SFB Workshop 2016 "DGD Days"
The 2016 edition of the annual DGD days on "Discretization in Geometry and Dynamics" will take place in Berlin on October 10-14. At the beginning of our second funding period, the DGD days 2016 will give our young investigators and new projects an opportunity to present their research within the SFB/Transregio. We are also very pleased to announce as our plenary speaker Prof. Dr. Bernd Sturmfels from U.C. Berkeley Mathematics. The DGD days program is organized in presentations and a framework of free time providing opportunities for ongoing discussions and exchange.
- Date: 10.10. - 14.10.2016
- Type: Workshop
- Location: TU Berlin, Institute of Mathematics, Strasse des 17. Juni 136, 10623 Berlin
Program
Please find below the scheduling overview for the „DGD-Days“.
Talks will be either 50 min presentation with 10 min discussion or 25 min presentation with 5 min discussion.
- Alexander Bobenko (TU Berlin)
- Folkmar Bornemann (TU München)
- Ulrich Bauer (TU München)
- Yuri Suris (TU Berlin)
- Günter Ziegler (FU Berlin)
Doyle spirals were discovered by Peter Doyle and then further investigated as special examples of circle packings. They can be considered as discrete analogues of the exponential map. In our talk we present a generalization of Doyle spirals and relate them to conformally equivalent triangular lattices. These examples belong to the class of conformally symmetric conformally equivalent triangular lattices. Finally, we reveal an analogue of the Schwarzian derivative which we have used to prove $C^\infty$-convergence for Dirichlet boundary value problems.
We give an overview of some recent results regarding the atomistic-to-continuum variational limit of some random lattice systems. In particular we discuss the problem of the stochastic homogenization of these systems under stationarity and/or ergodicity assumptions on the spatial distribution of the lattice points. Our analysis is motivated by microscopic models of polymeric materials giving rise in the continuum approximation to macroscopic theories for rubber elasticity and for the formation of Weiss domains in (possibly thin) magnetic composites.
We study the problem of finite crystallization in the discrete differential geometry framework; more precisely, we focus on a zero temperature, short range pairwise interactions 2D model introduced by Heitmann and Radin in 1980.
We show that for any $N \in \mathbb{N}$ and for any configuration X with N points, the Heitmann-Radin energy $\mathcal{E}_{HR}(X)$ decomposes as
$\mathcal{E}_{HR}(X) = -3 N + P(X) + \mu(X) + 3\chi(X)$
where the surface term P(X) is the perimeter of the bond graph generated by X, $\mu(X)$ is the defect term measuring how much this graph differs from a subset of the triangular lattice and $\chi(X)$ is the Euler characteristic of the bond graph.
Using a suitable notion of curvature defined on graphs and a discrete Gauss-Bonnet theorem, we provide a simple argument that shows that any minimizer $P + \mu + 3\chi$ (and hence of $\mathcal{E}_{HR}$) is a connected subset of the triangular lattice having simply closed polygonal boundary.
This is a joint work with Gero Friesecke (TU München).
We state and prove a tropical analog of the (discrete version of the) classical isoperimetric inequality. The planar case is elementary, but the higher-dimensional generalization leads to an interesting class of ordinary convex polytopes. This study is motivated by deep open complexity questions concerning linear optimization and its tropical analogs. This connection will be sketched briefly in the talk.
Joint work with Xavier Allamigeon, Pascal Benchimol, Jules Depersin and Stephane Gaubert.
X-ray crystallography is the method of choice for the analysis of molecular structures. Most known protein structures have been inferred from the characteristic discrete diraction patterns of their crystalline forms.
Based on a mathematical model for the diraction of time-harmonic radiation, concepts of X-ray crystallography can be generalized to certain non-crystalline molecular structures by designing suitable radiation (joint work with Gero Friesecke, TUM, and Richard James, University of Minnesota). Potentially, these ideas might help to analyze molecular structures that are not accessible by current methods.
In this talk I will explain the possibility of "double discreteness" of both a structure and its diraction pattern via a simple example structure. In particular, an abstract version of the Poisson summation formula and its non-abelian generalization play a central role.
Lagrangian coherent structures are maximal material subsets whose advective evolution is maximally persistent to weak diffusion. For their detection, we transform flow information from advection-diffusion dynamics into a deformed Riemannian geometry on the set of initial conditions/particles. Then, Lagrangian coherent structures express themselves as almost invariant sets, i.e., as those subsets of the manifold which are particularly slowly decaying under the heat flow induced by the geometry. We study and visualize the Riemannian geometry in detail, and discuss the connections to diffusion barriers. The talk is based on joint work with Johannes Keller.
