SFB-Seminar Berlin

There is a (weekly) seminar for the members of the SFB/Transregio and other interested people.

  • Next Occurrence: 21.03.2017, 14:15 - 15:15
  • Christian Klein: TBA
  • Type: Seminar
  • Location: TU Berlin

Contact: Ulrike Bücking
Room: MA 875
Time: Tuesday 14:15-15:15



  • 14:15 - 15:15 TBA,  Christian Klein (Université de Bourgogne)


  • 14:15 - 15:15 Asymptotic and Circular K-nets: a case study in discrete reparametrization and integrability, Andrew Sageman-Furnas (Universität Göttingen)
  • From an integrable perspective, surfaces of constant negative Gauß curvature are often given either in asymptotic- or curvature- line parametrization. In the smooth setting, a simple change of variables allows one to easily move between the two coordinate representations. However, the two corresponding discrete theories have remained mostly disparate.
    In this talk, I will start to shed some light on their relationship, by introducing a 2x2 Lax pair for circular K-nets that is tightly linked to the 4D consistency of the Lax pair for asymptotic K-nets given by Bobenko and Pinkall. The description naturally gives rise to an associated family and Bäcklund transformations for cK-nets. Each member of the resulting associated family — although no longer circular — has constant negative Gauß curvature. An unusual feature of the resulting Bäcklund transformation is that, while its double step compatibility cube is 3D consistent and agrees with one found by Schief using other methods, its single step compatibility cube is not 3D consistent. Nevertheless, we provide explicit solutions for the Bäcklund transformations of the vacuum (in particular, Dini, Kuen, and breather surfaces), together with their respective associated families.
    This is joint work with Tim Hoffmann.


  • 14:15 - 15:15 Hyperbolic earthquakes, Lara Skuppin (TU Berlin)
  • The aim of this talk is to present an introduction to earthquakes on hyperbolic surfaces. The material can be found in the article 'Earthquakes in two-dimensional hyperbolic geometry' by W.P. Thurston (1986).


  • 14:15 - 15:15 Critical Points and Local Normal Degree for Smooth and Polyhedral Surfaces in Three-Space, Thomas Banchoff (Brown University)
  • Following Heinz Hopf, we define critical point indices and local normal degree for height functions on a smooth or polyhedral surface embedded in three-space. We illustrate how these notions are different by showing that there is no smooth embedding of a torus into three-space with a height function that has exactly three critical points, although there is a polyhedral embedding of the torus with this property and also a smooth immersion of the torus with this property.


  • 14:15 - 15:15 How network topology determines condensation and transport properties of the antisymmetric Lotka-Volterra equation, Johannes Knebel (Ludwig-Maximilians-Universität München)
  • Condensation is a collective behavior of particles observed in both classical and quantum physics; both in and out of thermodynamic equilibrium. In our work, we studied condensation phenomena that occur for a driven and dissipative gas of bosons. Only recently has it been proposed that bosons in such a setup may not only condense into a single, but also into multiple non-degenerate states. This condensation is captured by a nonlinear dynamical system, the antisymmetric Lotka-Volterra equation. In our work, we applied an algebraic method to determine the states that become the condensates. This condensate selection is guided by the vanishing of relative entropy production. Our approach yields insights into and raises new questions about the interplay between network topology and transport properties of the antisymmetric Lotka-Volterra equation.


  • 14:15 - 15:15 On Thurston’s vision in geometry, topology, and dynamics — and aspects of current research, Dierk Schleicher (Jacobs University Bremen)
  • Since the 1980s, Bill Thurston has done fundamental work in apparently quite different areas of mathematics: in particular, on the geometry of 3-manifolds, on automorphisms of surfaces, and on holomorphic dynamics. In all three areas, he proved deep and fundamental theorems that turn out to be surprisingly closely connected both in statements and in proofs.
    In all three areas, the statements can be expressed that either a topological object has a geometric structure (the manifold is geometric, the surface automorphism has Pseudo-Anosov structure, a branched cover of the sphere respects the complex structure), or there is a well defined topological-combinatorial obstruction consisting of a finite collection of disjoint simple closed curves with specific properties. Moreover, all three theorems are proved by an iteration process in a finite dimensional Teichmüller space (this is a complex space that parametrizes Riemann surfaces of finite type).
    I will try to relate these different topics and at least explain the statements and their context. I will also try to outline current work on extending this work from rational to transcendental dynamics (joint with John Hubbard, Mitsuhiro Shishikura, and Bayani Hazemach).


  • 14:15 - 15:15 Smooth polyhedral surfaces, Felix Günther (Max-Planck-Institut für Mathematik, Bonn)
  • We study the geometry of polyhedral surfaces, which are fundamental objects in architectural geometry. The aim of this talk is to discuss suitable assessments of smoothness of polyhedral surfaces. A smooth reference surface which the polyhedral surface should approximate is not needed. To describe such properties, we analyze the Gaussian image of vertex stars and derive restrictions on its shape. By investigating the discrete Dupin indicatrix we will show that star-shapedness of the Gaussian images is a good indicator for smoothness in a region of non-vanishing discrete Gaussian curvature.
    (Joint work with Helmut Pottmann.)


  • 14:15 - 15:15 Two types of discrete isothermic surfaces in Minkowski space, Masashi Yasumoto (Kobe University, Japan)
  • In this talk we will discuss two types of discrete isothermic surfaces in Minkowski 3-space $\mathbb{R}^{2,1}$. In particular, we will see two types of discrete isothermic surfaces in $\mathbb{R}^{2,1}$ with mean curvature identically $0$, which.are called discrete maximal surfaces and discrete timelike minimal surfaces. Like in the case of discrete (isothermic) minimal surfaces in Euclidean 3-space, these surfaces admit Weierstrass-type representations. On the other hand, unlike in the case of discrete minimal surfaces, discrete maximal and timelike minimal surfaces in $\mathbb{R}^{2,1}$ genarally have certain singularities. We will introduce Weierstrass-type representations for discrete maximal and timelike minimal surfaces, and analyze their singularities.


