SFB-Seminar Berlin

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

  • Last Occurrence: 18.07.2023, 14:15 - 15:00
  • Type: Seminar
  • Location: TU Berlin

Contact: Niklas Affolter (affolter[at]math.tu-berlin.de)
Room: MA 875
Time: Tuesday 14:15-15:15



  • 14:15 - 15:00 (@A060) Inverse problem for electrical networks, Terrence George (University of Michigan)
  • A (circular planar) electrical network is a planar graph in a disk along with a function assigning to each edge a positive real number called its conductance. Associated to such an electrical network is its response matrix which encodes all electrical information that can be found by making measurements on the boundary of the disk. I will discuss how the problem of recovering conductances from the response matrix can be solved using an automorphism of the totally nonnegative Grassmannian called the twist.


  • 14:15 - 15:00 (@H 0106) Thoughts on (integrable) systems on hexagonal lattices, Wolfgang Schief (UNSW Sydney)
  • The notion of "consistency around the cube" of discrete equations defined on the quadrilaterals of Z^n lattices has turned out to be both theoretically useful and applicable in a practical sense. The study of the consistency of discrete equations which do not fall into the above category is still in its infancy. In this informal talk, I will present some ideas on the consistency of discrete equations which are defined on the hexagons of a honeycomb lattice. This will lead to questions about the connection between integrability and consistency.


  • 14:15 - 15:00 (H-Café) Planar networks and simple Lie groups, Anton Izosimov (U Arizona)
  • Elements of the matrix group GL(n) can be encoded by weighted planar networks (directed graphs with numbers written on edges) with n sources and n sinks. Such graphical representation is useful for studying matrix factorizations, totally positive matrices etc. Furthermore, Gekhtman, Shapiro, and Vainshtein showed that such networks also capture Poisson geometry of GL(n) endowed with a standard multiplicative Poisson bracket. In the talk I will present a similar graphical representations of simple Lie groups of type B and C.


  • 15:15 - 16:15 (H- Cafe) The Geometry of Discrete AGAG-Webs in Isotropic 3-Space, Christian Müller (TU Vienna)
  • We investigate webs from the perspective of the geometry of webs on surfaces in three dimensional isotropic space. Our study of AGAG-webs is motivated by architectural applications of gridshell structures where four families of manufactured curves on a curved surface are realizations of asymptotic lines and geodesic lines. We describe all discrete AGAG-webs in isotropic space and propose a method to construct them. Furthermore, we prove that some sub-nets of an AGAG-web are timelike minimal surfaces in Minkowski space and can be embedded into a one-parameter family of discrete isotropic Voss nets. This is a joint work with Helmut Pottman.


  • 14:15 - 15:15 (H- Cafe) Open problems on the dSKP lattice equation and related geometric systems, Paul Melotti (Université Paris-Saclay)
  • The dSKP lattice equation is a rather generic setup in which several dynamical geometric systems can be embedded: Miquel dynamics, P-nets, discrete holomorphic functions, polygon recutting, pentagram maps... It is also related to rich combinatorial structures: oriented dimers, trees and forests. Focusing on the example of P-nets, I will state this correspondence and show how it helps us describe the structure of singularities of the original geometric system. I will then state open problems that may be tackled by the same correspondence. Based on joint works with Niklas Affolter and Béatrice de Tilière.


  • 14:15 - 15:00 Discrete surfaces via binets: consistency, Niklas Affolter (TU Berlin)
  • We consider generalizations of binets defined on the vertices and faces of $\mathbb Z^N$ for $N > 2$. We need two new types of multi-dimensionally consistent nets: plane nets and line compounds. First, we consider pairs of a conjugate net and a plane net. As a special case, we obtain a definition of principal binets on $\mathbb Z^N$. It turns out that the Möbius lift of a principal binet is a polar binet, which leads to a simple proof of the consistency of principal binets. Second, we consider pairs of a conjugate net and a line compound. As a special case, we obtain a definition of Koenigs binets on $\mathbb Z^N$. As on $\mathbb Z^2$, a particular case of Koenigs binets on $\mathbb Z^N$ are pairs of Bobenko-Suris Koenigs nets and Doliwa Koenigs nets. As a byproduct, we show that line compounds provide a setup to obtain a consistent generalization of Doliwa Koenigs nets to $\mathbb Z^N$.


  • 14:15 - 15:15 (H-Café) Optimal discrete harmonic maps, Carl Lutz (TU Berlin)
  • Each embedding of a weighted graph into a surface of constant curvature is associated with a discrete Dirichlet energy. Minimizing the energy over all possible Euclidean/hyperbolic structures and over all embeddings within a fixed homotopy class leads to an (essentially) unique discrete harmonic map into a unique constant curvature surface. In this talk I will give an overview of a recent geometric characterization of these "optimal" harmonic maps in the case of flat tori given by W. Lam and discuss related questions in the case of hyperbolic surfaces.


  • 14:15 - 15:15 (H- Cafe) Construction of constant mean curvature surfaces with the DPW method, Thomas Raujouan (Leibniz  Universität Hannover)
  • The DPW method is a generalized Weierstrass representation of harmonic maps into symmetric spaces. As such, it can be used to construct conformal, constant mean curvature immersions into the Euclidean, hyperbolic, and spherical three-spaces. Studying the global geometry of the resulting surfaces (such as symmetries, periods, embeddedness, area) is hard because the method is local by nature. We will exhibit a procedure that allows for the construction of new examples of constant mean curvature surfaces via the DPW method and the Implicit Function Theorem and show how this procedure can deal with the global properties of the resulting surfaces.


  • 14:15 - 15:15 Grid Peeling and the Affine Curve-Shortening Flow, Gunter Röte (FU Berlin)
  • Grid Peeling is the process of taking the integer grid points inside a convex region and repeatedly removing the convex hull vertices. It has been observed by Eppstein, Har-Peled, and Nivasch, that, as the grid is refined, this process converges to the Affine Curve-Shortening Flow (ACSF), which is defined as a deformation of a smooth curve. As part of the M.Ed. thesis of Moritz Rüber, we have investigated the grid peeling process for special parabolas, and we could observe some striking phenomena. This has lead to a conjecture for the value of the constant that relates the two processes.


