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# Seminar - Abstracts

• 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 Riemann-Hilbert Point of View."

The classical Painlevé equations have been playing increasingly important role in physics since 1970s-1980s 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 Riemann-Hilbert 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 so-called 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. Courant-K. Friedrichs-H. Lewy for square lattices, by P. G. Ciarlet-P.-A. Raviart for special kite lattices, and by D. Chelkak-S. Smirnov for rhombic lattices. In particular, this proves the convergence for the Dirichlet problem on intrinsic Delauney triangulations studied by A. Bobenko-B. 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 alternating-current 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 contact-geometry of 2-spheres in the 4-sphere, generalizing Lie sphere geometry. Starting from the Pluecker quadric, the Lie quadric and the 4-sphere are identified by real structures as real sub-quadrics. I will define a discrete integrable system by Steiner cross-ratio in the Pluecker quadric that generalizes the quaternionic cross-ratio system in the 4-sphere. Then, I will show how the complex cross-ratio system in CP1 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 "half-touching" contact elements in the 4-sphere.

• Boris Hasselblatt "Legendrian knots and nonalgebraic contact Anosov flows on 3-manifolds"

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 3-manifolds.

• 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 translation-invariant, nonrelativistic, classical or quantal) Hamiltonian many-body 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 3-sphere can be described by its associated family of flat connections. This leads to an algebro-geometric 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 non-abelian 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 3-webs with holomorphic Chern connection and infinitesimal symmetries"

A finite collection of foliations form a web. Blaschke discovered that already for a 3-web in the plane, there is a nontrivial local invariant, namely the curvature form. Thus any local classification of 3-webs necessarily has functional moduli if no restriction on the class of webs is imposed. The most symmetric is a hexagonal 3-web when the curvature is supposed to vanish identically. In a regular point a hexagonal 3-web is locally diffeomorphic to 3 families of parallel lines. For singular points, where at least two foliations are not transverse, two hexagonal 3-webs are not necessarily locally diffeomorphic. We provide a complete classification of hexagonal singular 3-web 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 by-product, classification of hexagonal weighted homogeneous 3-webs is obtained.

• Dmitry Korotkin "Higher genus generalization of Weierstrass sigma-functions"

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 well-developed generalization to higher genus realized by Riemann theta-functions and related objects. On the other hand, the notion of higher genus sigma-functions remained essentially undeveloped until recently (except the hyperelliptic case considered by Klein and the very special class of so-called (n,s)-curves considered more recently by Buchstaber and his co-workers). In our talk we introduce a natural notion of higher genus sigma-function 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 sigma-function via Jacobi's theta-function θ1. 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 non-specialists. It is an extension of a result of Frenkel, Reshetikhin and Semenov-Tian-Shansky and therefore is expected to have ramifications in the study of lattice or quantum W-algebras. It is also expected to be a natural setting for discrete integrable systems analogous to those of so-called KdV-type.

• Elena Klimenko "Discrete groups of isometries in hyperbolic 3-space"

The full group of orientation-preserving isometries of hyperbolic 3-space H3 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 H3/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 two-generator discrete subgroups of PSL(2,C) and to show the corresponding manifolds and orbifolds.

• Vladimir Dragovic "From Poncelet porisms to quad-graphs: discrete differential-geometric structures approach"

We present discrete differential-geometric structures and configurations that appeared in our study of Poncelet porisms, like the Poncelet--Darboux grids and the Weyr chains. We give their natural higher-dimensional and higher-genera generalizations, we introduce a notion of discrete billiard geodesics, and we get generalizations of the Darboux theorem, the Griffiths-Harris space theorem, and the Poncelet theorem. This leads to progress in a thirty-year-old programme of Griffiths and Harris to understand higher-dimensional 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 differential-geometric notion of Darboux coordinates, the modern concept of $n$-valued Buchstaber--Novikov 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 Yang--Baxter 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 (Witten--Dijkgraaf--Verlinde--Verlinde) 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 Schnittkörper-Operatoren, 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 Laplace-Beltrami 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 2-manifolds.

• 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 Laplace-Operator 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 quad-equations"

We consider 3D consistent systems of six possibly different quad-equations assigned to the faces of a cube. The well-known classification of 3D consistent quad-equations, the so-called ABS-list, when all six equations are the same up to parameters, is included in this situation. The extension of these equations to the whole lattice Z3 is possible by reflecting the cubes. This construction allows to show the integrability of all quad-equations by deriving Bäcklund transformations and zero-curvature 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 Z3 and the procedure of deriving Bäcklund transformations and zero-curvature representations for quad-equations.