  • 14:15 - 15:15 Dodgson’s condensation method, octahedral equation and Burchnall-Chaundy polynomials, Alexander Veselov (Loughborough University)
  • The Burchnall-Chaundy polynomials $P_n(z)$ are determined by the differential recurrence relation $$P_{n+1}'(z)P_{n-1}(z)-P_{n+1}(z)P_{n-1}'(z)=P_n(z)^2$$ with $P_{-1}(z)=P_0(z)=1.$ The fact that this recurrence relation has all solutions polynomial is not obvious and is similar to the integrality of Somos sequences and the Laurent phenomenon.
    We discuss this parallel in more detail and extend it to the difference equation $$R_{n+1}(z+1)R_{n-1}(z-1)-R_{n+1}(z-1)R_{n-1}(z+1)=R^2_n(z),$$ related to Dodgson’s octahedral equation, describing the recursive way for computing determinants, known as condensation method. As a corollary we have a new form of the Burchnall-Chaundy polynomials in terms of the initial data $P_n(0)$, which is shown to be Laurent.
    The talk is based on a joint work with Ralph Willox (J. Phys. A 48 (2015) 205201).


  • 14:15 - 15:15 Elementary approach to closed billiard trajectories in asymmetric normed spaces, Arseniy Akopyan (IST Austria)
  • We apply the technique of Karoly Bezdek and Daniel Bezdek to study the billiards in convex bodies, when the length is measured with a (possibly asymmetric) norm. We give elementary proofs of some known results and prove an estimate for the shortest closed billiard trajectory, related to the non-symmetric Mahler problem.
    (Joint work with A. M. Balitskiy, R. N. Karasev, and A. Sharipova.)


  • 14:15 - 15:15 Equations of isomonodromic deformations of Fuchsian systems and canonical parameterization of coadjoint orbits, Mikhail Babich (Steklov Mathematical Institute, Moscow)
  • The connection between isomonodromic deformation of Fuchsian system of linear differential equations and Schlesinger system of equations will be considered. Namely, any Fuchsian system can be transformed in accordance with some Hamiltonian system, that is called the Schlesinger equations. Such deformation preserve the monodromy of the Fuchsian system. I will demonstrate that the space of all Fuchsian equations can be rationally projected on the standard symplectic space $(\mathbb{C}\times \mathbb{C})^M$ in such a way that the preimage of any point consists of the systems with the same monodromy. The Schlesinger flow can be projected on this symplectic space because the corresponding vector field has a property: the projections of the values of the field at points coincide, if the projections of the points coincide. The (non-linear) equations of the isomonodromic deformation are the Euler–Lagrange equations corresponding the resulting flow on the extended phase space.


  • 14:15 - 15:15 Integrability of limit shapes in the 6-vertex model, Nicolai Reshetikhin (UC Berkeley)
  • This talk is focused on the limit shape phenomenon in the 6-vertex model. First I will recall the definition of the model and the basic facts about the limit shape phenomenon. Then we will see that PDEs describing limit shapes have infinitely many integrals. Then we will focus on the free-fermionic point where Hamiltonians can be computed in terms of dilogarithms.


  • 14:15 - 15:15 Curves in $\mathbb{R}^d$ intersecting every hyperplane at most d+1 times, Imre Bárány (Hungarian Academy of Sciences and University College London)
  • A partial result: if a planar curve intersects every line in at most 3 points, then it can be partitioned into 4 convex curves. This result can be extended to $\mathbb{R}^d$: if a curve in $\mathbb{R}^d$ intersects every hyperplane at most d+1 times, then it can be split into M(d) convex curves. The extension implies a good, asymptotically precise, lower bound on a geometric Ramsey number. Joint result with the late Jiri Matousek and Attila Por.


  • 14:15 - 15:15 On variational systems on the root lattice $Q(A_{N})$, Raphael Boll (TU Berlin)
  • We present the theory of certain pluri-Lagrangian systems (i.e., integrable systems with variational origin) on the root lattice $Q(A_{N})$ and consider their relation to hyperbolic equations.
    In the two-dimensional case the considered hyperbolic equations are the quad-equations from the ABS-list and its asymmetric extension. These are less general than the variational equations, i.e., every solution of the system of quad-equations satisfies the corresponding system of variational systems, but not vice versa. Moreover, we demonstrate that some variational systems on $Q(A_{N})$ encodes several systems on $\mathbb{Z}^{N}$.
    In the three-dimensional case the considered hyperbolic equation is the discrete KP equation. It is a rather surprising fact that, in this case, the system of variational equations is, in a sense, equivalent to the system of hyperbolic equations if $N\geq4$. In addition, we demonstrate that the system of variational equations on $Q(A_{N})$ is more elementary than the one on the cubic lattice $\mathbb{Z}^{N}$.
    This is joint work with Matteo Petrera and Yuri B. Suris.