Fascinating and elegant shapes may be folded from a single planar sheet of material without stretching, tearing or cutting, if one incorporates curved folds into the design. Artists and designers have proposed a wide variety of different fold patterns to create a range of interesting surfaces. The creative process, design as well as fabrication, is usually only concerned with the static surface that emerges once folding has completed. Folding such patterns, however, is difficult as multiple creases have to be folded simultaneously to obtain a properly folded target shape. I will give an introduction to the geometric properties of curved folded surfaces and then focus on the most recent developments. I will introduce string actuated curved folded surfaces that can be shaped by pulling a network of strings thus vastly simplifying the process of creating such surfaces and making the folding motion an integral part of the design. This amounts to solving which surface points to string together and how to actuate them by locally expressing a desired folding path in the space of isometric shape deformations in terms of novel string actuation modes. Validity of the approach is demonstrated by computing string actuation networks for a range of well known crease patterns and testing their effectiveness on physical prototypes.
In applications, the information contained in a high-dimensional signal often results from a low-dimensional signal model. This can be due to sparsity of the signal (i.e. the signal lies on a union of a small number of planes) or due to the dependence of the signal on only few parameters (i.e. the signal lies on a low-dimensional submanifold of the ambient signal space). While sparse signal models are quite well understood, many questions are still open for manifold signal models. In this talk we discuss a result by Iwen/Maggioni on the approximation of points on low-dimensional manifolds and give an outlook how we planned to use this result for 1-bit quantization from manifold valued data.
In this talk I shall provide an overview of the area of fast-slow dynamical systems with a focus on the geometric approach to the resolution of singularities. In particular, I am going to illustrate the basic results regarding slow invariant manifolds, the blow-up method, the exchange lemma and apply these techniques to the resolution of the Olsen conjecture (joint work with P. Szmolyan) about oscillatory patterns. Furthermore, I am going to outline several connections of the geometry of multiscale dynamics to the DGD-SFB/TR.
Lovász proved in 2001 that realizations of 3-polytopes have an interpretation by eigenvectors of certain weighted adjacency matrices associated to their 1-skeleton. These matrices are examples of Colin-de-Verdière-matrices of a graph. Izmestiev partially generalized Lovász result in 2010: he associated a Colin-de-Verdière-matrix $M_P$ to a given realization of a d-polytope P but he was not able to relate eigenvectors of $M_P$ to the polytope P. In my talk, I will review the results of Lovász and Izmestiev and present various new families of polytopes in arbitrary dimension which illustrate the intimate relation between eigenvectors of $M_P$ and realizations of P.
We give a construction of completely integrable (2m)-dimensional Hamiltonian systems with cubic Hamilton functions. The construction depends on a constant skew-Hamiltonian matrix A, that is, a matrix satisfying $A^T J = J A$, where J is a non-degenerate skew-symmetric matrix defining the standard symplectic structure on the phase space $\mathbb{R}^{2m}$. Applying to any such system the so called Kahan-Hirota-Kimura discretization scheme, we arrive at a birational (2m)-dimensional map. We show that this map is symplectic with respect to a symplectic structure that is a perturbation of the standard symplectic structure on $\mathbb{R}^{2m}$, and possesses m independent integrals of motion, which are perturbations of the original Hamilton functions and are in involution with respect to the invariant symplectic structure. Thus, this map is completely integrable in the Liouville-Arnold sense. Moreover, under a suitable normalization of the original m-tuples of commuting vector fields, their Kahan-Hirota-Kimura discretizations also commute and share the invariant symplectic structure and the m integrals of motion.
The extension of discrete nets by smooth surface patches, most recently, the extension of discrete Q-nets with supercyclide patches, raised the following question: Why are the surface patches exactly the right ones for the extension? And, do they belong to a larger class of suitable surfaces?
Based on the (discrete) properties of hyperboloid, Dupin cyclide, and supercyclide patches, we investigate discrete nets (and smooth surfaces), that satisfy a given property, not only for elementary quadrilaterals, but also for general parameter rectangles. We classify these "multi-nets" in the case of Q-nets, circular nets, A-nets, and principle contact element nets. We see that the surface patches used previously for the extension of circular nets, A-nets, and Q-nets, are the simplest in these new classes to allow for tangent plane continuity along common boundary curves.
Bernd Sturmfels: Nearest points on Toric Varieties
This talk concerns the following optimization problem: given a data point, find its best approximation in a model that is parametrized by monomials. This algebraic complexity of this problem is given by the Euclidean distance degree of a projective toric variety. We present a formula for this degree. It extends a formula of Matsui and Takeuchi for the degree of the A-discriminant in terms of Euler obstructions. The motivation for this work is the development of our optimization problem. A key ingredient is the study of characteristic classes such as the Chern-Mather class. This is joint work with Martin Helmer.