  • 14:15 - 15:15 (H- Cafe) Semi-discrete pluri-Lagrangian structures and the Toda hierarchy, Mats Vermeeren (University of Leeds)
  • The first part of this talk will be an introduction to pluri-Lagrangian systems (Lagrangian multiforms) in the context of continuous integrable systems. The role of a Lagrange function in this theory is played by a differential d-form in a higher-dimensional multi-time, which describes the system of interest together with its symmetries. I will highlight some appealing features of this approach and connections to more common notions of integrability. In the second part I will present the newly developed theory of semi-discrete Lagrangian 2-forms. The main example will be the Toda lattice and the hierarchy of ODEs which it is part of. There is more to this hierarchy than meets the eye: by constructing a semi-discrete Lagrangian 2-form, we will find a hierarchy of PDEs hiding within.


  • 15:00 - 16:00 (MA 415) Geometrische Analysis auf singulären Räumen, Prof. Dr. Boris Vertman (Universität Oldenburg)
  • Nach einer Einleitung in die Spektralgeometrie, gehe ich auf meine Forschung ein: speziell die singulären Räume und die mikrolokale Beschreibung des entsprechenden Wärmeleitungsoperators mittels Aufblasung. Anschliessend stelle ich einige Anwendungen meiner Forschung vor, insbesondere zu spektralen Invarianten und diskreter Geometrie, geometrischen Evolutionsgleichungen und mathematischer Physik, sowie singulären Deformationen und Graphen.


  • 12:00 - 13:00 (MA 415) The mathematics of ``hearing the shape of a drum", Prof. Dr. Julie Rowlett (Chalmers University)
  • Have you heard the question, ``Can one hear the shape of a drum?" Do you know the answer? In 1966, M. Kac's article of the same title popularized the inverse isospectral problem for planar domains. Twenty five years later, Gordon, Webb, and Wolpert demonstrated the answer, but many naturally related problems remain open today. For example, in my research, I have recently demonstrated with co‐authors that ``one can hear the corners of a drum'' and ``one can realistically hear the symmetry of polygons.'' I will discuss this field and some of my contributions and the techniques used to obtain them. In the spirit of Kac's article, these techniques combine geometry with mathematical physics.


  • 10:00 - 11:00 (MA 415) Camels and amoebas in symplectic and algebraic geometry, Prof. Dr. Grigory Mikhalkin (Université de Genève)
  • The celebrated "Symplectic Camel" theorem of Gromov describes and detects persistent bottleneck phenomena in Hamiltonian dynamical systems, imposed by the intrinsic geometry of the phase space. The theorem gave birth to the exceptionally powerful technique of pseudoholomorphic curves. Now this technique serves as the main working tool of modern symplectic geometry. In the talk, we'll look at instances when the behavior of pseudoholomorphic curves is governed by the classical complex algebraic geometry. In these cases the symplectic problems get reduced to study of genuine algebraic curves and their logarithmic images known as "amoebas", and further to tropical geometry.


  • 12:00 - 13:00 (MA 415) Lorentzian spectral zeta functions and spacetime dynamics, Prof. Dr. Michal Wrochna (Universität Cergy Paris)
  • The spectral theory of the Laplace–Beltrami operator on Riemannian manifolds is known to be intimately related to geometric invariants. This kind of relationships has inspired many developments in relativistic physics, but a priori it only applies to the case of Euclidean signature. In contrast, the physical setting of Lorentzian manifolds has remained problematic for very fundamental reasons. In this talk I will present results that demonstrate that there is a well‐posed Lorentzian spectral theory nevertheless, and moreover, it is related to Lorentzian geometry in a way that has striking analogies with the Euclidean case. In particular, we show that the scalar curvature can be obtained as the pole of a spectral zeta function. The proof indicates that a key role is played by the dynamics of the null geodesic flow and its asymptotic properties. Furthermore, I will present consequences for the description of spacetime dynamics and outline new perspectives at the interface of general relativity and quantum physics.


  • 11:30 - 12:30 (MA 313) Discrete subgroups of semisimple Lie groups: Anosov groups, higher rank Teichmüller theories and beyond, Prof. Dr. Beatrice Pozzetti (Universität Heidelberg)
  • In recent years, the study of discrete subgroups of semisimple Lie groups has undergone exciting developments inspired from different areas of mathematics: geometric and combinatorial group theory, low dimensional topology, complex and symplectic geometry, mathematical physics, dynamical systems, real algebra. I will discuss the framework in which to encompass these achievements as well as some of my contributions to the area.


  • 12:00 - 13:00 (MA 415) Symplektische Billards, Prof. Dr. Peter Albers (Universität Heidelberg)
  • Symplektische Billiards sind ein sowohl durch elementare geometrische als auch physikalische Überlegungen motiviertes diskretes dynamisches System. Nach einer Einführung im 2‐dimensionalen Fall präsentieren wir Resultate für glatte Tische, wie z.B. Existenz periodischer Orbiten und mögliche Kontinuumslimiten, und polygonale Tische mit überraschender Dynamik. Hier hat ein Visualisierungprojekt zu Theoremen (mit theoretischen Beweisen) geführt. Außerdem werden wir das Wort „symplektisch“ im Titel erklären.


  • 10:00 - 11:00 (MA 415) Die Witten-Deformation und das Cheeger-Müller-Theorem für singulare Räume, Prof. Dr. Ursula Ludwig (Universität Munster)
  • Das Leitmotiv der globalen Analysis ist das Studium analytischer und topologischer Invarianten glatter Mannigfaltigkeiten. Einer der berühmtesten Vergleichssätze solcher Invarianten ist das Cheeger-Müller-Theorem aus den 70er Jahren, welches die analytische und die topologische Torsion einer glatten kompakten Mannigfaltigkeit miteinander vergleicht. In diesem Vortrag werde ich eine Verallgemeinerung des Cheeger-Müller-Theorems für singuläre Räume vorstellen. Eine zentrale Rolle hierbei spielen Wittens Ideen eines analytischen Beweises der Morse-Ungleichungen mithilfe des harmonischen Oszillators der Quantenmechanik.


