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

Next Occurrence:
18.10.2019, 10:30  11:30

JeanMarc Schlenker: Circle packings and Delaunay circle patterns on surfaces with complex projective structures

Type: Seminar
 Location: TU Berlin
Contact: Niklas Affolter (affolter[at]math.tuberlin.de)
Room: MA 875
Time: Tuesday 14:1515:15
18.10.2019

10:30  11:30
(@TUB)
Circle packings and Delaunay circle patterns on surfaces with complex projective structures,
JeanMarc 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).
08.10.2019

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.
17.09.2019

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.
13.09.2019

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.
10.09.2019

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 discretetime dynamical system on circle patterns defined by using the classical Miquel sixcircles 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).
09.07.2019

14:15  15:15
(@TUB)
On an integrable multidimensionally consistent 2n+2ndimensional heavenlytype 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
multidimensionally consistent 2n+2ndimensional 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
wellknown integrable equations such as the dispersionless KP and Hirota
equations and various avatars of the heavenly equation governing
selfdual Einstein spaces.
28.06.2019

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.
21.05.2019

14:15  15:15
(@TUB)
How to find a pluriLagrangian 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 pluriLagrangian 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 pluriLagrangian property characterizes integrability, but it remains a challenge to prove relations to more common notions of integrability, or to construct a pluriLagrangian structure for a given integrable system.
This talk will give a gentle introduction to pluriLagrangian systems and discuss some recent progress on the construction of continuous pluriLagrangian structures. I will present two construction techniques: one using variational symmetries (joint work with Matteo Petrera) and one using continuum limits of discrete pluriLagrangian systems.
23.04.2019

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.
05.03.2019

14:15  15:15
(@TUB)
Symmetry through Geometry ,
Nalini Joshi (The University of Sydney)
+
 Discrete integrable equations can be considered in two, three or Ndimensions, as equations fitted together in a selfconsistent way on a square, a cube or an Ndimensional cube. We show to find their symmetry reductions (and other properties) through a geometric perspective.
26.02.2019

14:15  15:15
Leave, flowers, and seaslugs: 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 NonEuclidean sheets, I
will discuss the role of nonsmooth isometries in explaining the
observed morphologies of thin NonEuclidean sheets. I will highlight
the role of DDG methods in constructing the relevant nonsmooth
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.
19.02.2019

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 semiautomatic 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.
18.12.2018

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.
10.12.2018

14:15  15:15
(Monday!)
Variational models for microstructures in shapememory alloys,
Barbara Zwicknagl (TU Berlin)
+
 Shapememory alloys are special materials that undergo a martensitic phase
transition,
that is, a diffusionless first order solidsolid phase transformation. Pattern
formation in
shapememory 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
needlelike microstructures and stressfree inclusions.
27.11.2018

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 nearoptimal recovery guarantees.
Secondly, we introduce a mathematical theory that complements recent analogtodigital 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 lowcost 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 noncoherent multiuser communication. Our results establish for the first time nearoptimal 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).
25.09.2018

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, Osurfaces.
In this talk we investigate this concept for triply orthogonal systems and discuss the
subclass of Guichard nets, which are special coordinate systems of 3dimensional conformally
flat hypersurfaces. On the way, we introduce Gsurfaces, a rather unknown class
of surfaces originally defined by Calapso in 1921, which arise as coordinate surfaces of
Guichard nets.
21.09.2018

11:00  12:00
(Friday and exceptional time!)
Cone Angles, Gram’s relation, and zonotopes,
Raman Sanyal (GoetheUniversität Frankfurt)
+
 The EulerPoincare 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 $(n2)\pi$. 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 flagangles as a
semidiscrete generalization of flagvectors and their linear relations. This is
joint work with Spencer Backman and Sebastian Manecke.
17.07.2018

14:15  15:15
Polytopal surfaces in Fuchsian manifolds,
Roman Prosanov (Université de Fribourg)
+
 There are several Alexandrovtype 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.
26.06.2018

