) the induced integrator preserves energy and
an exact discrete Kelvin circulation theorem holds for it.
Moreover it works on arbitrary simplicial grids. In this way
spectacularly realistic fluid animations can be achieved. As a
part of my diploma thesis I will implement this scheme, whose
basic ideas will be introduced in this talk.
 Keenan Crane "Lie Group Integrators for
Animation and Control of Vehicles"
In this talk we address the animation and control of vehicles
with complex dynamics such as helicopters, boats, and cars.
Motivated by recent developments in discrete geometric mechanics
we develop a general framework for integrating the dynamics of
holonomic and nonholonomic vehicles by preserving their
statespace geometry and motion invariants. We demonstrate that
the resulting integration schemes are superior to standard
methods in numerical robustness and efficiency, and can be
applied to many types of vehicles. In addition, we show how to
use this framework in an optimal control setting to
automatically compute accurate and realistic motions for
arbitrary userspecified constraints.
 Yuri Suris "On geometry, algebra and
combinatorics of discrete integrable 3d systems"
There exist several fundamental discrete integrable systems on
3d lattices, known under the (jargon) names of discrete AKP,
BKP, CKP, DKP equations. The first letters A,B,C,D refer to the
classical series of simple Lie algebras (or to their root
systems). A brief overview of these systems and their geometric
and algebraic properties will be given. Nonexperts in
integrable systems are cordially invited in view of numerous
(envisaged) interrelations with other disciplines.
 Ulrich Brehm "A Universality Theorem for
Realization Spaces of Maps"
A universality theorem for maps in R^{3} is shown,
stating essentially that every semialgebraic set can occur as a
realization space of some map (with a distinguished set of
vertices). In the talk we give an outline of the proof and the
main ideas for the construction.
Full details of the proof will be given in the subsequent talks:
"Encoding a Semialgebraic Set by a Graph and Collinear Triples
of Points", Thursday 12.03.2009 at 15:00 in MA 645
"A Universal Extension of Partial Maps with Respect to Linear
Embedings", Friday 13.03.2009 at 14:00 in MA 645
A detailed abstract in pdf.
 Ivan Izmestiev "Dual metrics of convex
polyhedral cusps"
In a paper of 1993, Rivin and Hodgson characterized convex
hyperbolic polyhedra in terms of their dual metrics. Here, the
dual metric is obtained by gluing the Gauss images of the
vertices. The RivinHodgson theorem generalizes Andreev's
theorem that describes acuteangled hyperbolic polyhedra in
terms of their dihedral angles.
We will present the result of our joint work with
François Fillastre, where we prove an analog of the
RivinHodgson theorem for hyperbolic cusps. A specialization in
the spirit of Andreev's theorem can be formulated in terms of
circle patterns on the torus.
The proof uses a variational principle close to that used in the
proof of Minkowski theorem on the existence of a convex
polyhedron with given face normals and face areas.
 Carsten Lange "Geometry and combinatorics
of associahedra"
Classical associahedra are polytopes which are closely related
to Catalan numbers and visible in different branches throughout
mathematics.
The focus of this talk will be to describe different
constructions of these polytopes and how to distinguish these
realisations using combinatorial information. This paves the way
to another description of these constructions in terms of
Minkowski sums and differences of simplices.
Finally, generalisations of these constructions to "generalised
associahedra" will be mentioned. These objects were introduced
by S. Fomin and A. Zelevinsky in the context of cluster
algebras.
 Michael Joswig "Generalized Quadrangles
and Polar Spaces"
Given a polarity on a projective space, the induced incidence
geometry of the absolute points, lines, etc. is a polar space.
If there are no absolute planes, this yields a generalized
quadrangle. These spaces seem to play an important role in
Bobenko's and Suri's work on discrete integrability. The purpose
of this talk is to give an introduction to the topic mostly from
an incidence geometry point of view. Relationships to Lie theory
will be mentioned in passing.