  • 12:00 - 13:00 (MA 415) Gauge theoretic approach to constructing harmonic maps from Riemann surfaces, Jun.-­‐Prof. Dr. Lynn Heller (Universität Hannover)
  • In my talk (based on joint work with Sebastian Heller and Martin Traizet) I want to introduce a new set of ideas to explicitly constructing harmonic maps from compact Riemann surfaces into 3‐space using an implicit function theorem type argument. When the target space is the round 3‐sphere we obtain complete families of high genus embedded constant mean curvature surfaces deforming Lawson’s minimal surfaces. Moreover, we compute the Taylor expansion of their area at g=\infty which remarkably relates to values of the Riemann zeta function and multiple‐Polylogarithms. Similar ideas for harmonic maps into hyperbolic 3‐space leads to more explicitness of the non‐ abelian Hodge correspondence for the 4‐punctured sphere. This correspondence is a real analytic diffeomorphism between the moduli space of Higgs bundles and the moduli space of flat connections. Each space is hereby equipped with a complex structure that is Kähler with respect to the same Riemannian metric. We identify the rescaled limit moduli space of harmonic maps to be the Eguchi‐Hanson space of (complex) dimension two and we explicitly compute the limit non‐abelian Hodge correspondence as well as its first order derivatives. Also in this case multiple‐Polylogarithms appear in the Taylor expansions and the geometry of the problem leads to identities of certain values of multiple‐Polylogarithms.


  • 10:00 - 11:00 Integrable dynamics and beyond, Prof. Dr. Sonja Hohloch (Universität Antwerpen)
  • After a brief overview of my works in nonautonomous Hamiltonian systems, related PDEs, and symplectic geometry, this talk will focus on the recent progress around symplectic classifications and important examples of so‐called semitoric and hypersemitoric integrable systems and their interactions with known classifications of S^1‐actions, toric systems, and singularities.


  • 14:15 - 15:15 (@TUB) Canonical tessellations of decorated hyperbolic surfaces (Part 2), Carl Lutz (TU Berlin)
  • A decoration of a hyperbolic surface is a choice of circle, horocycle or hypercycle about each cone point, cusp of flare of the surface, respectively. Each decoration induces a canonical tessellation and dual decomposition of the underlying surface. They exhibit a rich geometric structure. In this follow-up talk, I am going to focus on the flip algorithm and the configuration space of decorations.


  • 14:15 - 15:15 (@TUB) Canonical tessellations of decorated hyperbolic surfaces, Carl Lutz (TU Berlin)
  • A decoration of a hyperbolic surface is a choice of circle, horocycle or hypercycle about each cone point, cusp of flare of the surface, respectively. Each decoration induces a canonical tessellation and dual decomposition of the underlying surface. They exhibit a rich geometric structure. Amongst others: a characterisation in terms of the hyperbolic geometric analogues of Delaunay's empty discs and Laguerre's tangent-distance (aka power-distance); connections to convex hulls in Minkowski space (Epstein-Penner convex hull construction); they can be computed using a flip algorithm (Weeks' flip algorithm). In this talk, I am going to introduce the (planar) hyperbolic geometric tools needed to analyse decorated hyperbolic surfaces and sketch the derivation of the properties mentioned above.


  • 14:15 - 15:00 (@zoom) Universality of spin correlations in the Ising model on isoradial graphs, Rémy Mahfouf (ENS Paris)
  • We prove universality of spin correlations in the scaling limit of the planar Ising model on isoradial graphs and Z–invariant weights. Specifically, we show that in the massive scaling limit, (i. e. as the mesh size tends to zero at the same rate as the inverse temperature goes to the critical one) the two-point spin correlations converges to a rotationally invariant function, which is universal among isoradial graphs and independant of the local geometry. We also give a simple proof of the fact that the infinite-volume sub-critical magnetization is independent of the site and the local geometry of the lattice. If there is some time ramaining, we will discuss a geometrical interpretation of the correlation length using the formalism of s-embeddings introduced recently by Chelkak. Based on a joint work (arXiv:2104.12858) with Dmitry Chelkak (ENS), Konstantin Izyurov (Helsinki).


  • 14:15 - 15:00 (@zoom) Geometric structures motivated by architecture, Christian Müller (TU Wien)
  • We will study discrete structures that have been motivated by architecture from a discrete differential geometry point of view. In particular we will have a look at discrete surfaces that are isometric to surfaces of revolution, surfaces with a constant ratio of principal curvatures, tensegrities which are frameworks from cables and bars in static equilibrium and supportstructures with tangential quadrics.


  • 14:15 - 15:00 (@Zoom) Triply Periodic Minimal Surfaces, Hao Chen (Georg-August-Universität Göttingen)
  • Triply Periodic Minimal Surfaces (TPMSs) are fantastic geometric objects that combine the beauty of bubble films and crystals. I identify two golden ages on the topic, one highlights the pioneering works of Schwarz in the 19th century, and the other highlights the interdisciplinary collaborations between mathematicians and physicists in the 20th century. I will talk about recent and ongoing discoveries of new TPMSs, and how the new insights could possibly lead to an eventual classification of all TPMSs of genus 3.


  • 14:15 - 15:15 (@Zoom) Elliptic dimer models and genus 1 Harnack curves, David Cimasoni (Université de Genève)
  • The aim of this talk is to present a joint project with Cédric Boutillier and Béatrice de Tilière about the dimer model on minimal graphs with Fock’s elliptic weights. I will start by introducing the principal characters, namely minimal graphs and elliptic weights, before explaining our main results. The first one is an explicit local expression for a two-parameter family of inverses of the corresponding Kasteleyn operator. When the minimal graph satisfies a natural condition, we then construct a family of dimer Gibbs measures from these inverses and describe the associated phase diagram. Finally, in the periodic case, we establish a correspondence between these models and Harnack curves of genus 1.


  • 16:00 - 17:00 (canceled!) The limit point of the pentagram map and infinitesimal monodromy (canceled), Anton Izosimov (University of Arizona)
  • The pentagram map T takes a planar polygon P to a polygon T(P) whose vertices are intersection points of consecutive shortest diagonals of P. In his 1992 paper on the subject, Richard Schwartz showed that successive images T(P), T(T(P)), ... of a convex polygon P under the pentagram map converge exponentially to a point. One of the open problems in that paper asked if coordinates of that limit point are analytic functions of the coordinates of vertices of P. This question was recently answered affirmatively by Max Glick who showed that the limit point can be computed as an eigenvector of a certain (defined by an explicit formula and rather mysterious) operator associated with the polygon. In the talk, I will try to somewhat demystify Glick's construction. Namely, I will show that his operator can be interpreted as what we called the "infinitesimal monodromy" of the polygon. The talk is based on joint work with Quinton Aboud.