  • 14:15 - 15:15 On the classification of 4D consistent maps, Matteo Petrera (TU Berlin)
  • It is nowadays a well-established fact that integrability of 2D discrete equations can be identified with their 3D consistency. Our aim is to turn our attention to integrability of 3D discrete systems, now understood as 4D consistency. The most striking feature is that the number of integrable systems drops dramatically with the growth of the dimension: only half a dozen of discrete 3D systems with the property of 4D consistency are known and all of them are of a geometric origin.
    In this talk I will present a classification of 4D consistent maps given by (formal or convergent) series of the following kind: $$ T_k x_{ij}=x_{ij} + \sum_{m=2}^\infty A_{ij ; \, k}^{(m)}(x_{ij},x_{ik},x_{jk}), $$ where $A_{ij;\, k}^{(m)}$ are homogeneous polynomials of degree $m$ of their respective arguments. Here $T_k$ denotes the unit shift in the $k$-th coordinate direction.
    The result of the classification is that the only non-trivial 4D consistent map is given by the known symmetric Darboux system $$ T_k x_{ij}=\frac{x_{ij}+x_{ik}x_{jk}}{\sqrt{1-x_{ik}^2}\sqrt{1-x_{jk}^2}}. $$ I will present a new geometric interpretation of such a system, thus showing that its dynamics can be interpreted as a suitable iteration of spherical triangles on the unit sphere.
    Finally, I will show that the symmetric Darboux system is also Arnould-Liouville integrable, thus possessing two functionally independent integrals of motion and a family of compatible Poisson structures. Its solvability in terms of elliptic functions is also established.


  • 14:15 - 15:15 Hyper-ideal circle patterns and discrete uniformization of finite branch covers over the Riemann sphere, Nikolay Dimitrov (TU Berlin)
  • With the help of hyper-ideal circle pattern theory, I propose a discrete version of the classical uniformization theorem for compact Riemann surfaces represented as finite branch covers over the Riemann sphere.
    The talk will be divided into two parts. In the first part I will briefly introduce the necessary tools and present the main result. In the second part, I will propose a tentative proof of the claim that, in the context of branch covers, discrete uniformization via hyper-ideal circle patterns always exists and is unique (up to isometry). It is fair to say that this is work in progress and the talk serves as a proof-verification of the latter claim.


  • 14:15 - 14:45 Approximation by planar elastica, Toke Nørbjerg (Technical University of Denmark)
  • New technology under development in the building industry requires the segmentation of a CAD surface into pieces that are then approximated by surfaces foliated by planar elastic curves. A critical part of this problem is to approximate a given "arbitrary" curve segment by a segment of an elastic curve. Doing this depends on choosing an appropriate parameterization for the space of elastic curve segments. The success of the method I will describe depends on the fact that the curvature of a planar elastica is an affine function of the distance along a special direction in the plane.
  • 14:45 - 15:15 Generating new examples of integrable surfaces from curves, David Brander (Technical University of Denmark)
  • Integrable surfaces, such as constant mean and Gauss curvature surfaces, Willmore surfaces, etc., are all associated to nonlinear PDE that can be solved using loop group methods. Variants of the DPW method generalize the Weierstrass representation for minimal surfaces; however, the utitility of this method is somewhat tempered by the loss of geometric information in loop group decompositions. Recent work allows us to preserve all geometric information along an entire curve, thereby providing the possibility to easily produce many new examples of these surfaces in a geometrically controlled manner. I will explain the essential idea briefly, and show some examples of its use.


  • 14:15 - 15:15 Linearly Constrained Evolutions of Critical Points and Adaptive Anisotropic Remeshing in Brittle Fracture Simulation, Massimo Fornasier (TU München)
  • The quasistatic evolution of a fracture in continuum mechanics is driven by the instantaneous minimization of an energy functional which is typically nonsmooth and nonconvex. The minimizers may posses singularities corresponding to physically interesting modes of fracture and thus the capture of such configurations plays a pivotal role in the simulation of realistic processes. Additionally, time-dependent boundary conditions, modeling external forces acting on the bending and fracturing body, must be taken into account, resulting in linear constraints to be satisfied by the critical points. To efficiently realize numerically such type of evolutions, we focused on designing a new algorithm to search for critical points of nonconvex functionals, under very mild smoothness assumptions and with convergence guarantees We reformulated the problem as a sequence of locally quadratic perturbations which are solved by means of the classical non-stationary augmented Lagrangian method and we proved its unconditional (or global) convergence to critical points of the objective functional. Besides the challenging analysis, we performed extensive numerical tests to validate the procedure in several interesting cases. In particular we considered the well-known free-discontinuity model of brittle fracture by Francfort-Marigo, which requires the minimization of an energy balancing the elastic energy and the fracture one. However, the minimization of the nonconvex and nonsmooth functional involving unknown functions and sets makes the numerical realization very challenging. A smooth phase field $\Gamma$-approximation of the energy functional is given by the Ambrosio and Tortorelli functional whose minimization can be realized now by an alternating minimization. Bourdin, Francfort, and Marigo showed that the discretization of such an alternating minimization can produce reliable fracture simulation only by using very fine grids. More recently Burke, Ortner, and Süli proposed a fully adaptive scheme based on isotropic mesh refinements, leading though to the generation of extremely fine adapted meshes as reported in their paper. The results we obtained are built upon the latter work, but using anisotropic 'mesh adaptation'. Relevant features of the anisotropic mesh adaptation are: 1. The number of degrees of freedom and the computational times are dramatically reduced; 2. The remeshing does not alter the energy profile evolution; 3. On the crack tip the automatically generated mesh is nearly isotropic and does not constitute an artificial bias for the crack evolution. As a consequence of 2. and 3. we obtain always physically acceptable crack evolutions beyond state of the art simulations. Eventually not only we addressed successfully several benchmark tests outperforming previous attempts in efficiency and accuracy, but also we could perform correctly simulation where former algorithms failed.


  • 14:15 - 15:15 On the Construction of cmc 1 surfaces in $H^3$ with platonic symmetries, Jonas Ziefle (Universität Tübingen)
  • In this talk we introduce a Weierstrass representation in which cmc 1 surfaces in the hyperbolic space are represented as the Hopf differential and the Schwarzian derivative of the hyperbolic Gauss map. In order to construct cmc 1 surfaces, we have to unitarize the monodromy representation of a Fuchsian system. First we see how this is done for Trinoids (i.e. three ends) by using the spherical triangular inequalities, then we use symmetry to reduce the unitarization problem for surfaces with more ends again to the spherical triangular inequalities.