We first construct an exact mathematical frame of the discrete spin transformation by introducing a set of discrete surfaces called the edge-constraint nets. These surfaces are not restricted to the triangulated or quad nets, but is allowed to have planar or non-planar polygons as their faces. Given an immersed surface, the discrete spin transformation produces an immersed discrete surface by adding a prescribed integrated mean curvature on each face. By solving the discrete extrinsic Dirac equation the closeness condition of faces is guaranteed. In this frame the discrete minimal surfaces and their associated family arise naturally. The notion spin cross-ratio is then developed as the criteria of the spin equivalence. In the second part we focus on the more classic intrinsic Dirac operator. First it will be explained what exactly is the structure of a discrete intrinsic surfaces and then the discrete version of spinor bundle and connection, spin structure and intrinsic Dirac operator are constructed. By solving the intrinsic Dirac operator over an intrinsic surface it gives rise to an immersed surface with prescribed integrated mean curvature.
Hotel reservation for external participants:
Novum Hotel Gates Berlin
Knesebeckstr. 8-9
Breakfast included
Walking Information:
How to get to the Novum Hotel Gates (Knesebeckstr. 8-9, 10623 Berlin) and
the TU Institute of Mathematics (Strasse des 17. Juni 136, 10623 Berlin):
By car
Exit the A 100 at the Kaiserdamm exit and drive along Kaiserdamm street (it later changes ist name into Bismarckstrasse) for approximately 2 km in the direction of Ernst-Reuter-Platz, until you reach the roundabout.
For the hotel: Take the first exit in the roundabout onto Hardenbergstrasse and shortly after turn into the second street on the right onto Knesebeckstrasse. After a few metres you will see the Hotel Gates on the right-hand side.
For the Math Institute: Take the second exit in the roundabout onto Strasse des 17. Juni and follow the street for a few metres until shortly before the bridge and make a U-turn there. Follow the street for a few metres back; the Math Dept. will be on your right-hand side. There is a public parking area directly at the Math Building.
By train to Berlin central station (Hauptbahnhof)
From Berlin Hauptbahnhof take either the S7 or S75 tube line towards Potsdam (S7) or Westkreuz (S75) as far as Zoologischer Garten. Here change to the U2 tube line towards Ruhleben/Theodor-Heuss-Platz and exit at Ernst-Reuter-Platz.
For the hotel: Exit the underground station in the direction of “Renaissance-Theater”; the second street on the right is Knesebeckstrasse.
For the Math Institute: Exit in the direction of “Technische Universität, Campus Charlottenburg”, go straight and cross the Strasse des 17. Juni at the next light; turn right and keep going for a few metres.
Travel time approx. 20 minutes; required single ticket: travel zones AB for EURO 2,70.
If you take a taxi from Hauptbahnhof, travel time will be about 15 minutes; costs for a taxi will be around EURO 13.
By plane to Tegel airport
From Tegel airport please take Expressbus X9 towards Zoologischer Garten and get off at Ernst-Reuter-Platz.
For the hotel: Continue in the direction the bus was going and turn right onto Knesebeckstrasse.
For the Math Institute: Turn the opposite direction and cross the Hardenbergstrasse at the first light. Go straight and cross the Strasse des 17. Juni at the next light and turn right.
Travel time approx. 20 minutes; required single ticket: travel zones AB for EURO 2,70.
If you take a taxi from Tegel airport, you will arrive within 15-20 minutes; costs for a taxi will be around EURO 20.
By plane to Schönefeld airport
The fastest connection is from Schönefeld train/S-tube station (walk approx. 6 minutes). From there take either train RE7 or RB14 towards Bad Belzig/Dessau (RE7) or Nauen (RB14) as far as Zoologischer Garten. Here change to the U2 tube line towards Ruhleben/Theodor-Heuss-Platz and get off at Ernst-Reuter-Platz.
For the hotel: Exit the underground station in the direction of “Renaissance-Theater”; the second street on the right is Knesebeckstrasse.
For the Math Institute: Exit in the direction of “Technische Universität, Campus Charlottenburg”, go straight and cross the Strasse des 17. Juni at the next light; turn right and keep going for a few metres.
Travel time approx. 55 minutes; required single ticket: travel zones ABC for EURO 3,30.
Please note that this connection is only available every 20-40 minutes. In between there are additional S- and U-tube connections with a double change of lines; they take approx. 10 minutes longer.
If you take a taxi from Schönefeld aiport, travel time will be about 30-40 minutes; costs for a taxi will be around EURO 45.
For detailed timetables for public transportation please consult the homepage of our local transportation system (English available): http://www.bvg.de/