  • 14:15 - 15:15 (@Zoom) Monge-Ampère equations and inverse problems in optics, Boris Thibert (Université Grenoble Alpes)
  • Non-imaging optics is a field of optics which is interested in designing optical components, such as mirrors or lenses, that transfer a given source light to a prescribed target. The goal is not to simulate the trajectory of the light through an optical component, which would be the direct problem, but instead to build the shape of a mirror or a lens that transfers a source light to a given target light. This inverse problem amounts in different settings to solving Monge-Ampère type equations. In this talk, I will show how these equations are connected to optimal transport and can be solved using a geometric discretization called semi-discrete. I will also present the design of different kinds of mirrors or lenses that allow to transfer any point or parallel source light to any target. This work is in collaboration with Quentin Mérigot and involves Jocelyn Meyron.


  • 14:15 - 15:00 (@Zoom) Variational integrators for stochastic dissipative Hamiltonian systems, Tomasz Tyranowski (IPP Garching)
  • Variational integrators are derived for structure-preserving simulation of stochastic forced Hamiltonian systems. The derivation is based on a stochastic discrete Hamiltonian which approximates a type-II stochastic generating function for the stochastic flow of the Hamiltonian system. The generating function is obtained by introducing an appropriate stochastic action functional and considering a stochastic generalization of the deterministic Lagrange-d'Alembert principle. Our approach presents a general methodology to derive new structure-preserving numerical schemes. The resulting integrators satisfy a discrete version of the stochastic Lagrange-d'Alembert principle, and in the presence of symmetries, they also satisfy a discrete counterpart of Noether's theorem. Furthermore, mean-square and weak Lagrange-d'Alembert Runge-Kutta methods are proposed and tested numerically to demonstrate their superior long-time numerical stability and energy behavior compared to non-geometric methods. The Vlasov-Fokker-Planck equation is considered as one of the numerical test cases, and a new geometric approach to collisional kinetic plasmas is presented.


  • 14:15 - 15:00 (@Zoom) The Maxwell Cremona correspondence and weighted Delaunay triangulations, Nina Smeenk (TU Berlin)
  • In the Euclidean, spherical and hyperbolic geometry a framework is realization of a graph i.e. an embedding of the vertices into the Euclidean, spherical or hypoerbolical plane. The Maxwell Cremona correspondence gives an equivalence of the following properties of such an embedding: (i) The existence of a self stress (ii) The existence of a dual framework with orthogonal dual edges (iii) The existence of a lift to a polytope in 3-space A self stress is an assignment of real numbers to edges that fulfill a static equilibrium condition. The aim of the talk is to show, that the three equivalent conditions can be extended by a fourth condition: (iv) The framework is a weighted Delaunay triangulation For this we will specialize the original Maxwell Cremona correspondence to frameworks where all faces of the embedding of the graph are convex. We will investigate the Euclidean case to find important geometric conditions and properties for the construction. Subsequently we investigate corresponding constructions and the Maxwell Cremona correspondence in the hyperbolic plane. The talk is based on the results of my Bachelor thesis from 2019 wich itself was based on a paper by I.Izmestiev (arXiv:1707.02172).


  • 14:15 - 15:00 (@Zoom) Semitoric systems and the computation of symplectic invariants, Jaume Alonso (Uni Antwerpen)
  • Semitoric systems are a special class of completely integrable systems defi ned on four-dimensional symplectic manifolds. One of the reasons that make these systems interesting is the fact that they have been classi fied by Pelayo and Vu Ngoc in terms of five symplectic invariants. In the last years, many e fforts have been made in order to extend this classification to broader settings, to generate more examples and to compute their invariants. In this talk we will discuss some of the most important properties of semitoric systems and introduce some families of systems with one and more focus-focus singularities. We will also show how the symplectic invariants of these systems change as we move the parameters of the families and how they can be computed using mathematical software. This is a joint work with H. Dullin, S. Hohloch and J. Palmer.


  • 14:15 - 15:00 (@TUB) A discrete version of Liouville's theorem on conformal maps, Boris Springborn (TU Berlin)
  • Liouville's theorem says that in dimension greater than two, all conformal maps are Möbius transformations. We prove an analogous statement about simplicial complexes, where two simplicial complexes are considered discretely conformally equivalent if they are combinatorially equivalent and the lengths of corresponding edges are related by scale factors associated with the vertices.


  • 14:15 - 15:00 (@TUB) Cube flips in statistical mechanics and in planar geometry, Paul Melotti (Université de Fribourg)
  • An important property in the study of exactly solvable statistical models is the existence of a local transformation of the underlying graph and parameters, such that long range observables remain unchanged. This can often be seen as a "cube flip" (or star-triangle, or Yang-Baxter equation). On the other hand, there has been recently an intense line of research concerning canonical embeddings of such models, giving rise to "s-embeddings" related to the Ising model, "circle patterns" for the dimer model, and others. In this correspondence, local transformations of the model should be conjugated to theorems of planar (or projective) geometry on these embeddings. I will present a few cases of this correspondence, and of the following reciprocal question: can one find theorems of planar geometry that should be related to the local "flip" of such a model? This leads to the introduction of new classes of homogeneous quadrilaterals. This is based on a joint work in progress with Sanjay Ramassamy and Paul Thévenin.


  • 10:30 - 11:30 (@TUB) Circle packings and Delaunay circle patterns on surfaces with complex projective structures, Jean-Marc Schlenker (University of Luxembourg)
  • We consider circle packings and, more generally, Delaunay circle patterns - arrangements of circles arising from a Delaunay decomposition of a finite set of points - on surfaces equipped with a complex projective structure. Motivated by a conjecture of Kojima, Mizushima and Tan, we prove that the forgetful map sending a complex projective structure admitting a circle packing with given nerve (resp. a Delaunay circle pattern with given nerve and intersection angles) to the underlying complex structure is proper. The talk will contain an introduction to complex projective structures on surfaces. Joint with Andrew Yarmola (Princeton University).