• 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 so-called 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 Hertrich-Jeromin "Special Lie-applicable surfaces"

I will aim to describe how linear Weingarten surfaces in space forms appear as special Lie-applicable 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 Yang-Baxter equation and classical discrete integrable equations"

We obtain a new solution of the star-triangle 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 temperature-like 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 Q4-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 Kashiwara-Miwa 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 3D-sphere. 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 2D-sphere, the averaged system has one degree of freedom and is therefore integrable and admits a complete description in terms of phase portraits. In the 3D-sphere 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 Hilbert-Einstein functional"

An infinitesimal isometry of a surface in R3 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 Hilbert-Einstein 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 Fisher-Hartwig singularities. The Riemann-Hilbert Approach."

We will discuss some new asymptotic results obtained via the integrable systems techniques, notably the Riemann-Hilbert method, in the area of Hankel and Toeplitz determinants whose symbols possess Fisher-Hartwig singularities on a smooth background. Specifically, we will discuss the proof of the Basor-Tracy conjecture concerning the Toeplitz determinants, the asymptotics of Hankel and Toeplitz + Hankel determinants on a finite interval, and the Painlevé-type crossover formulae describing a transition between the Szegö and the Fisher-Hartwig 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 Cahn-Hilliard equation and the Hele-Shaw 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 infinite-dimensional space of densities that induces the desired metric tensor and allows to define and analyse the gradient flow of the free energy functional. A by-product of this is a method to construct weak solutions by means of time-discrete "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 Manifold-Valued 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 manifold-valued 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 non-linearity, 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, physics-like 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 "Self-Linking, Inflections, and the Normal Euler Class for Smooth and Polyhedral Surfaces in Four-Space"

The normal Euler class for a simplicial surface in four-space 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 self-linking 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 differential-algebraic 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 log|f'| 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, Kohlenstoff-Nanotuben) 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 Monge-Amp\{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 Fermat-like 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 Non-Symmetric Discrete Relativistic Toda Systems and Their Relation to Quad-Graphs"

In this talk, we establish a relation of non-symmetric discrete relativistic Toda type equations to 3D consistent systems of quad-equations . Moreover, we will present a master equation which includes both symmetric and non-symmetric discrete relativistic Toda type equations as particular or limiting cases. In addition, we will give a survey at adequate derivations of non-symmetric 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 "Hirota-Kimura 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 Hirota-Kimura 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 4-connected 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 "Energy-preserving 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 state-space 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 user-specified 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. Non-experts 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 R3 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 Rivin-Hodgson theorem generalizes Andreev's theorem that describes acute-angled 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 Rivin-Hodgson 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"

Three-dimensional 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 non-convex 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 non-convex 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 Schwarz-Christoffel formula for such polygons, involving theta-functions.
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 2k-manifolds as subcomplexes of the boundary complex of a regular polytope. We call such a subcomplex k-Hamiltonian if it contains the full k-skeleton 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 so-called super-neighborly triangulations) we focus on the case of the cross polytope and the sporadic regular 4-polytopes. By our results the existence of 1-Hamiltonian surfaces is now decided for all regular polytopes. Furthermore we investigate 2-Hamiltonian 4-manifolds in the d-dimensional 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 S2 × S2. By this example all regular cases of n vertices with n < 19 or, equivalently, all cases of regular d-polytopes 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 higher-dimensional 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 3-dimensional lattices"

We study geometric consistency relations between angles on 3-dimensional (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 ultra-local'' 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 Yang-Baxter equation). These solutions generate an infinite number of non-trivial solutions of the Yang-Baxter 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 quasi-classical 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 graph-theoretic version of the Riemann-Roch 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 well-known edge-based 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 Hertrich-Jeromin "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 3-Polytopes on Small Integer Grids"

We present a constructive method for embedding a 3-connected planar graph as 3-polytope with small integer coordinates. The embedding needs no coordinate greater than O(27.55n). 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 Maxwell-Cremona Theorem. We use the Matrix-Tree 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 form-finding 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 multilayer-constructions 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 3k-2 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 Hertrich-Jeromin and Susana Santos. The central goal is a definition for discrete CMC surfaces in any of the three space forms (Euclidean 3-space, spherical 3-space and hyperbolic 3-space), 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 3-space 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 1-skeleton of triangulated surfaces with common length one of all edges.

• Michael Baake "Coincidence site lattices and their generalizations"

It is well-known 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 d-manifolds. 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 d-manifold is the partial unfolding of some combinatorial d-sphere.

• Max Wardetzky "Discrete bending energies"

Efficient computation of curvature-based 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 time-integration of cloth dynamics, and near-interactive 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 k-point metrics is an open ball. (This space is also known to be the moduli space of certain CMC surfaces, the coplanar k-unduloids.) Second, we can give a moebius-invariant version of the four-vertex 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 non-existence 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 non-linear fourth order PDE. The PDE is factored into two linear systems, which have to be solved repeatedly. For achieving fast computations, the Laplace-Beltrami operator is approximated with the graph Laplacian, so that the factorization can be computed in a pre-process. 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 Richter-Gebert "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 3-polytopes.

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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 Ivan Izmestiev . 02.05.2012.