 H. Si, J. Fuhrmann, K. Gaertner "Delaunay
mesh generation and some related discrete geometry problems"
Threedimensional boundary conforming Delaunay meshes allow to
use the the Voronoi box based finite volume method for
discretizing parabolic or elliptic nonlinear coupled systems of
partial differential equations. For this method, it is possible
to prove that qualitative properties of the continuous problem
like existence of bounded steady state solutions, dissipativity,
uniqueness at equilibrium and others hold as well for the
discrete system independent of the size of the time step and the
mesh spacing.
There are still many challenging problems in 3D boundary
conforming Delaunay mesh generation. A characteristic property
of many problems of interest is the occurrence of boundary
layers. It is necessary to resolve them using aligned,
anisotropic meshes. Proper mesh adaptation in this sense should
be based on the use of the solutions of adjoint and other
auxiliary problems which allow to define point sets which serve
as the input to the Delaunay mesh generation algorithm.
This talk should stimulate to look at the problem of generating
3D boundary conforming Delaunay meshes from different
mathematical points of view. The present status and recent
progress with respect to the numerical analysis and the
practical mesh generation will be reported [1].
We will present two discrete geometry problems which we
encountered for the above problem, which are (1) convex
decomposition of 3D nonconvex polyhedra; and (2) update 3D
triangulations by bistellar flips. We will present some recent
work on these two problems [2] [3].
We will conclude by discussing open issues and possible future
directions of boundary conforming Delaunay mesh generation in
the context of the demands from the numerical methods described
above.
References:
[1] H. Si, J. Fuhrmann, K. Gaertner, Boundary conforming
Delaunay mesh generation, Journal of Computational Mathematics
and Mathematical Physicals, submitted, 2008.
[2] H. Si, On the existence of triangulations of nonconvex
polyhedra without new vertices, WIAS Preprint No. 1329, 2008
[3] J. R. Shewchuk, Updating and constructing constrained
Delaunay and constrained regular triangulations by flips, Proc.
19th Annu. Sympos. Comput. Geom., 2003.
 Matthias Weber "Triply periodic Minimal
surfaces  a unified approach"
In joint work with Shoichi Fujimori, we obtain many classical
and new embedded triply periodic minimal surfaces. Our surfaces
have in common that vertical symmetry planes cut them into
simply connected domains. The Weierstrass data of these surfaces
can be understood in terms of the conformal geometry of periodic
planar polygons. On the way, we develop a new
SchwarzChristoffel formula for such polygons, involving
thetafunctions.
The limits of the surfaces we obtain suggest new gluing methods
for minimal surfaces, which have been used by my student Peter
Connor to construct doubly periodic minimal surfaces, with which
I will close my talk.
 Dirk Ferus "Reilly's proof of the
Alexandrov soap bubble theorem"
Alexandrov's theorem states that the only soap bubbles are round
spheres. Besides the original paper (Vestnik Leningrad Univ.
1958) or its english translation (AMS Transl. 1962) the standard
reference is the more elaborate proof by Heinz Hopf (Springer
Lecture Notes in Math vol. 1000). I shall present a quite
different proof by Robert Reilly (1970). It rests on a deep, but
well established existence theorem for a Poisson problem, but is
otherwise completely elementary. And I think it deserves a
broader publicity.
 Felix Effenberger "Hamiltonian
submanifolds of regular polytopes"
We investigate polyhedral 2kmanifolds as subcomplexes of the
boundary complex of a regular polytope. We call such a
subcomplex kHamiltonian if it contains the full
kskeleton of the polytope. Since the case of the cube is well
known and since the case of a simplex was also previously
studied (these are socalled superneighborly
triangulations) we focus on the case of the cross
polytope and the sporadic regular 4polytopes. By our results
the existence of 1Hamiltonian surfaces is now decided for all
regular polytopes. Furthermore we investigate 2Hamiltonian
4manifolds in the ddimensional cross polytope. These are the
``regular cases'' satisfying equality in Sparla's inequality. In
particular, a new example with 16 vertices which is highly
symmetric with an automorphism group of order 128 will be
presented. Topologically it is homeomorphic to a connected sum
of 7 copies of S^{2} × S^{2}. By this
example all regular cases of n vertices with n < 19 or,
equivalently, all cases of regular dpolytopes with d < 8 are
now decided.