  • 11:15 - 12:15 (Friday) Flexibility of hyperbolic polyhedra and compact domains with prescribed convex boundary metrics in quasi-Fuchsian manifolds, Dmitriy Slutskiy (Université de Strasbourg)
  • 1. We construct an infinitesimally flexible polyhedron in hyperbolic 3-space such that its volume is not stationary under the infinitesimal flex.
    2. We obtain a necessary condition for flexibility of suspensions in hyperbolic 3-space.
    3. We show the existence of a convex compact domain in a quasi-Fuchsian manifold such that the induced metric on its boundary coincides with a prescribed surface metric of curvature K≥−1 in the sense of A. D. Alexandrov.


  • 14:15 - 15:15 Challenges in discrete nonholonomic mechanics: Preservation of volumes, Hamiltonization and Integrability, Luis García-Naranjo (IIMAS-UNAM, México)
  • In the first part of the talk I will review the structure of the equations of nonholonomic mechanics and the discretization of the Lagrange-D'Alembert principle proposed by J. Cortés and S. Martínez in 2001. I will then proceed to describe particular discretizations of some classic nonholonomic systems on Lie groups paying special attention to three important aspects: volume preservation, Hamiltonization and integrability.


  • 14:15 - 15:15 Discrete Complex Line Bundles over Simplicial Complexes, Felix Knöppel (TU Berlin)
  • We classify all discrete complex line bundles with connection over finite simplicial complexes. As a result we present a discrete analogue of a theorem of André Weil.


  • 14:15 - 15:15 (with live broadcast to TU München) DGD Gallery - Storage, Sharing, and Publication of Research Data, Stefan Sechelmann (TU Berlin)
  • Projects within the DGD collaborative research center frequently produce digital data. This data is usually not published along with corresponding publications. Often is is not even stored in a structured fashion for later reference or collaboration.
    The DGD Gallery solves this problem by offering an online web service for the storage, sharing, and publication of digital research data. We present the basic ideas and usage of the service and show examples taken from projects of the research center.


  • 14:15 - 15:15 Construction of embedded n-periodic surfaces in $\mathbb{R}^n$, Susanne Kürsten (TU Darmstadt)
  • It is possible to construct complete minimal surfaces by applying the Schwarz reflection principle. When choosing a Jordan curve $J$ along the edges of a $3$-dimensional cube and considering the solution of Plateau's problem with boundary $J$, Schwarz reflections across the boundary edges lead to a complete minimal surface. A prominent example for this kind of minimal surfaces is the well known Schwarz D-surface. The resulting surfaces in $\mathbb{R}^3$ are known to be embedded, $3$-periodic and minimal.
    In this talk I will explain under which conditions the analog construction in $\mathbb{R}^n$ leads to an embedded, $n$-periodic minimal surface. The main problem is to ensure that the resulting surface is embedded. This question is not related to the minimality of the surface. It is possible to consider nearly arbitrary embedded surfaces $f$ with boundary $J$, where $J$ is a Jordan curve along the edges of an $n$-dimensional cube and $f$ lies in this cube. By reflection across the boundary edges a surface is constructed. I will give concrete criteria for the embeddedness of this surface, which are easy to check in applications.


  • 14:15 - 15:15 Regular and anti-regular generalized quadrangles, Michael Joswig (TU Berlin)
  • Generalized quadrangles are special cases of spherical buildings, which, e.g., provide geometric models for the simple Lie groups. In this talk we survey known results about two very special families of generalized quadrangles related to symplectic and orthogonal groups.


  • 14:15 - 15:15 Hyper-ideal circle patterns and discrete uniformization of surfaces with non-positive curvature, Nikolay Dimitrov (TU Berlin)
  • In this talk I will try to explain how hyper-ideal circle patterns can be used to construct a discrete version of the uniformization theorem for polyhedral surfaces with cone points of non-poisitive curvature. I will also show how the same method can be applied to discretely uniformize hyper-elliptic Riemann surfaces.


  • 14:15 - 15:15 Random Geometry , Steffen Rohde (University of Washington)
  • How does a triangulation of the sphere, chosen uniformly among all triangulations with n triangles, look like when n is large? I will explain the answer (due to Le Gall and Miermont) in the realm of metric spaces, and will discuss speculation and partial results when the triangulation is viewed as a Riemann surface. Along the way, I will describe the Loewner equation and how it yields an algorithm to compute dessins d'enfants.


  • 14:15 - 15:15 Discrete Riemann surfaces, Felix Günther (IHÉS, France)
  • I present a linear theory of discrete Riemann surfaces based on quadrilateral cellular decompositions and continue previous work of Mercat, Bobenko and Skopenkov. On a purely combinatorial level I discuss a discrete Riemann-Hurwitz formula and show how several neighboring branch points can be merged to one branch point of higher order. Using the medial graph to define operators on discrete differential forms, a discrete theory of Abelian differentials can be developed in analogy to the classical case. Inter alia, I prove a discrete Riemann-Roch theorem that includes double poles of discrete differentials. Since the dimension of discrete holomorphic differentials is twice as high as in the smooth case, I comment why the spin-holomorphic functions defined by Chelkak and Smirnov might help to reduce the dimension.


  • 14:15 - 15:15 Isothermic triangulated surfaces, Wai-Yeung Lam (TU Berlin)
  • In differential geometry, many interesting surfaces are isothermic, such as surfaces of revolution, quadrics and constant mean curvature surfaces.
    Motivated from the smooth theory, we define a triangulated surface in Euclidean space to be isothermic if there exists an infinitesimal rigid deformation preserving the integrated mean curvature. This definition is Moebius invariant and has several equivalent formulations. They relate the notions of length cross ratio, circle pattern and self-stress. We will see how this definition generalizes the discrete isothermic nets with quadrilaterals.
    As an application, we have a simple way to find the reciprocal-parallel meshes of an inscribed triangular mesh, which are regarded as discrete minimal surfaces.


