SFB Seminare und Abstracts

Dienstags 14.15 Uhr   Raum MI 02.06.011

17-07-18

Kolloquium aus Berlin mit Live-Übertragung nach München

Programm:

14.15      Martin Schmidt (Universität Mannheim) @ TUB
"Deformations of spectral curves"

15.30      Luis García Naranjo (IMAS-UNAM, Mexico) @ TUB
"The geometric discretisation of the Suslov problem: a case study of consistency for nonholonomic integrators"


Martin Schmidt: Deformations of Spectral curves

The talk reviews some results on spectral curves of integrable systems. First we show, how the theory of the Whitham equations for spectral curves fit into the theory of deformations of algebraic curves. In the context of the algebraic geometric correspondence between spectral data and integrable systems with Lax operators there exists different notions of spectral curves. They are all one-sheeted coverings of the multiplier curve and have the same normalization. If the spectral curves are chosen to be locally planar, then the theory of Whitham deformations yields a smooth universal deformation. As an application we consider spectral curves of closed curves and CMC tori in three-dimensional Euclidean space. There are two different density results related to these two examples: Firstly, the closed curves of finite type in three dimensional Euclidean space are dense in all closed curves. Secondly, for fixed spectral genus the closure of the spectral curves of CMC tori in three dimensional Euclidean space inside the spectral curves of CMC planes can be determined. These results are the first steps toward the soul conjecture of Ulrich Pinkall.

Luis García Naranjo: The geometric discretisation of the Suslov problem: a case study of consistency for nonholonomic integrators

In mechanics, constraints that restrict the possible configurations of the system are termed holonomic. A simple example is the fixed length of the rod of a pendulum. Mechanical systems with constraints on the velocities that are not consequences of constraints on positions are called nonholonomic. These often arise in rolling systems, like a sphere rotating without slipping on a table.

The geometric discretisation of nonholonomic systems is a an open research subject that goes back to a paper by Cortés and Martinez from 2001. In their approach, it is necessary to propose a discrete version of the constraints and of the Lagrangian of the system.

In this talk I will first present a general introduction to nonholonomic systems and their discretisation. Next, I will present recent results on the geometric discretisation of the Suslov problem, which is one of the simplest examples of a nonholonomic system that exhibits some of the main features that distinguish nonholonomic from Hamiltonian systems. Our results indicate that the so-called consistency condition on the choice of the discrete constraints and the discrete Lagrangian, is not relevant in the design of nonholonomic geometric integrators with optimal properties.



17-06-20

Kolloquium aus München mit Live-Übertragung nach Berlin

Programm:

14.15      Caroline Moosmüller (University of Passau)@TUM
"Iterative refinement processes for manifold-valued data"

15.30      Rayan Saab (University of California, San Diego)@TUM
"Near-optimal quantization and encoding under various measurement models"


Caroline Moosmüller: Iterative refinement processes for manifold-valued data

We consider iterative refinement processes which operate on discrete data and produce a smooth curve or surface in the limit. Most results on such processes, also called subdivision schemes, concern data in vector spaces and rules which are linear. We are interested in studying manifold-valued data and refinement rules which are solely defined by the intrinsic geometry of the underlying manifold. In this talk we focus on a particular class of subdivision schemes, called Hermite schemes. These algorithms successively refine discrete point-vector data and, via a limit process, produce a curve and its derivatives. We give an introduction to linear Hermite subdivision schemes and present adaptations to manifolds using geodesics and parallel transport. Furthermore, we analyse the resulting nonlinear algorithms with respect to convergence and C1 smoothness.

Rayan Saab: Near-optimal quantization and encoding under various measurement models

In the era of digital computation, data acquisition consists of a series of steps. A sampling or measurement process is typically followed by quantization, or digitization, which allows digital storage and transmission of data. In turn, quantization is often followed by encoding, or compression, to efficiently represent the quantized data. In this talk, we discuss quantization and encoding schemes for a variety of measurement processes, along with their associated reconstruction algorithms.

We show results for classically oversampled band-limited functions, oversampled linear measurements of finite dimensional signals, and compressed sensing measurements of sparse and compressible signals. The encoding methods we discuss are practical, rely on noise-shaping quantization schemes such as Sigma-Delta quantization, and also work in the extreme case of 1-bit quantization. Moreover, they yield near-optimal approximation accuracy as a function of the bit-rate.



17-05-23

Kolloquium aus München mit Live-Übertragung nach Berlin

Programm:

15.30      Mathieu Carrière (INRIA, France)@TUM
"A theoretical framework for the analysis of Mapper"


Mathieu Carrière: A theoretical framework for the analysis of Mapper

Mapper is probably the most widely used TDA (Topological Data Analysis) tool in the applied sciences and industry. Its main application is in exploratory analysis, where it provides novel data representations that allow for a higher-level understanding of the geometric structures underlying the data. The output of Mapper takes the form of a graph, whose vertices represent homogeneous subpopulations of the data, and whose edges represent certain types of proximity relations. Nevertheless, the inherent instability of the output and the difficult parameter tuning make the method rather difficult to use in practice. This talk will focus on the study of the structural properties of the graphs produced by Mapper, together with their partial stability properties, with a view towards the design of new tools to help users set up the parameters and interpret the outputs.