Reference: F. Effenberger and W. Kühnel, Hamiltonian
submanifolds of regular polytopes, 25 p. (2008,
Preprint), arXiv 0809.4168
 Vladimir Dragovic "Poncelet Porisms and
Beyond"
First, we will briefly review 260 years of the history of the
subject. Then, we will report on progress in the thirty years
old programme of Griffiths and Harris of understanding of
higherdimensional analogues of Poncelet type problems and
synthetic approach to higher genera addition theorems as it has
been settled and completed in our recent paper. Reference: V.
Dragovic, M. Radnovic, Hyperelliptic Jacobians as Billiard
Algebra of Pencils of Quadrics: Beyond Poncelet
Porisms//Advances in Mathematics, in press, arXiv: 0710.3656
 Nikolai Dolbilin "Parallelohedra:
Classical Theorems by Minkowski, Voronoi and Beyond"
A parallelohedron is defined as a convex polyhedron which can
fill Euclidean space by its translates. The concept was
introduced by E.Fedorov (1885). The parallelohedron is one of
basic notions in Geometry of Numbers and Crystallography (d=3).
Basic theorems on parallelohedra have been found by H.
Minkowskii (1897). G. Voronoi (1908) has developed a deep
algorithm computing for any given dimension all combinatorial
types of a special class of parallelohedra (now called Voronoi
parallelohedra), and outlined a plan of how his theory could be
applied for computing types of general parallelohedra. His
conjecture on affinity of each parallelohedron to some Voronoi
parallelohedron was investigated in works by B.Delone, A.
Zhitomirski, B. Venkov (Sr.), A.Alexandrov, P.McMullen,
S.Ryshkov, and others. Recently the author received a result
which enforces one of the two Minkowski theorems and gives new
information about Voronoi parallelohedra.
 Vladimir Bazhanov "Quantum Geometry of
3dimensional lattices"
We study geometric consistency relations between angles on
3dimensional (3D) circular quadrilateral lattices  lattices
whose faces are planar quadrilaterals inscribable into a circle.
We show that these relations generate canonical transformations
of a remarkable ``ultralocal'' Poisson bracket algebra defined
on discrete 2D surfaces consisting of circular quadrilaterals.
Quantization of this structure leads to new solutions of the
tetrahedron equation (the 3D analog of the YangBaxter
equation). These solutions generate an infinite number of
nontrivial solutions of the YangBaxter equation and also
define integrable 3D models of statistical mechanics and quantum
field theory. The latter can be thought of as describing quantum
fluctuations of lattice geometry. The classical geometry of the
3D circular lattices arises as a stationary configuration giving
the leading contribution to the partition function in the
quasiclassical limit.
 Silke Möser "Cubical projectivities"
Consider a pure cubical complex $K$ and two facets $\sigma$ and
$\sigma'$ of $K$ that share a common ridge $\rho$. Then we can
define the \textit{perspectivity} $\langle \sigma,\sigma'
\rangle$ to be a certain combinatorial isomorphism
$\sigma\to\sigma'$. Moreover, we define the
\textit{projectivity} along a facet path (i.e., a sequence of
facets such that each pair of consecutive facets share a common
ridge) to be the concatenation of the corresponding
perspectivities. The set of all projectitivies along closed
loops based at a fixed base facet $\sigma$ forms a group, the
\textit{group $\Pi(K,\sigma)$ of projectivities} of $K$ with
respect to the base facet $\sigma$. In the talk we will
introduce different types of groups of projectivities and give
an overview over some results and applications.
 Sergey Norin "Divisors on graphs"
Finite graphs can be viewed, in many respects, as discrete
analogues of algebraic curves. In this talk, we consider this
analogy in the context of linear equivalence of divisors on a
graph. We will state a graphtheoretic version of the
RiemannRoch theorem, and outline its proof. Additionally, we
will discuss harmonic morphisms between graphs and connections
of our results to tropical geometry. This is joint work with
Matt Baker.