  • 14:15 - 15:15 Matching centroids by a projective transformation, Ivan Izmestiev (FU Berlin)
  • Let K and L be two subsets of R^d. Does there exist a projective transformation f such that the centroids of f(K) and f(L) coincide? We allow each of K and L to be a point, a finite set of points, or a d-dimensional body, and find in each case a functional whose critical points correspond to solutions. Under certain assumptions the transformation f is unique modulo post-composition with affine transformations.
    Connections arise with the algebraic polarity, Moebius centering of polytopes, Santalo points, and Hilbert geometry.
    The talk is based on the arxiv preprint 1409.6176.


  • 14:15 - 15:15 Periodic conformal maps, Thilo Rörig and Stefan Sechelmann (TU Berlin)
  • We present a new method to obtain periodic conformal parameterizations of surfaces with cylinder topology and describe applications to architectural design and rationalization of surfaces. The method is based on discrete conformal maps from the surface mesh to a cylinder or cone of revolution. It accounts for a number of degrees of freedom on the boundary that can be used to obtain a variety of alternative panelizations. We illustrate different choices of parameters for NURBS surface designs. Further, we describe how our parameterization can be used to get a periodic boundary aligned hex-mesh on a doubly-curved surface and show the potential on an architectural facade case study. Here we optimize an initial mesh in various ways to consist of a limited number of planar regular hexagons that panel a given surface.


  • 14:15 - 15:15 Li-Yau inequalites on finite graphs, Florentin Münch (Friedrich-Schiller-Universität Jena)
  • In 1986, Li and Yau proved a logarithmic gradient estimate for manifolds with non-negative Ricci-curvature, later known as Li-Yau inequality. Since then, great effort was made to establish an analog result on graphs. To handle the concept of a Ricci-curvature on graphs, one introduces curvature bound conditions, since no suitable explicit definition is known yet. The calculus of Bakry and Émery is used to formulate such curvature bound conditions. A breakthrough was made in 2013, as a gradient estimate, which is very similar to the Li-Yau inequality, was established on graphs. In this talk, we will prove the original Li-Yau inequality on graphs and we will give a new notion of curvature.


  • 14:15 - 15:15 Ricci curvature on triangulations, using optimal transport, following Ollivier, Pascal Romon (Université Paris-Est Marne-la-Vallée)
  • The problem of defining relevant geometric objects, such as the curvature, is a hard one in discrete geometry. In 2009, Ollivier defined a notion of curvature applicable to a wide category of measured metric spaces, including graphs and polyhedral surfaces, which coincides with classical (smooth) Ricci curvature when the space is a smooth manifold. Lin, Lu and Yau and Jost and Liu have used and extended this notion for graphs, giving estimates for the curvature and, hence, the diameter, in terms of the combinatorics (Myers’ theorem). I will explain how optimal transport can be used to define such a curvature, and how one actually computes it for polyhedral surfaces, as well as some applications. Joint work with Benoît Loisel.


  • 14:15 - 15:15 Robust discrete complex analysis: a toolbox, Dmitry Chelkak (Steklov Institute St.Petersburg and ETH Zurich)
  • We prove a number of double-sided estimates relating discrete counterparts of several classical conformal invariants of a quadrilateral: cross-ratios, extremal lengths and random walk partition functions. The results hold true for any discrete quadrilateral (simply connected domain with four marked boundary vertices) and are uniform with respect to geometric properties of the configuration. Moreover, due to recent results of Angel, Barlow, Gurel-Gurevich and Nachmias, those uniform estimates are fulfilled for domains drawn on any infinite "properly embedded" planar graph (e.g., any parabolic circle packing) whose vertices have bounded degrees.


  • 14:15 - 15:15 Circular coordinates and dynamical systems, Vin de Silva (Pomona College)
  • High-dimensional data sets often carry meaningful low-dimensional structures. There are different ways of extracting such structural information. The classic (circa 2000, with some anticipation in the 1990s) strategy of nonlinear dimensionality reduction (NLDR) involves exploiting geometric structure (geodesics, local linear geometry, harmonic forms etc.) to find a small set of useful real-valued coordinates. The classic (circa 2000, with some anticipation in the 1990s) strategy of persistent topology calculates robust topological invariants based on a parametrized modification of homology theory. In this talk, I will describe a marriage between these two strategies, and show how persistent co-homology can be used to find circle-valued coordinate functions. Such coordinates can be used empirically to study periodicity phenomena in dynamical systems. This is joint work with Dmitry Morozov, Primoz Skraba, and Mikael Vejdemo-Johansson.


  • 14:15 - 15:15 Singular spectral curves and orthogonal curvilinear coordinate systems, Iskander A. Taimanov (Sobolev Institute of Mathematics, Novosibirsk)
  • We shall expose the extension of Krichever's finite-gap procedure for constructing orthogonal curviliner coordinate systems to the case of singular spectral curves and demonstrate how therewith one may obtain some classical coordinate systems such as polar and spherical by this method.