  • 14:15 - 15:15 (@TUB) Quadratic differentials and circle patterns on complex projective tori, Wai Yeung Lam (Université du Luxembourg)
  • In the smooth theory, holomorphic quadratic differentials parametrize the space of complex projective structures on a Riemann surface via the Schwarzian derivate. It motivates the study of circle patterns on surfaces with complex projective structures, where circle patterns with prescribed intersection angles play a role of discrete conformal structures. Given a triangulation of a closed surface, we consider a cross ratio system that assigns a complex number to every edge satisfying certain polynomial equations per vertex. Every cross ratio system induces a complex projective structure together with a circle pattern. In particular, there is an associated conformal structure. We show that for any triangulated torus, the projection from the space of cross ratio systems with prescribed Delaunay angles to the Teichmueller space of the closed torus is a covering map with at most one branch point. Our approach is based on a notion of discrete holomorphic quadratic differentials.


  • 14:15 - 15:15 (@TUB) Towards Thurston theorem for dynamics of iterated trigonometric functions, Sergey Shemyakov (Jacobs Universität, Bremen)
  • Thurston topological characterization problem is one of important problems in the world of holomorphic dynamics. It talks about connecting topology and geometry in a dynamical context and has powerful implications and extensions around the field. Take some family of functions, e.g. rational functions, trigonometric functions a cos(z)+b sin(z) (which I study in my work), polynomials, etc. Topological characterization problem wants to describe all functions in the family with the help of topological (in some sense combinatorial) data. One half of the task is to describe a holomorphic function with a simplified topological model (which is done with the help of Hubbard trees in the scope of another research). Another half is to determine when a topological model gives rise to an entire function, and that is exactly the Thurston characterization problem. In my talk I will explain the Thurston characterization problem generally and in the particular case of trigonometric functions which I study. I will try to give some background for people working away from holomorphic dynamics, make connections to other related geometrical problems. I will introduce some of the important ingredients in the proof, Teichmueller space being one example. I will try to convey the sense of value of my results in a way appealing to a broader audience, as well as to outlook further directions of research.


  • 13:15 - 14:15 (@TUB) Variational principles for discrete $\vartheta$-conformal maps as interpolation between circle patterns and conformal equivalence, Ulrike Bücking (FU Berlin)
  • For pairs of immersed triangulations in the plane - considered as preimage and image of a discrete map - conformal equivalence and circle patterns are known notions of discrete conformality. Both have related variational principles. We present an interpolation between these two notions and focus especially on the existence of a variational principle for the interpolated notions of conformality.


  • 14:15 - 14:45 (@TUB) Miquel dynamics on circle patterns and the dimer model, Sanjay Ramassamy (École normale supérieure, Paris)
  • Circle patterns are a way to draw a graph in the plane such that every face is cyclic. Miquel dynamics is a discrete-time dynamical system on circle patterns defined by using the classical Miquel six-circles theorem. Its integrability is obtained by establishing a novel connection between circle patterns and the dimer model from statistical mechanics. Joint work with Richard Kenyon (Yale University), Wai Yeung Lam (University of Luxembourg) and Marianna Russkikh (Massachussetts Institute of Technology).


  • 14:15 - 15:15 (@TUB) On an integrable multi-dimensionally consistent 2n+2n-dimensional heavenly-type equation, Wolfgang  Schief (The University of New South Wales)
  • Based on the commutativity of scalar vector fields, an algebraic scheme is presented which leads to a privileged multi-dimensionally consistent 2n+2n-dimensional integrable partial differential equation with the associated eigenfunction constituting an infinitesimal symmetry. The “universal” character of this novel equation of vanishing Pfaffian type is demonstrated by retrieving and generalising to higher dimensions a great variety of well-known integrable equations such as the dispersionless KP and Hirota equations and various avatars of the heavenly equation governing self-dual Einstein spaces.


  • 13:15 - 14:15 Geometry and dynamical degree for some birational maps, Yuri Suris (Technische Universität Berlin)
  • I will describe a class of birational planar maps of a beautiful geometric origin (Pascal configurations and pencils of elliptic curves). Then, I will give a brief introduction to the notion of dynamical degree (or algebraic entropy) and show how to find this quantity for the class of maps discussed in the first part of the talk.


  • 14:15 - 15:15 (@TUB) How to find a pluri-Lagrangian structure for an integrable hierarchy, Mats Vermeeren (Technische Universität Berlin)
  • Compared to the importance of Hamiltonian formulations in integrable systems, there is a striking absence of Lagrangians in the literature on integrability. The theory of pluri-Lagrangian systems has been developed in an attempt to correct this. It is applicable both to integrable difference equations and to hierarchies of PDEs. There is strong evidence that the pluri-Lagrangian property characterizes integrability, but it remains a challenge to prove relations to more common notions of integrability, or to construct a pluri-Lagrangian structure for a given integrable system. This talk will give a gentle introduction to pluri-Lagrangian systems and discuss some recent progress on the construction of continuous pluri-Lagrangian structures. I will present two construction techniques: one using variational symmetries (joint work with Matteo Petrera) and one using continuum limits of discrete pluri-Lagrangian systems.


  • 14:15 - 15:15 (@TUB) Commuting Hamiltonian Flows of Space Curves, Albert Chern (TU Berlin)
  • Starting from the vortex filament flow introduced in 1906 by Da Rios, there is a hierarchy of commuting geometric flows on space curves. The traditional approach relates those flows to the nonlinear Schrödinger hierarchy satisfied by the complex curvature function of the space curve. Rather than working with this infinitesimal invariant, we describe the flows directly as vector fields on manifolds of space curves, which carries a canonical symplectic form introduced by Marsden and Weinstein. The flows are precisely the symplectic gradients of a natural hierarchy of invariants, beginning with length, total torsion, and elastic energy. There are a number of advantages of our geometric approach. For instance, the spectral curve is geometrically realized as the motion of the monodromy axis when varying total torsion. This insight provides a new explicit formula for the hierarchy of Hamiltonians. We also complete the hierarchy of Hamiltonians by adding area and volume. These allow for the characterization of elastic curves as solutions to an isoperimetric problem: elastica are the critical points of length while fixing area and volume.