 Ulrich Bauer "Uniform convergence of
discrete curvatures on nets of curvature lines"
For the discretization of a smooth surface by a discrete net of
curvature lines, we prove uniform pointwise convergence of a
broad class of wellknown edgebased discrete curvatures to
smooth principal curvatures. Our proofs use explicit error
bounds, with constants depending only on the maximum curvature,
the derivative of curvature of the smooth surface, and the form
regularity of the discrete net. One important aspect of our
result is that the error bound is independent of the geodesic
curvature of the curvature lines, and therefore is also
applicable in the vicinity of umbilical points. This is the
first pointwise convergence result for discrete curvatures that
is applicable to closed discrete surfaces. Joint work with Max
Wardetzky and Konrad Polthier.
 Dmitri Chelkak "Universality for the Ising
model on the isoradial graphs"
We consider the Ising model in discrete domains (finite parts of
isoradial graphs on the plane). The main result is the conformal
invariance of the interfaces (convergence of some random curves
to SLE) in the limit as mesh of the lattice goes to zero and the
macroscopic shape of the domain is fixed. This holds true
uniformly with respect to the choice of the underlying lattice
(the property known in statistical physics as universality of
the model). Joint work with S. Smirnov.
 Peter Schröder "Everything you always
wanted to know about numerical algorithms but never dared to
ask!"
In this 3 lecture set I will cover at a high level some
fundamental ideas of numerical algorithms with an eye towards
giving users of numerical algorithms some guidance. Focus will
be on the basic ideas and some of the keywords in terms of
algorithmic concepts so that users can quickly find the right
algorithms for a given task. The selection of concepts and
algorithms will be unabashedly colored by my own experience and
prejudices.
 Kristoffer Josefsson "Mathematics in
architecture"
This is a report from the SmartGeometry 2008 conference in
Munich. Using tools primarily created for computer animation,
architects face new problems when attempting to realize their
computer models as buildings. New 'parametric' modelling
programs and new manufacturing processes makes the distinction
between the architect and the structural engineer blurry. I will
give an overview of recent trends in computational and
generative architecture and the mathematical challenges faced by
architects now and in the future.
 Udo HertrichJeromin "Discrete CMC
surfaces in space forms"
We shall discuss a definition of discrete cmc surfaces in space
forms as special "special discrete isothermic nets". We will
briefly touch upon the transformations of these nets, including
a "Baecklund transformation" and discuss some characterizations
in terms of those. This provides a point of contact with earlier
notions of discrete cmc surfaces in Euclidean space, for
example.
 Christophe DM Barlieb "Hidden Dimensions"
Hidden Dimensions refers to spatial constructs and fictions
challenging our formal, functional and historical preconceptions
of space; by shifting and playing off our experiences and
environments. Carl Jung speaks of the collective unconscious,
the repository of man symbols as a library for interpreting and
giving meaning. The projects presented here are codes bridging
and binding emotional and rational spaces. They are by no means
absolutes! They are visions, glimpses of a timeless
architectural space; one rooted far deeper in our souls where
the 'meaning' is its driving force.
 Joachim Linn "Discrete rods from geometric
finite differences"
We present a model of flexible rods, based on Kirchhoff's
geometrically exact theory, which is suitable for the fast
simulation of quasistatic deformations within VR (vitual
reality) or FDMU (functional digital mock up) applications. Our
computational approach is based on a variational
formulation, combined with a finite difference
discretization of the continuum model. Our specific
choice of the finite difference expressions approximating the
curvature measures is guided primarily by geometric
considerations in the spirit of Robert Sauer's ``Differenzengeometrie''
rather than by formal numerical aspects (like e.g.~a high
approximation order of the difference scheme). We obtain
approximate solutions of the mechanical equilibrium equations
for sequentially varying boundary conditions by means of energy
minimization, using unconstrained optimization methods
like nonlinear CG or BFGS. The computational performance of our
rod model proves to be sufficient for the purpose of an
interactive manipulation of flexible cables in assembly
simulation. Apart from the ability to produce reasonably
accurate rod deformations, the forces and moments
computed from our discrete model approximate those of analytical
reference solutions (like Euler's plane ``Elastica'' or
Kirchhoff's spatial ``helices'') as well as numerical benchmarks
(here: against the ``Cosserat'' type rods provided by ABAQUS)
surprisingly well. Using an appropriate time stepping scheme, we
observe the same also for the dynamical version of our
model, which accounts for inertial effects and is well suited
for a coupling to multibody dynamics simulation (MBS) software.