  • 14:15 - 15:15 Supercyclidic extension of Q-nets, Emanuel Huhnen-Venedey (TU Berlin)
  • Supercyclides are surfaces in $RP^3$ that posses a conjugate parametrization for which all parameter lines are conics, where the tangent planes along each conic envelop a quadratic cone. Restriction of such a parametrization to a closed rectangle yields a "supercyclidic patch", bounded by pieces of conic sections, for which the four vertices are automatically coplanar. Conversely, given a planar quadrilateral, there is a 10 parameter family of supercyclidic patches with those prescribed vertices. In this talk we introduce and discuss the according extension of Q-nets (quadrilateral nets with planar faces) by supercyclidic patches, such that each 2D layer becomes a piecewise smooth $C^1$-surface, the resulting objects being called "supercyclidic nets". The extension of Q-nets to supercyclidic nets is multidimensionally consistent, i.e., discrete integrable, and induces fundamental transforms of supercyclidic nets that possess the usual Bianchi permutability properties. Moreover, supercyclidic nets induce in a natural way piecewise smooth conjugate coordinates on open subsets of $RP^3$ and also abitrary Q-refinements of the initial supporting Q-net. It is to be noted that the extension of a Q-net to a supercyclidic net is inherently linked with the extension of Q-nets to "fundamental line systems", that is, assigning to each vertex of a Q-net a line through that vertex such that lines at adjacent vertices intersect and the focal nets of the obtained line system are again Q-nets. This is joint work with Alexander I. Bobenko and Thilo Rörig.


  • 14:15 - 15:15 Integrable discretisation of hodograph-type systems, hyperelliptic integrals and Whitham equations, Wolfgang K. Schief (UNSW Sydney, Australia)


  • 14:15 - 15:15 Quasiconformal distortion of projective maps, Boris Springborn (TU Berlin)


  • 14:15 - 15:15 (with live broadcast to TU München) Curvature flow and lattice dislocations, Ken Stephenson (University of Tennessee)
  • 2d lattices take physical form in modern material science, as with buckyballs, buckytubes, and graphene. These lattices often involve random or intentional dislocations and take concrete geometrical forms determined by the laws of physics. We propose a naive model for these lattices based on circle packing, where the basic unit is a ring of atoms rather than single atoms. Circle packing brings not only suggestive embeddings, but notions of "curvature flow" which may help model physical relaxation processes. The talk will review circle packing and illustrate curvature flows.


  • 14:15 - 15:15 Polar actions and homogeneous compact geometries, Linus Kramer (Universität Münster)
  • In Riemannian geometry, an isometric group action is called polar if it admits a cross section which intersects all orbits orthogonally. The cross-section leads then to an interesting combinatorial structure which is transversal to the orbits and which looks locally "building-like". In my talk I will explain the classification of these building-like geometries, and the resulting classification of polar actions on symmetric spaces. This is joint work with Alexander Lytchak.


  • 14:15 - 15:15 Dispersionless integrable systems in 3D and their dispersive deformations, Evgeny Ferapontov (Loughborough University)
  • I will give a brief review of the classification of dispersionless integrable systems in 3D within particularly interesting subclasses. Integrable dispersive deformations thereof will also be discussed. Our approach is based on the method of hydrodynamic reductions.


  • 14:15 - 15:15 Formfinding with statics for polyhedral meshes, Johannes Wallner (TU Graz)
  • We report on recent progress in the efficient modeling and computation of polyhedral meshes or otherwise constrained meshes, in particular meshes to be used in architectural and industrial design. As it turns out, in many cases the constraint equations can be rewritten to allow almost-standard numerical methods to converge quickly, with appropriate regularization taking care of constraints which are both redundant and under-determined. We also demonstrate how equilibrium forces, with or without compression-only constraints, are part of the formfinding process. This is joint work with C.-C. Tang, X. Sun, A. Gomes and Helmut Pottmann.


  • 14:15 - 15:15 Classification of discrete 3D Hirota-type equations, Ilia Roustemoglou (Loughborough University)
  • We have recently proposed a novel approach to the classification of integrable differential/difference equations in 3D based on the method of deformations of hydrodynamic reductions. This approach is now extended to the fully discrete case. We classify discrete 3D integrable Hirota-type equations within various particularly interesting subclasses. The method can be viewed as an alternative to the conventional multi-dimensional consistency approach.


  • 14:15 - 15:15 Stability of minimal Lagrangian submanifold and $L^2$ harmonic 1-forms, Reiko Miyaoka (Tohoku University)
  • It is well-known that a compact stable minimal Lagrangian submanifold L in a Kaehler manifold M with positive Ricci curvature satisfies $H^1(L,R)=0$ (Y.G. Oh, `90). We generalize it to a non-compact complete stable minimal Lagrangian submanifold in M, showing that there are no $L^2$ harmonic 1-forms on L. It gives a topological and conformal obstruction to L. We give some other important facts and problems in this field. This is a joint work with my PhD student S. Ueki.


  • 14:15 - 15:15 Discrete complex analysis - the medial graph approach, Felix Günther (TU Berlin)
  • I discuss a new formulation of the linear theory of discrete complex analysis on planar quad-graphs based on its medial graph. It generalizes the theory on rhombic quad-graphs developed by Duffin, Mercat, Kenyon, Chelkak and Smirnov. In this talk, I provide discrete counterparts of the most fundamental objects in complex analysis such as holomorphic functions, derivatives, differential forms, wedge products, Hodge star, and the Laplacian. Also, I consider discrete versions of important fundamental theorems such as Green's first and second identity, and Cauchy's integral formulae for a holomorphic function and its derivative.


  • 14:15 - 15:15 An Introduction to Amoeba Theory, Timo de Wolff (Universität Saarbrücken)
  • Given an Laurent polynomial $f$ in $C[z^{±1}_1, . . . , z^{±1}_n ]$ the amoeba $A(f)$ (introduced by Gelfand, Kapranov, and Zelevinsky 1994) is the image of its variety $V(f)$ under the Log-map $Log : (C^*)^n to R^n, Log(z_1, . . . , z_n)=(log |z_1|, . . . , log |z_n|) $, where $V(f)$ is considered as a subset of the algebraic torus $(C^*)^n$. Amoebas carry an amazing amount of structural properties of and are related to applications in various mathematical subjects (including complex analysis, the topology of real algebraic curves and dynamical systems). In particular, they can be regarded as the canonical connection between classical algebraic geometry and tropical geometry. In this talk I give an introduction to amoeba theory by presenting an overview about selected key theorems and current key problems.