14:15  15:15
Constraintbased 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 vertexbased 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 stateoftheart methods in experiments
15.06.2018

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.
22.05.2018

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).
20.02.2018

14:15  15:15
Differentialgleichungen und unendlichdimensionale Liegruppen,
Helge Glöckner (Universität Paderborn)
30.01.2018

14:15  15:15
What is a tropical period matrix?,
Mario Kummer (TU Berlin)
+
 Tropical geometry is a piecewise linear version of algebraic geometry that can be used to study degenerations of classical objects. We explain tropical curves, 1forms and Jacobians for an audience new to tropical geometry but familiar with the theory of Riemann surfaces.
23.01.2018

14:15  15:15
(Live broadcast from TU Berlin to TU München)
EulerArnold 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 infinitedimensional manifolds has been used to study the so called EulerArnold 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).
22.01.2018

16:00  16:45
(Monday and exceptional time!)
Higher solutions of Hitchin's selfduality equations,
Sebastian Heller (Universität Tübingen)
+
 Solutions of Hitchin's selfduality equations correspond to special real sections
of the DeligneHitchin moduli space  twistor lines. A question posed by Simpson in 1995
asks whether all real sections give rise to global solutions of the selfduality 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 3space (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 selfduality theory. The talk is based on joint work with L. Heller.
16.01.2018

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.
19.12.2017

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 semiclassicalorthogonal polynomials to the solutions of discrete and differential Painlevé equations. I shall also briefly discuss multiple orthogonal polynomials.
05.12.2017

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.
28.11.2017

14:15  15:15
Linearization of planar 3webs,
Sergey I. Agafonov (Federal University of Paraíba (Brazil))
+
 A planar dweb 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 dweb 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 3webs. In particular, we develop a projectively invariant description of planar linear 3webs and discuss Gronwall's conjecture.
29.08.2017

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 BonnetMyers and Synge's theorems. We will discuss some fundamental concepts, examples of lowdimensional orbifolds and some of these generalized theorems.
22.08.2017

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.
10.07.2017

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 1dimensional and in therefore all compact trajectories are periodic.
27.06.2017

14:15  15:15
Bethe ansatz for Gaudin models,
Evgeny Mukhin (Indiana UniversityPurdue 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 singlesurface 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.
16.05.2017

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.
21.03.2017

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 twodimensional 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. Multidimensional 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 nonhyperelliptic Riemann surfaces.
14.02.2017

14:15  15:15
Asymptotic and Circular Knets: a case study in discrete reparametrization and integrability,
Andrew SagemanFurnas (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 Knets that is tightly linked to the 4D consistency of the Lax pair for asymptotic Knets given by Bobenko and Pinkall. The description naturally gives rise to an associated family and Bäcklund transformations for cKnets. 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.
22.11.2016

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 twodimensional hyperbolic geometry' by W.P. Thurston (1986).
13.09.2016

14:15  15:15
Critical Points and Local Normal Degree for Smooth and Polyhedral Surfaces in ThreeSpace,
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 threespace. We illustrate how these notions are different by showing that there is no smooth embedding of a torus into threespace 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.
21.06.2016

14:15  15:15
How network topology determines condensation and transport properties of the antisymmetric LotkaVolterra equation,
Johannes Knebel (LudwigMaximiliansUniversität 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 nondegenerate states.
This condensation is captured by a nonlinear dynamical system, the antisymmetric LotkaVolterra 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 LotkaVolterra equation.
14.06.2016

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 3manifolds, 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 PseudoAnosov structure, a branched cover of the sphere respects the complex structure), or there is a well defined topologicalcombinatorial 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).
07.06.2016

14:15  15:15
Smooth polyhedral surfaces,
Felix Günther (MaxPlanckInstitut 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 starshapedness of the
Gaussian images is a good indicator for smoothness in a region of
nonvanishing discrete Gaussian curvature.
(Joint work with Helmut Pottmann.)
03.05.2016