  • 14:15 - 15:15 (@TUB) Symmetry through Geometry , Nalini Joshi (The University of Sydney)
  • Discrete integrable equations can be considered in two, three or N-dimensions, as equations fitted together in a self-consistent way on a square, a cube or an N-dimensional cube. We show to find their symmetry reductions (and other properties) through a geometric perspective.


  • 14:15 - 15:15 Leave, flowers, and sea-slugs: From discrete geometry to discrete mechanics, Shankar Venkataramani (The University of Arizona)
  • I will talk about some geometric questions that arise in the study of soft/thin objects with negative curvature. After reviewing basic ideas from the mechanics of Non-Euclidean sheets, I will discuss the role of non-smooth isometries in explaining the observed morphologies of thin Non-Euclidean sheets. I will highlight the role of DDG methods in constructing the relevant non-smooth solutions and discuss potential extensions of DDG ideas in order to make the jump from solving for isometries to solving for the mechanical equilibrium, i.e force, and moment balance.
    This is joint work with Toby Shearman and Ken Yamamoto.


  • 14:15 - 15:15 (broadcast to TU Munich) Quad Meshing and Vector Field Design - A Computer Graphics Perspective, Olga Diamanti (TU Berlin)
  • In the domain of computer graphics, triangle meshes are typically the representation of choice for geometric data - their simplicial nature makes storage, rendering and processing convenient. Even so, quadrilateral meshes are better tailored for other domains - the quad structure allows for alignment to principal curvature directions, which are heavily used to guide computational surface modeling and animation, or for representing directions of stress or strain, commonly considered in architectural design and simulation. As such, automatic and semi-automatic quad mesh generation from triangle meshes remains an active research area in graphics. This talk will outline some of the popular methods in geometry processing for this task, with a particular focus in works involving vector fields that express the modeler's intent of how the quads should be laid out. We will comparatively look at some of the objectives and quality criteria commonly used, among which reliability, robustness, efficiency and expressiveness, and state some of the remaining open problems.


  • 14:15 - 15:15 Tractrices and Hyperbolic Geometry, Niklas Affolter (TU Berlin)
  • I will talk about joint work in progress with Jan Techter. We will explain a relation between tractrices and geodesics in a developable surface in hyperbolic three space. The relation holds in both the discrete and the continuous setting.


  • 14:15 - 15:15 (Monday!) Variational models for microstructures in shape-memory alloys, Barbara Zwicknagl (TU Berlin)
  • Shape-memory alloys are special materials that undergo a martensitic phase transition, that is, a diffusionless first order solid-solid phase transformation. Pattern formation in shape-memory alloys is often studied in the framework of the calculus of variations. The formation of microstructures is typically explained as result of a competition between a bulk elastic energy and a higher order surface energy. I shall discuss some recent analytical results on such variational problems, in particular regarding needle-like microstructures and stress-free inclusions.


  • 13:00 - 13:45 (Exceptional time and room MA 313!!!) Structure and Randomness in Signal Processing, Felix Krahmer (TU München)
  • In this talk, we will discuss various examples how mathematically inspired measurement design in signal processing applications yields improved performance and allows for provable error guarantees, crucial for critical applications. Firstly, motivated by computational imaging applications, we discuss the problem of sparse recovery from subsampled random convolutions. We advance techniques related to the theory of empirical processes to establish near-optimal recovery guarantees.
    Secondly, we introduce a mathematical theory that complements recent analog-to-digital converter designs to allow for the reconstruction of bandlimited signals even when the dynamic range is limited. Our results are driven by the need for cheap quantizers to design low-cost sensors and cameras.
    Lastly, motivated by applications in wireless communication, we consider the problem of simultaneous blind demixing and deconvolution for randomly encoded signals, as it arises in the context of sporadic non-coherent multi-user communication. Our results establish for the first time near-optimal parameter dependence.
    These are joint works with the speaker’s PhD student Dominik Stöger, as well as with Shahar Mendelson (Technion), Holger Rauhut (RWTH Aachen), Peter Jung (TU Berlin/HHI), Ayush Bhandari (MIT), and Ramesh Raskar (MIT).


  • 14:15 - 15:15 Combescure transformations – a helpful tool to study Guichard nets, Gudrun Szewieczek (TU Wien)
  • Two surfaces are Combescure transformations of each other if they have parallel tangent planes along corresponding curvature lines. The existence of particular Combescure transformations can be used to characterize various classes of integrable surfaces, e. g. the Christoffel transformation for isothermic surfaces or, more general, O-surfaces.
    In this talk we investigate this concept for triply orthogonal systems and discuss the subclass of Guichard nets, which are special coordinate systems of 3-dimensional conformally flat hypersurfaces. On the way, we introduce G-surfaces, a rather unknown class of surfaces originally defined by Calapso in 1921, which arise as coordinate surfaces of Guichard nets.


  • 11:00 - 12:00 (Friday and exceptional time!) Cone Angles, Gram’s relation, and zonotopes, Raman Sanyal (Goethe-Universität Frankfurt)
  • The Euler-Poincare formula is a cornerstone of the combinatorial theory of polytopes. It states that the number of faces of various dimensions of a convex polytope satisfy a linear relation and it is the only linear relation (up to scaling). Gram’s relation generalizes the fact that the sum of (interior) angles at the vertices of a convex $n$-gon is $(n-2)\pi$. In dimensions $3$ and up, it is necessary to consider angles at all faces. This gives rise to the interior angle vector of a convex polytope and Gram’s relation is the unique linear relation (up to scaling) among its entries. In this talk, we will consider generalizations of “angles” in the form of cone valuations. It turns out that the associated generalized angle vectors still satisfy Gram’s relation and that it is the only linear relation, independent of the notion of “angle”! To prove such a result, we rely on a very powerful connection to the combinatorics of zonotopes. The interior angle vector of a zonotope is independent of the chosen cone valuation and depends only on the associated lattice of flats. If time permits, we discuss flag-angles as a semi-discrete generalization of flag-vectors and their linear relations. This is joint work with Spencer Backman and Sebastian Manecke.