 Maria Ares Ribo Mor "Embedding 3Polytopes
on Small Integer Grids"
We present a constructive method for embedding a 3connected
planar graph as 3polytope with small integer coordinates. The
embedding needs no coordinate greater than O(2^{7.55n}).
We also give bounds for polytopes with a triangular facet and
polytopes with a quadrilateral facet. Finding a 2d embedding
which supports an equilibrium stress is the crucial part in the
construction. We have to guarantee that the size of the
coordinates and the stresses are small. This is achieved by
extending Tutte's spring embedding method. The approach is based
on the lifting given by the MaxwellCremona Theorem. We use the
MatrixTree Theorem and some new upper bounds for the number of
spanning trees of a planar graph.
 Ivan Izmestiev "Projective properties of
the infinitesimal rigidity of frameworks"
A framework is a discrete structure formed by rigid bars joined
together at their ends by universal joints. A framework is
called infinitesimally rigid iff its joints cannot be flexed so
that the lengths of the bars remain constant in the first order.
We present proofs of two classical theorems. The first one, due
to Darboux and Sauer, states that infinitesimal rigidity is a
projective invariant; the other one establishes relations
(infinitesimal Pogorelov maps) between the infinitesimal motions
of a Euclidean framework and of its hyperbolic and spherical
images. We start by explaining the duality between infinitesimal
and static rigidity.
 Ivan Izmestiev "Connecting geometric
triangulations by stellar moves"
In 1996 and 1997, Morelli and Włodarczyk proved the weak Oda
conjecture in the theory of toric varieties. A crucial lemma in
both proofs claims that any two triangulations of a convex
polytope can be connected by a sequence of stellar moves. In the
talk we present a translation of Morelli's proof from the
language of algebraic geometry into the language of discrete
geometry.
 Helmut Pottmann "Architectural Geometry"
Geometry lies at the core of the architectural design process.
It is omnipresent, from the initial formfinding stages to the
final construction. Modern geometric computing provides a
variety of tools for the efficient design, analysis, and
manufacture of complex shapes. This opens up new horizons for
architecture. On the other hand, the architectural application
also poses new problems to geometry. Architectural geometry is
therefore an entire research area, currently emerging at the
border between applied geometry and architecture. The speaker
will report on recent progress in this field, putting special
emphasis on the design of architectural freeform structures.
Important practical requirements on such structures such as
planarity of panels, complexity of nodes in the underlying
supporting structure or properties of multilayerconstructions
can be elegantly treated within the framework of discrete
differential geometry. In fact such architectural applications
have also led to advances in mathematical research. The
computation of discrete architectural freeform structures is a
challenging topic since the underlying mesh geometry needs to be
optimized to a much higher aesthetic and functional level than
meshes used for typical Computer Graphics applications.
 Nikolaus Witte "Knotted Tori"
For every knot K with stick number k there is a knotted
polyhedral torus of knot type K with 3k vertices. We prove that
at least 3k2 vertices are necessary.
 Jerrold Marsden "Discrete Exterior
Calculus and Asynchronous Variational Integrators for
Computational Electromagnetism"
This talk, based on work with Ari Stern, Mathieu Desbrun and
Yiying Tong, will show how to merge DEC and AVI techniques in
the context of computational electromagnetism. In particular,
the Yee scheme falls into this framework and allows a
generalization of the scheme to unstructured meshes with
asynchronous time steps.