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 3space $\mathbb{R}^{2,1}$. 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 3space, these
surfaces admit Weierstrasstype 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 Weierstrasstype representations for
discrete maximal and timelike minimal surfaces, and analyze their
singularities.
05.04.2016

14:15  15:15
Dodgson’s condensation method, octahedral equation and BurchnallChaundy polynomials,
Alexander Veselov (Loughborough University)
+
 The BurchnallChaundy polynomials $P_n(z)$
are determined by the differential recurrence relation
$$P_{n+1}'(z)P_{n1}(z)P_{n+1}(z)P_{n1}'(z)=P_n(z)^2$$
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
$$R_{n+1}(z+1)R_{n1}(z1)R_{n+1}(z1)R_{n1}(z+1)=R^2_n(z),$$
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 BurchnallChaundy 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).
26.01.2016

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
nonsymmetric Mahler problem.
(Joint work with A. M. Balitskiy, R. N. Karasev, and A. Sharipova.)
15.12.2015

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 (nonlinear) equations of the isomonodromic deformation are the Euler–Lagrange equations corresponding the resulting flow on the extended phase space.
27.10.2015

14:15  15:15
Integrability of limit shapes in the 6vertex model,
Nicolai Reshetikhin (UC Berkeley)
+
 This talk is focused on the limit shape phenomenon in the 6vertex 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 freefermionic point where
Hamiltonians can be computed in terms of dilogarithms.
30.06.2015

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.
23.06.2015

14:15  15:15
On variational systems on the root lattice $Q(A_{N})$,
Raphael Boll (TU Berlin)
+
 We present the theory of certain pluriLagrangian systems (i.e., integrable
systems with variational origin) on the root lattice $Q(A_{N})$ and consider
their relation to hyperbolic equations.
In the twodimensional case the considered hyperbolic equations are the
quadequations from the ABSlist and its asymmetric extension. These are less
general than the variational equations, i.e., every solution of the system of
quadequations 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 threedimensional 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.
09.06.2015

14:15  15:15
On the classification of 4D consistent maps,
Matteo Petrera (TU Berlin)
+
 It is nowadays a wellestablished 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 nontrivial 4D consistent map is
given by the known symmetric Darboux system
$$
T_k x_{ij}=\frac{x_{ij}+x_{ik}x_{jk}}{\sqrt{1x_{ik}^2}\sqrt{1x_{jk}^2}}.
$$
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 ArnouldLiouville
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.
02.06.2015

14:15  15:15
Hyperideal circle patterns and discrete uniformization of finite branch covers over the Riemann sphere,
Nikolay Dimitrov (TU Berlin)
+
 With the help of hyperideal 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 hyperideal 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 proofverification of the
latter claim.
12.05.2015

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.
05.05.2015

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, timedependent 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 nonstationary 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 wellknown freediscontinuity model of brittle fracture by FrancfortMarigo, 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.
14.04.2015

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.
26.03.2015

11:15  12:15
(Friday)
Flexibility of hyperbolic polyhedra and compact domains with prescribed convex boundary metrics in quasiFuchsian manifolds,
Dmitriy Slutskiy (Université de Strasbourg)
+
 1. We construct an infinitesimally flexible polyhedron in hyperbolic 3space such that its volume is not stationary under the infinitesimal flex.
2. We obtain a necessary condition for flexibility of suspensions in hyperbolic 3space.
3. We show the existence of a convex compact domain in a quasiFuchsian 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.
24.03.2015

14:15  15:15
Challenges in discrete nonholonomic mechanics: Preservation of volumes, Hamiltonization and Integrability,
Luis GarcíaNaranjo (IIMASUNAM, México)
+
 In the first part of the talk I will review the structure of the equations of nonholonomic mechanics and the discretization of the LagrangeD'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.
24.02.2015

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.
17.02.2015

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.
20.01.2015

14:15  15:15
Construction of embedded nperiodic 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 Dsurface. 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.
06.01.2015

14:15  15:15
Regular and antiregular 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.
09.12.2014