  • 14:15 - 15:15 Polytopal surfaces in Fuchsian manifolds, Roman Prosanov (Université de Fribourg)
  • There are several Alexandrov-type theorems about isometric polytopal realization of metrics on surfaces with singularities. Two approaches to problems of this type are known. The first one is an indirect method, which was proposed by Alexandrov himself. Another option is to transform the initial problem to a variational one and then resolve it. This approach is more constructive and was realized by A. Bobenko, I. Izmestiev and B. Springborn.
    We consider hyperbolic metrics with cusps on surfaces of genus at least 2. Such a metric can be uniquely realized as the induced metric on the boundary components of an ideal symmetric Fuchsian polytope.
    In our talk we will give a variational proof of this result and will discuss possible perspectives of this method in applications to other problems.


  • 14:15 - 15:15 Constraint-based Point Set Denoising, Marting Skrodzki (FU Berlin)
  • In many applications, point set surfaces are acquired by 3D scanners. During this process, noise and outliers are inevitable. For a high fidelity surface reconstruction from a noisy point set, a feature preserving point set denoising operation has to be performed to remove noise and outliers from the input point set. We introduce an anisotropic point set denoising algorithm in the normal voting tensor framework. The method consists of three different stages that are iteratively applied: in the first stage, noisy vertex normals are processed using a vertex-based normal voting tensor and binary eigenvalues optimization. In the second stage, feature points are categorized into corners, edges, and surface patches using a weighted covariance matrix, which is computed based on the processed vertex normals. In the last stage, vertex positions are updated using restricted quadratic error metrics. Finally, we show our method to be robust and comparable to state-of-the-art methods in experiments


  • 12:15 - 13:15 (Exceptional date and time!) Conformal Tilings of the Plane: Foundations, Theory, and Practice, Ken Stephenson (University of Tennessee)
  • Certainly one of the most famous tilings of the plane is the Penrose tiling. The common "kite/dart" form is an aperiodic tiling having just two euclidean tile shapes - it is visually quite fascinating. This is also a "subdivision tiling", one associated with a rule for subdividing tiles into subtiles of the same shapes. What happens if one disregards the geometry entirely and considers the "combinatorics" alone - the abstract pattern of tiles, who is next to whom? We start with such combinatorics, impose canonical conformal structures, and find a new family of geometric tilings, the "conformal" tilings. I will introduce these, mention some basic theory, and show how to visualize themq in practice via circle packing. Using the resulting experimental capabilities, we will discover that the geometry of traditional tilings, like the Penrose, emerge spontaneously from their combinatorics. We will, indeed, find a whole new playground for those captivated by these intricate objects.


  • 14:15 - 15:15 Discrete Gaussian distributions via theta functions, Daniele Agostini (MPI MIS Leipzig)
  • Maximum entropy probability distributions are important for information theory and relate directly to exponential families in statistics. Having the property of maximizing entropy can be used to define a discrete analogue of the classical continuous Gaussian distribution. We present a parametrization of such a density using the Riemann Theta function, use it to derive fundamental properties and exhibit strong connections to the study of abelian varieties in algebraic geometry. This is joint work with Carlos Améndola (TU Munich).


  • 14:15 - 15:15 Differentialgleichungen und unendlich-dimensionale Liegruppen, Helge Glöckner (Universität Paderborn)


  • 14:15 - 15:15 What is a tropical period matrix?, Mario Kummer (TU Berlin)
  • Tropical geometry is a piece-wise linear version of algebraic geometry that can be used to study degenerations of classical objects. We explain tropical curves, 1-forms and Jacobians for an audience new to tropical geometry but familiar with the theory of Riemann surfaces.


  • 14:15 - 15:15 (Live broadcast from TU Berlin to TU München) Euler-Arnold theory for SPDEs?, Alexander Schmeding (TU Berlin)
  • In 1966 V. Arnold demonstrated that Euler's equations for an ideal fluid can be understood as the geodesic equation on the group of volume preserving diffeomorphisms with respect to a suitable Riemannian metric. Subsequently this bridge between PDEs on finite dimensional manifolds and ODEs on infinite-dimensional manifolds has been used to study the so called Euler-Arnold equations (e.g. Ebin/Marsden 1970).
    In this talk we will give a short tour to this theory and its key ideas. Our aim is to discuss a possible extensions of these techniques to certain SPDEs which have recently been considered in Fluid dynamics (Crisan, Flandoli, Holm 2017).
    This is joint with K. Modin (Chalmers, Gothenburg) and M. Maurelli (TU Berlin).


  • 16:00 - 16:45 (Monday and exceptional time!) Higher solutions of Hitchin's self-duality equations, Sebastian Heller (Universität Tübingen)
  • Solutions of Hitchin's self-duality equations correspond to special real sections of the Deligne-Hitchin moduli space - twistor lines. A question posed by Simpson in 1995 asks whether all real sections give rise to global solutions of the self-duality equations. An affirmative answer would allow for a complex analytic procedure to obtain solutions of the self- duality equations. The purpose of my talk is to explain the construction of counter examples given by certain (branched) Willmore surfaces in 3-space (with monodromy) via the generalized Whitham flow. Though these higher solutions do not give rise to global solutions of the self- duality equations on the whole Riemann surface M, they are solutions on an open dense subset of it. This suggest a deeper connection between Willmore surfaces, i.e., rank 4 harmonic maps theory, with the rank 2 self-duality theory. The talk is based on joint work with L. Heller.


  • 14:15 - 15:15 Quanta of Discrete Spacetime, Alex Goeßmann (TU Berlin)
  • Space and time - Our daily experience leads to think about those as a continuous manifold. But does this picture remain true, if we look at small length scales inaccessible to our senses? Attempts to unify the fundamental physical descriptions of the world propose that space and time consist of discrete quanta. In the talk, I will present the associated mathematical framework of tensor networks and group fields. Quantum Field Theories of these quanta will be defined and the resulting Feynman diagrams interpreted to carry the geometry of spacetime.


  • 14:15 - 15:15 Orthogonal polynomials and the Painlevé equations, Galina Filipuk (University of Warsaw)
  • In this talk I shall review some of my recent results on the connection of recurrence relation coefficients of semi-classical-orthogonal polynomials to the solutions of discrete and differential Painlevé equations. I shall also briefly discuss multiple orthogonal polynomials.