 Wayne Rossman "Conserved quantities and
discrete surfaces"
I will talk about a joint work with Fran Burstall, Udo
HertrichJeromin and Susana Santos. The central goal is a
definition for discrete CMC surfaces in any of the three space
forms (Euclidean 3space, spherical 3space and hyperbolic
3space), using linear conserved quantities. We justify this
definition by looking at related results in the smooth case (by
Burstall and Calderbank), and prove that this definition
generalizes other known definitions in the discrete case (by
Bobenko and Pinkall and others). The case of hyperbolic 3space
with mean curvature having absolute value less than 1 is the
case that has not been defined before. If time allows, I hope to
also include some comments and results about polynomial
conserved quantities, constituting a bigger class of discrete
surfaces than just CMC.
 Nico Düvelmeyer "Embeddings of
partially given metric spaces"
We will study the question, whether a given set $X$ of $n$
points admits a mapping $\phi$ from $X$ to $\R^d$, such that
some of the distances between the images take predefined values:
$\norm{\phi(x)\phi(y)}=\rho_{x,y}$ for all $\{x,y\}\in A$,
where $A\subseteq\Pow_2(X)$. Thus we are interested in metric
embeddings of  possibly only partially  given metrics into
$\R^d$. Besides the usual Euclidean way to measure lengths, we
will also consider arbitrary norms in $\R^d$. In particular we
focus on partially given metrics representing the 1skeleton of
triangulated surfaces with common length one of all edges.
 Michael Baake "Coincidence site lattices
and their generalizations"
It is wellknown in material science that coincidence site
lattices provide a useful tool for the understanding of grain
boundaries in crystals and other solids. Mathematically, one
deals with the group of isometries that map a given lattice to a
commensurate copy. An analogous problem emerges for
quasicrystals, which requires more abstract methods from algebra
for its solution. Guided by examples that are used in practice,
the talk will give an introduction to this type of questions and
a survey of the results obtained so far. A nice common feature
of systems in dimensions 2 and 3 with large symmetry groups is
the encapsulation of the full combinatorial problem in a
Dirichlet series generating function that is closely related to
various Dedekind zeta functions.
 Julian Pfeifle "Gale duality bounds for
the zeros of polynomials with nonnegative coefficients"
The set of polynomials of degree at most d in one variable forms
a vector space, which comes with several interesting bases. For
several relevant families of polynomials, for example Ehrhart
polynomials or chromatic polynomials, the coordinates with
respect to one of these bases are nonnegative. We present a
general method based on Gale duality to provide sharp bounds on
the location of the roots of such polynomials. Linear
inequalities between the coordinates (which arise, for example,
in the Ehrhart case) can also be accommodated, and provide
further bounds on the locations of the roots.
 Nikolaus Witte "Constructing simplicial
branched covers"
Branched covers are applied frequently in topology  most
prominently in the construction of closed oriented PL
dmanifolds. In particular, the number of sheets and the
topology of the branching set are known for dimension d≤4
(Hilden, Montesinos, Piergallini, Iori). On the other hand,
Izmestiev and Joswig described how to obtain a simplicial
covering space (the "partial unfolding") of a given simplicial
complex, thus obtaining a simplicial branched cover. We ask,
which simplicial branched covers can be constructed via the
partial unfolding. In particular, for d≤4 every closed oriented
PL dmanifold is the partial unfolding of some combinatorial
dsphere.
 Max Wardetzky "Discrete bending energies"
Efficient computation of curvaturebased energies is important
for geometric modeling and physical simulation. In this talk, we
present an axiomatic approach for bending energies of discrete
thin plates. These energies arise from linear models of mean
curvature, which in turn are constructed from a class of
discrete Laplace operators. Under the assumption of isometric
surface deformations, these energies are shown to be quadratic
in surface positions. The corresponding linear energy gradients
and constant energy Hessians constitute an efficient model for
computing bending forces and their derivatives, enabling fast
timeintegration of cloth dynamics, and nearinteractive rates
for Willmore smoothing of large meshes. This is joint work with
Miklos Bergou, Akash Garg, Eitan Grinspun, David Harmon, and
Denis Zorin.
 John Sullivan "Medial axes in Moebius
geometry"
We consider the medial axis of a disk with a moebius structure.