  • 14:15 - 15:15 The Moutard transformation of two-dimensional Schroedinger operators, Iskander A. Taimanov (Sobolev Institute of Mathematics, Russia)
  • We demonstrate how the Moutard transformation which was invented and widely used in the 19th century surface theory can be applied for constructing explicit examples of two-dimensional potentials of Schroedinger operators with interesting spectral properties.


  • 14:15 - 15:15 Quantum entanglement and finite gap integration, Alexander Its (IUPUI)


  • 14:15 - 15:15 Poisson geometry of difference Lax operators, Michael Semenov-Tian-Shansky (Université de Bourgogne)


  • 10:15 - 11:15 (Friday) How to prove Steinitz's theorem, Igor Pak (UCLA)
  • Steinitz's theorem is a classical but very remarkable result characterizing graphs of convex polytopes in $R^3$. In this talk, I will first survey several known proofs, and present one that is especially simple. I will then discuss the quantitative version and recent advances in this direction. Joint work with Stedman Wilson.


  • 16:15 - 17:15 (MA 144) Knots, algorithms and linear programming: the quest to solve unknot recognition in polynomial time, Benjamin Burton (University of Queensland)
  • In this talk we explore new approaches to the old and difficult computational problem of unknot recognition. Although the best known algorithms for this problem run in exponential time, there is increasing evidence that a polynomial time solution might be possible. We outline several promising approaches, in which computational geometry, linear programming and greedy algorithms all play starring roles. We finish with a new algorithm that combines techniques from topology and combinatorial optimisation, which is the first to exhibit "real world" polynomial time behaviour: although it is still exponential time in theory, exhaustive experimentation shows that this algorithm can solve unknot recognition for "practical" inputs by running just a linear number of linear programs. This includes joint work with Melih Ozlen.


  • 16:15 - 17:15 Hamiltonian dynamics of rigid bodies and point vortices, Steffen Weißmann (TU Berlin)
  • In this talk we introduce the equations of motion of several rigid bodies in a 2–dimensional inviscid and incompressible fluid, whose vorticity field is given by point vortices. We derive the Hamiltonian formulation via symplectic reduction from a canonical Hamiltonian system, in the spirit of Arnold's geometric description of fluid dynamics. From the Hamiltonian formulation we deduce a Lagrangian of the system, which we use to develop a family of geometric, variational time-integrators.


  • 16:15 - 17:15 Regular maps: some ways to visualize them, Faniry Razafindrazaka (FU Berlin)
  • Take six squares made from elastic fabric, glue them at their edges such that each three of them have a common vertex. The result is a cube. Now, take 24 heptagons and glue them at their edges such that each three of them have a common vertex. The resulting surface is the so called Klein quartic. It is a closed surface of genus three which can be realized on the tube frame of the edges of a tetrahedron. These are examples of regular maps. A regular map is a family of equivalent polygons glued together to form a closed 2-manifold which is, topologically, vertex-, edge- and face-transitive. They can be realized as a triangular tiling group in the hyperbolic space by making the correct identifications at the boundary. The challenge is to find a 3D realization which should preserve the transitivity properties. In this talk, we are going to present first, a group theoretical approach which gives beautiful realizations of some of these maps and second, a conceptual geometrical approach which reduces the problem to finding a "nice" map between two universal coverings of 4g-gons in hyperbolic space.


  • 16:15 - 17:15 Computing hyperbolic structures on 3-manifolds, Stephan Tillmann (The University of Sydney)
  • It is known, by remarkable work of Thurston and Perelman, that all three-dimensional spaces can be decomposed into pieces admitting uniform geometric structures. However a constructive way of finding these structures is not known. I will briefly describe the geometrisation theorem for 3-manifolds and the special role played by hyperbolic structures. After this, I will describe an algorithm to compute the hyperbolic structures, which uses a surprising "discretisation" of the problem. The main ingredients of this algorithm are normal surface theory, Groebner bases and computation of the Lobachevsky function. Part of this talk is based on joint work with Feng Luo (Rutgers) and Tian Yang (Rutgers).


  • 16:15 - 17:15 Globally Optimal Smooth Direction Fields, Felix Knöppel (TU Berlin)
  • Rotationally symmetric direction fields, so called n-RoSy fields, serve as input for applications in computer graphics such as non-photorealistic rendering or remeshing. Here one is especially interested in optimal smooth line or cross fields which are usually aligned with a given guidance field. It turns out that such fields can be handled in the most natural way if they are treated as vector fields in a Hermitian line bundle. The talk presents a finite element approach to these spaces for triangle meshes of arbitrary genus. Further, the Dirichlet energy of direction fields is infinite, in general. We introduce a well-defined energy on direction fields. This allows to produce direction fields which are globally optimal smooth in this sense. Comparison shows that they are also optimal with respect to a state-of- the-art smoothness measure while the results are obtained in much less time. This is joint work with K. Crane, U. Pinkall, and P. Schröder.


  • 16:15 - 17:15 Amoebas and Ronkin functions of algebraic curves with punctures, Igor Krichever (Columbia University)
  • Recently a notion of amoebas, Ronkin functions of plane algebraic curves have become central in the theory of real algebraic curves and the theory of dimer models. In the talk their generalizations for algebraic curves with marked points function will be presented. Connections with the spectral theory of difference operators will be discussed.