  • 14:15 - 15:15 Graphene and Discrete Differential Geometry, Edmund O. Harriss (University of Arkansas)
  • Two dimensional carbon molecules go back to Harry Kroto’s discovery of Fullerenes, such as the classic C60, where the atoms lie at the vertices of a truncated icosahedron. Hidden in the same soot as the Fullerenes, however lay other strange molecules, in particular single layers of Graphite, now called Graphene, discovered by the simple act of apply sticky tape to a pice of graphite. Since then other 2d crystals have been discovered such as phosphorene, an allotrope of phosphorus and various compounds. The initial models for these molecules first embedded the atoms into a continuum surface and then used the tools of crystallography to work on that surface. This works well for large scale questions but cannot look at an atomic scale. With Salvador Barraza Lopez and his group at the University of Arkansas we have worked instead with modelling using a mesh, with atoms as vertices and bonds as edges. This model has proved effective and we are developing the tools and connections further in collaboration with Dr. Van Horn Morris, also at University of Arkansas.


  • 14:15 - 15:15 Linearization of planar 3-webs, Sergey I. Agafonov (Federal University of Paraíba (Brazil))
  • A planar d-web is a superposition of d foliations in the plane. If the leaves of all foliations are rectilinear then the web is called linear. A linearization of a planer d-web is a local diffeomorphism mapping the web to a linear one.
    In this talk we present a recent progress in the old and difficult problem of linearization of planar 3-webs. In particular, we develop a projectively invariant description of planar linear 3-webs and discuss Gronwall's conjecture.


  • 14:15 - 15:15 An introduction to the Riemannian Geometry of Orbifolds, Christodoulos Savva (Imperial College London)
  • Manifolds are locally modelled on R^n. In the same sense, orbifolds are locally modelled on R^n modulo a finite group action. By introducing a Riemannian metric on orbifolds we have the generalization of various results like Bonnet-Myers and Synge's theorems. We will discuss some fundamental concepts, examples of low-dimensional orbifolds and some of these generalized theorems.


  • 14:15 - 15:15 Cerf theory for graphs,  Max Krause (FU Berlin)
  • Following the 1998 paper by Karen Vogtmann and Allen Hatcher, we study a deformation theory for families of pointed marked graphs with fixed fundamental group, inspired by established Cerf theory for smooth functions on manifolds. We will explain how these notions translate to the setting of pointed marked graphs and give an outlook on a basic application – a stability proof for the rational homology of the automorphism group of the free group with n generators.


  • 14:15 - 15:15 Superintegrable systems, Nicolai Reshetikhin (University of California,  Berkeley, USA)
  • Hamiltonian integrability is a well known property in Hamiltonian mechanics. For integrable systems the trajectories are contained in level surfaces of Poisson commuting Hamiltonians, i.e. in Lagrangian subspaces of the symplectic manifold (phase space of the system). In superintegrable systems trajectories are contained in isotropic submanifolds. In the extreme case of maximal superintegrability these isotropic sub manifolds are 1-dimensional and in therefore all compact trajectories are periodic.


  • 14:15 - 15:15 Bethe ansatz for Gaudin models, Evgeny Mukhin (Indiana University-Purdue University Indianapolis)
  • Gaudin Hamiltonians are commuting linear operators acting in tensor products of representations of simple Lie algebras. Bethe ansatz is a "physics motivated" method which allows to find eigenvectors and eigenvalues of the Hamiltonians. In this talk I will explain the basics of the gl(n) Bethe ansatz and multiple connections to real and complex algebraic geometry, Fuchsian differential operators, orthogonal polynomials, Selberg type integrals, etc.
  • 15:30 - 16:30 Double Polyhedra – Weaving, knots and helices, Rinus Roelofs (Hengelo, Netherlands)
  • Art can help to like and to understand mathematics.
    In 1509 the mathematician Luca Pacioli published his book “La Divina Proportione” in which he introduced the idea of adding a second layer to a polyhedron. With “Elevation” he defined an operation with which the extra skin could be created. About a century later, Kepler showed that we can create polyhedra which have a second skin themselves. With Kepler’s “Stellation” we enter the field of multilayer single-surface structures.
    This part of mathematics has attracted many artists, sometimes with the goal of making mathematics visible. Leonarda da Vinci made all the illustrations of Pacioli’s book. But also other artists like Albrecht Dürer, Maurits Cornelis Escher and Kenneth Snelson were inspired by these ideas. Based on their works, I started my own investigations, resulting in new mathematical ideas and new sculptures, which I would like to present.


  • 14:15 - 15:15 The Success Story of Wavelets: Honoring Yves Meyer's Abel Prize, Gitta Kutyniok and Philipp Petersen (TU Berlin)
  • This year's Abel price was awarded to Yves Meyer "for his pivotal role in the development of the mathematical theory of wavelets." To honor Yves Meyer, we will give an introduction to the central parts of his outstanding work. We will present Meyer's motivation, his key insights, and his main contributions on this topic. Particularly, we start with the beginning of wavelet theory in applications of geophysics, present the construction of a smooth orthogonal wavelet basis, and demonstrate how this naturally leads to the concept of multiresolution analysis​. Finally, we discuss some applications demonstrating the impact of these ideas.


  • 14:15 - 15:15 Computational approach to compact Riemann surfaces, Christian Klein (Université de Bourgogne)
  • A purely numerical approach to compact Riemann surfaces starting from plane algebraic curves is presented. The critical points of the algebraic curve are computed via a two-dimensional Newton iteration. The starting values for this iteration are obtained from the resultants with respect to both coordinates of the algebraic curve and a suitable pairing of their zeros. A set of generators of the fundamental group for the complement of these critical points in the complex plane is constructed from circles around these points and connecting lines obtained from a minimal spanning tree. The monodromies are computed by solving the defining equation of the algebraic curve on collocation points along these contours and by analytically continuing the roots. The collocation points are chosen to correspond to Chebychev collocation points for an ensuing Clenshaw–Curtis integration of the holomorphic differentials which gives the periods of the Riemann surface with spectral accuracy. At the singularities of the algebraic curve, Puiseux expansions computed by contour integration on the circles around the singularities are used to identify the holomorphic differentials. The Abel map is also computed with the Clenshaw–Curtis algorithm and contour integrals. Multi-dimensional theta functions are computed with an approximate transformation to the Siegel fundamental domain. As an application of the code, solutions to the Kadomtsev–Petviashvili equation are computed on non-hyperelliptic Riemann surfaces.


  • 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)