14:15  15:15
Hyperideal circle patterns and discrete uniformization of surfaces with nonpositive curvature,
Nikolay Dimitrov (TU Berlin)
+
 In this talk I will try to explain how hyperideal circle
patterns can be used to construct a discrete version of the
uniformization theorem for polyhedral surfaces with cone
points of nonpoisitive curvature. I will also show how
the same method can be applied to discretely uniformize
hyperelliptic Riemann surfaces.
02.12.2014

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.
11.11.2014

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 RiemannHurwitz 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 RiemannRoch 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 spinholomorphic functions defined by Chelkak and
Smirnov might help to reduce the dimension.
04.11.2014

14:15  15:15
Isothermic triangulated surfaces,
WaiYeung 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 selfstress. We will see
how this definition generalizes the discrete isothermic nets with
quadrilaterals.
As an application, we have a simple way to find the reciprocalparallel
meshes of an inscribed triangular mesh, which are regarded as discrete
minimal surfaces.
28.10.2014

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
ddimensional body, and find in each case a functional whose critical
points correspond to solutions. Under certain assumptions the
transformation f is unique modulo postcomposition 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.
14.10.2014

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 hexmesh on a doublycurved 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.
30.09.2014

14:15  15:15
LiYau inequalites on finite graphs,
Florentin Münch (FriedrichSchillerUniversität Jena)
+
 In 1986, Li and Yau proved a logarithmic gradient estimate for manifolds with nonnegative Riccicurvature, later known as LiYau inequality. Since then, great effort was made to establish an analog result on graphs. To handle the concept of a Riccicurvature 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 LiYau inequality, was established on graphs. In this talk, we will prove the original LiYau inequality on graphs and we will give a new notion of curvature.
16.09.2014

14:15  15:15
Ricci curvature on triangulations, using optimal transport, following Ollivier,
Pascal Romon (Université ParisEst MarnelaVallée)
+
 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.
09.09.2014

14:15  15:15
Robust discrete complex analysis: a toolbox,
Dmitry Chelkak (Steklov Institute St.Petersburg and ETH Zurich)
+
 We prove a number of doublesided estimates relating discrete counterparts of several classical conformal invariants of a quadrilateral: crossratios, 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, GurelGurevich 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.
15.07.2014

14:15  15:15
Circular coordinates and dynamical systems,
Vin de Silva (Pomona College)
+
 Highdimensional data sets often carry meaningful lowdimensional 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 realvalued 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 cohomology can be used to find circlevalued 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 VejdemoJohansson.
08.07.2014

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 finitegap 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.
01.07.2014

14:15  15:15
Supercyclidic extension of Qnets,
Emanuel HuhnenVenedey (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 Qnets (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 Qnets 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 Qrefinements of the initial supporting Qnet. It is to be noted that the extension of a Qnet to a supercyclidic net is inherently linked with the extension of Qnets to "fundamental line systems", that is, assigning to each vertex of a Qnet a line through that vertex such that lines at adjacent vertices intersect and the focal nets of the obtained line system are again Qnets.
This is joint work with Alexander I. Bobenko and Thilo Rörig.
17.06.2014

14:15  15:15
Integrable discretisation of hodographtype systems, hyperelliptic integrals and Whitham equations,
Wolfgang K. Schief (UNSW Sydney, Australia)
10.06.2014

14:15  15:15
Quasiconformal distortion of projective maps,
Boris Springborn (TU Berlin)
03.06.2014

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.
13.05.2014

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 crosssection leads then to an interesting combinatorial structure which is transversal to the orbits and which looks locally "buildinglike". In my talk I will explain the classification of these buildinglike geometries, and the resulting classification of polar actions on symmetric spaces. This is joint work with Alexander Lytchak.
22.04.2014

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.
08.04.2014

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 almoststandard numerical methods to converge quickly, with appropriate regularization taking care of constraints which are both redundant and underdetermined. We also demonstrate how equilibrium forces, with or without compressiononly constraints, are part of the formfinding process. This is joint work with C.C. Tang, X. Sun, A. Gomes and Helmut Pottmann.
18.03.2014