This leads to two interesting results. First, we can show (using
also the combinatorics of the associahedron) that the space of
spherical kpoint metrics is an open ball. (This space is also
known to be the moduli space of certain CMC surfaces, the
coplanar kunduloids.) Second, we can give a moebiusinvariant
version of the fourvertex theorem, which in the polygonal case
avoids the need for Dahlberg's local regularity condition. (We
will also revisit an old conjecture of Pinkall.)
 Kristoffer Josefsson "The four vertex
theorem for polygons"
We look at a discrete notion of curvature for polygons in the
plane, inspired by the correspondence of curvature and the
radius of osculating circles in the smooth case. A purely
geometric proof due to Dahlberg shows that under mild extra
conditions, every simple closed polygon has two local minima and
two local maxima of its curvature function.
 Boris Springborn "There are no
(5,7)triangulations of the torus, and similar theorems.
Function theoretic proofs."
There is no torus triangulation with only two irregular vertices
with degrees 5 and 7 and all other vertices of degree 6.
Strangely, no purely combinatorial proof is known for this and a
few similar theorems. We present function theoretical proofs.
The nonexistence of the torus triangulations described above
appears as a consequence of the fact that there are no elliptic
functions with only one simple zero and one simple pole. This is
joint work with Ivan Izmestiev, Rob Kusner, Günter Rote and
John Sullivan.
 Marc Alexa "Fair triangulated surfaces
from positional constraints at interactive rates"
We aim at computing fair triangulated surfaces in fractions of
second for interactive modeling environments. We assume the
combinatorics of the triangulation and positional constraints
for a subset of the vertices are given. Then, the positions for
the vertices are defined as an approximate solution to a
nonlinear fourth order PDE. The PDE is factored into two linear
systems, which have to be solved repeatedly. For achieving fast
computations, the LaplaceBeltrami operator is approximated with
the graph Laplacian, so that the factorization can be computed
in a preprocess. This approximation, however, requires that the
edges in the triangulation have equal lengths. The main idea to
accommodate for this is to prescribe edge vectors with edge
lengths that are the result of a diffusion process over the
surface. The resulting equations are also linear and can be
integrated into the original linear system with negligible extra
computation. This results in an iterative process that quickly
converges and produces fair triangulated surfaces in fractions
of a second.
 Jürgen RichterGebert "Recognition of
computationally contructed loci"
Curve recognition is a fundamental task in computer vision.
There algebraic curves are used to approximate shapes that have
been extracted from camera pictures in order to recognize the
shapes (a spoon, a fork, a nail, etc). In the talk we will
report on work in progress on a different branch of algebraic
curve recognition. Our problem is much more mathematically
motivated and has several crucial different features from the
computer vision problem. We consider curves that are generated
by a mathematical computer program like a computer algebra
system or a dynamic geometry program. The curves are given by a
collection of numerical sample points on the curve usually with
high arithmetic precision. The curve generating program itself
is treated as a black box or as an oracle that does not give a
priori knowledge on the curve. The ultimate goal of our research
is an algorithm that is able to the algebraic degree, a
parameter set, and if possible a classifying name to the curve
(like limacon, lemniscate, Watt curve,...). The talk will
describe how algebraic data can be reconstructed from the sample
points, how invariant properties of the curve can be extracted
and how randomization techniques can be used to stabilize the
obtained results. The talk will also contain software
demonstrations of the partial implementation of the proposed
algorithm.
 Ivan Izmestiev "The Colin de Verdiere
graph parameter and rigidity of convex polytopes"
The Colin de Verdiere parameter μ(G) is a number derived from
the spectral properties of matrices associated with the graph G.
A classical result says that the planar graphs are characterized
by the inequality μ(G) ≤ 3. In 2001 Lovász, basing on his
previous work with Schrijver, introduced a construction that
relates Colin de Verdiere matrices of corank 3 to skeleta of
convex 3polytopes.
In this talk we give a more direct geometric interpretation to
Lovász's construction. We show that it is connected to
the theory of mixed volumes. The technique developed provides
yet another proof of the infinitesimal rigidity of convex
polytopes.
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