  • 16:15 - 17:15 On the Circumcenter of Mass, Arseniy Akopyan (Russian Academy of Science, Moscow)
  • We study the circumcenter of mass which is an affine combination of the circumcenters of the simplices in a triangulation of a polytope. In the talk we give simple proofs of recent results on the cirumcenter of mass of S. Tabachnikov and E. Tsukerman. In particular we give a simple explanation of existence of the Euler line in polytopes.


  • 16:15 - 17:15 (E-N 053) Gradient flows and Ricci curvature in discrete analysis, Jan Maas (Universität Bonn)
  • video broadcasting from TU Munich


  • 16:15 - 17:15 (Monday) Linearization through symmetries for discrete equations, Decio Levi (Roma Tre University)


  • 16:15 - 17:15 Idempotent biquadratics, Yang-Baxter maps and birational representations of Coxeter groups, James Atkinson (The University of Sydney)
  • What we call the "idempotent biquadratic" has its roots in the theory of elliptic functions. It is the formula for trisection of the periods expressed algebraically via the addition law. Interesting properties of this formula in part correspond to integrability of two quad-graph dynamical systems (namely the quadrirational Yang-Baxter map FI and the multi-quadratic quad equation Q4*), but go also beyond that. I will explain this new interpretation and describe the resulting generalised dynamics in terms of a birational representation of a particular sequence of Coxeter groups.


  • 16:15 - 17:15 Optimal Topological Simplification of Discrete Functions on Surfaces, Carsten Lange (Université Pierre et Marie Curie, Paris VI)
  • Given a function $f$ and a tolerance $\delta>0$, find a pertubation $f_\delta$ with the minimum number of critical points such that $|| f - f_\delta ||_\infty < \delta$. The presented algorithm solves this problem for functions on discrete surfaces and relies on a connection between discrete Morse theory and persistent homology. A survey of discrete Morse theory and persistent homology will be included.


  • 16:15 - 17:15 Discrete projective-minimal surfaces, Wolfgang K. Schief (UNSW Sydney, Australia)
  • Minimal surfaces in projective differential geometry may be characterised in various different ways. Based on discrete notions of Lie quadrics and their envelopes, we propose a canonical definition of (integrable) discrete projective-minimal surfaces. We discuss various algebraic and geometric properties of these surfaces. In particular, we present a classification of discrete projective-minimal surfaces in terms of the number of envelopes of the associated Lie quadrics. It turns out that this classification is richer than the classical analogue and sheds new light on the latter.


  • 16:15 - 17:15 Effective rational approximation and computations in moduli spaces of curves, Andrei Bogatyrev (Institute of Numerical Mathematics, Moscow)
  • Problems of conditional minimization of the uniform norm of polynomials or rational functions arise in different branches of science and technology. The solutions for those problems are very specific functions as they satisfy the so called equiripple property. This means that the vast majority of critical values (but not all of them!) lie in the set of (say) just two elements. Such polynomials/rational functions are described by the algebro-geometric Chebyshev construction. Effectivization of the theory grounds on the computations with Schottky groups and their moduli.


  • 16:15 - 17:15 Weingarten transformations of hyperbolic nets, Emanuel Huhnen-Venedey (TU Berlin)
  • Classically, Weingarten transformations are the transformations associated with surfaces parametrized along asymptotic lines. Hyperbolic nets provide a discretization of surfaces parametrized along asymptotic lines that extends the discretization of such surfaces by discrete A-nets. Accordingly, the theory of Weingarten transformations of hyperbolic nets can is an extension of the corresponding theory for discrete A-nets. Weingarten pairs of hyperbolic nets are described in terms of 3-dimensional A-nets with crosses attached to elementary quadrilaterals that have so satisfy certain incidence geometric properties. Algebraically, these crosses are described by a scalar function $\rho$ at vertices of the supporting A-net $x$ and the geometric conditions on crosses translate to algebraic conditions on $\rho$ in terms of Moutard invariants of $x$. This yields a relation between functions $\rho$ that describe Weingarten pairs of hyperbolic nets and potentials $\tau$ that parametrize the Moutard coefficients of Lelieuvre representations of the supporting A-nets. One obtains the general solution for such $\rho$ in terms of $\tau$.


  • 16:15 - 17:15 The discretization of surfaces parametrized along asymptotic lines by hyperbolic nets, Emanuel Huhnen-Venedey (TU Berlin)
  • Quadrilateral nets with planar vertex stars, so-called discrete A-nets, are a well-known discretization of smooth A-surfaces. A hyperbolic net is the extension of a supporting discrete A-surface $x$, obtained by fitting hyperboloid surface patches into the (generically skew) quadrilaterals of $x$, such that the resulting surface has a unique tangent plane in each point. On a geometric level, this corresponds to equipping elementary quadrilaterals of $x$ with crisscrossing lines that satisfy certain incidence relations. The algebraic description of this construction makes use of a strictly positive discrete scalar function $\rho$ at vertices of $x$, which has to satisfy an algebraic condition induced by the Moutard invariants of the supporting A-surface.


  • 16:15 - 17:15 Tarski-Bang type theorems for partitions of a convex body, Arseniy Akopyan (Russian Academy of Science, Moscow)
  • Alfred Tarski proved that for any covering of the unit disk by planks (the sets $a\le \lambda(x) ≤ b$) for a linear function $\lambda$ and two reals $a


  • 16:15 - 17:15 How many times can two polygons intersect?, Felix Günther (TU Berlin)
  • We determine the maximum number of intersections between two polygons with $p$ and $q$ vertices in the plane. The cases where $p$ or $q$ is even or the polygons do not have to be simple are quite easy and already known, but when $p$ and $q$ are both odd and both polygons are simple, the problem is more difficult. We prove that the conjectured maximum $(p-1)(q-1)+2$ is correct for all odd $p$ and $q$.


  • 16:15 - 17:15 News from Advances in Architectural Geometry 2012, Thilo Rörig and Stefan Sechelmann (TU Berlin)