14:15  15:15
Classification of discrete 3D Hirotatype 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 Hirotatype equations within various particularly interesting subclasses. The method can be viewed as an alternative to the conventional multidimensional consistency approach.
25.02.2014

14:15  15:15
Stability of minimal Lagrangian submanifold and $L^2$ harmonic 1forms,
Reiko Miyaoka (Tohoku University)
+
 It is wellknown 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 noncompact complete stable minimal Lagrangian submanifold in M, showing that there are no $L^2$ harmonic 1forms 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.
11.02.2014

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 quadgraphs based on its medial graph. It generalizes the theory on rhombic quadgraphs 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.
28.01.2014

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 Logmap $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.
14.01.2014

14:15  15:15
The Moutard transformation of twodimensional 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 twodimensional potentials of Schroedinger operators with interesting spectral properties.
03.12.2013

14:15  15:15
Quantum entanglement and finite gap integration,
Alexander Its (IUPUI)
29.10.2013

14:15  15:15
Poisson geometry of difference Lax operators,
Michael SemenovTianShansky (Université de Bourgogne)
09.08.2013

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.
23.07.2013

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.
09.07.2013

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 timeintegrators.
25.06.2013

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 2manifold which is, topologically, vertex, edge and facetransitive. 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 4ggons in hyperbolic space.
18.06.2013

16:15  17:15
Computing hyperbolic structures on 3manifolds,
Stephan Tillmann (The University of Sydney)
+
 It is known, by remarkable work of Thurston and Perelman, that all threedimensional 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 3manifolds 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).
28.05.2013

16:15  17:15
Globally Optimal Smooth Direction Fields,
Felix Knöppel (TU Berlin)
+
 Rotationally symmetric direction fields, so called nRoSy fields, serve as input for applications in computer graphics such as nonphotorealistic 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 welldefined 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 stateof theart smoothness measure while the results are obtained in much less time. This is joint work with K. Crane, U. Pinkall, and P. Schröder.
21.05.2013

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.
14.05.2013

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.
05.02.2013

16:15  17:15
(EN 053)
Gradient flows and Ricci curvature in discrete analysis,
Jan Maas (Universität Bonn)
+
 video broadcasting from TU Munich
04.02.2013

16:15  17:15
(Monday)
Linearization through symmetries for discrete equations,
Decio Levi (Roma Tre University)
29.01.2013

16:15  17:15
Idempotent biquadratics, YangBaxter 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 quadgraph dynamical systems (namely the quadrirational YangBaxter map FI and the multiquadratic 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.
22.01.2013

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.
11.12.2012

16:15  17:15
Discrete projectiveminimal 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 projectiveminimal surfaces. We discuss various algebraic and geometric properties of these surfaces. In particular, we present a classification of discrete projectiveminimal 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.
04.12.2012

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 algebrogeometric Chebyshev construction. Effectivization of the theory grounds on the computations with Schottky groups and their moduli.
20.11.2012

16:15  17:15
Weingarten transformations of hyperbolic nets,
Emanuel HuhnenVenedey (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 Anets. Accordingly, the theory of Weingarten transformations of hyperbolic nets can is an extension of the corresponding theory for discrete Anets. Weingarten pairs of hyperbolic nets are described in terms of 3dimensional Anets 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 Anet $x$ 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 Anets. One obtains the general solution for such $\rho$ in terms of $\tau$.
13.11.2012

16:15  17:15
The discretization of surfaces parametrized along asymptotic lines by hyperbolic nets,
Emanuel HuhnenVenedey (TU Berlin)
+
 Quadrilateral nets with planar vertex stars, socalled discrete Anets, are a wellknown discretization of smooth Asurfaces. A hyperbolic net is the extension of a supporting discrete Asurface $x$, 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 Asurface.
06.11.2012

16:15  17:15
TarskiBang 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
23.10.2012

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 $(p1)(q1)+2$ is correct for all odd $p$ and $q$.
16.10.2012

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