17-02-07

Kolloquium aus München mit Live-Übertragung nach Berlin

Programm:

14.15      Axel Voigt (TU Dresden)@TUM
"Orientational order on surfaces - the coupling of topology, geometry and dynamics"

15.30      Frederic Chazal (INRIA, France)@TUM
"Persistent homology for data: stability and statistical properties"


Axel Voigt: Orientational order on surfaces - the coupling of topology, geometry and dynamics

We consider the numerical investigation of surface bound orientational order using unit tangential vector fields by means of a gradient-flow equation of a weak surface Frank-Oseen energy. The energy is composed of intrinsic and extrinsic contributions, as well as a penalization term to enforce the unity of the vector field. Four different numerical discretizations, namely a discrete exterior calculus approach, a method based on vector spherical harmonics, a surface finite-element method, and an approach utilizing an implicit surface description, the diffuse interface method, are described and compared with each other for surfaces with Euler characteristic 2. We demonstrate the influence of geometric properties on realizations of the Poincare-Hopf theorem and show examples where the energy is decreased by introducing additional orientational defects.

Frederic Chazal: Persistent homology for data: stability and statistical properties

Computational topology has recently seen an important development toward data analysis, giving birth to Topological Data Analysis. Persistent homology appears as a fundamental tool in this field. It is usually computed from filtrations built on top of data sets sampled from some unknown (metric) space, providing "topological signatures" revealing the structure of the underlying space. When the size of the sample is large, direct computation of persistent homology often suffers two issues. First, it becomes prohibitive due to the combinatorial size of the considered filtrations and, second, it appears to be very sensitive to noise and outliers. The goal of the talk is twofold. First, we will briefly introduce the notion of persistent homology and show how it can be used to infer relevant topological information from metric data through stability properties. Second, we will present a method to overcome the above mentioned computational and noise issues by computing persistent diagrams from several subsamples and combining them in order to efficiently infer robust and relevant topological information.



17-01-10

Kolloquium mit Live-Übertragung zwischen Berlin und München

Programm:

14.15      Boris Springborn (TU Berlin) @TUB
"The hyperbolic geometry of Markov's theorem on Diophantine approximation and quadratic forms"

Coffee break

15.30      Felix Krahmer (TU München) @TUM
"On the connection between A/D conversion and the roots of Chebyshev polynomials"


Boris Springborn: The hyperbolic geometry of Markov's theorem on Diophantine approximation and quadratic form

Markov's theorem classifies the worst irrational numbers (with respect to rational approximation) and the binary quadratic forms whose values stay far away from zero (for nonzero integer arguments). I will present a proof of Markov's theorem that uses hyperbolic geometry. The main ingredients are a dictionary to translate between algebra/arithmetic and hyperbolic geometry, and some basic ideas from decorated Teichmüller theory. This will explain what Diophantine approximation has to do with simple closed geodesics, and also with the question: How far can a straight line crossing a triangle stay away from the vertices?

Felix Krahmer: On the connection between A/D conversion and the roots of Chebyshev polynomials

Sigma-Delta modulation is a popular approach for coarse quantization of audio signals. That is, rather than taking a minimal amount of samples and representing them with high resolution, one considers redundant representations and works with a low resolution. The underlying idea is to employ a feedback loop, incorporating the prior evolution of the sampling error. In this way, the representation of a sample can partially compensate for errors made in previous steps. The design of the filter at the core of the feedback loop is crucial for stability and hence for performance guarantees. Building on work of Güntürk (2003) who proposed to use sparse filters, we optimize the sparsity pattern, showing that a distribution mimicking the roots of Chebyshev polynomials of the second kind is optimal. The focus of this talk will be on the interplay between complex variables, orthogonal polynomials, and signal processing in the proof. This is joint work with Percy Deift and Sinan Güntürk (Courant Institute of Mathematical Sciences, NYU), derived as a part of my doctoral dissertation.



16-12-13

Kolloquium aus München mit Live-Übertragung nach Berlin

Programm:

14.15      Richard D. James (University of Minnesota)@TUM
"Atomistically inspired origami"


Richard D. James: Atomistically inspired origami

“Objective Structures” are structures generated as orbits of discrete groups of isometries. We comment on their unexpected prevalence in nanoscience, materials science and biology and also explain why they arise in a natural way as distinguished structures in quantum mechanics, molecular dynamics and continuum mechanics. The underlying mathematical idea is that the isometry group that generates the structure matches the invariance group of the differential equations. Their characteristic features in molecular science imply highly desirable features for macroscopic structures, particularly 4D structures that deform. We illustrate the latter by constructing some “objective origami” structures.

Origami Graphic



16-11-15

Kolloquium aus München / Berlin mit Live-Übertragung

Programm:

14.15      Igor Dolgachev (University of Michigan at Ann Arbor; John von Neumann Guest Professor at TUM)@TUM
"Salem numbers and discrete groups of automorphisms of algebraic surfaces"

15.30      Franz Pedit (UMass/Amherst) @TUB
"Constant mean curvature surfaces in $\mathbb{R}^3$ with Delaunay ends and Hilbert's 21st problem"


Igor Dolgachev: Salem numbers and discrete groups of automorphisms of algebraic surfaces

Abstract: A Salem number is a real algebraic integer greater than one whose all conjugates have absolute value at most one and at least one of them has absolute value one. In particular, a Salem number is a root of a reciprocal monic polynomial with integer coefficients. In complex dynamics the logarithms of Salem numbers are realized as topological entropy of an automorphism of an algebraic surface. In my talk I will explain when such an automorphism exists and which Salem numbers occur in this way.

Franz Pedit: Constant mean curvature surfaces in $\mathbb{R}^3$ with Delaunay ends and Hilbert's 21st problem

We will discuss the classification problem for constant mean curvature surfaces with Delaunay ends, its relation to surface group representations of punctured compact Riemann surfaces into a loop group, and Fuchsian connections (differential equations) with coefficients in a loop algebra.



16-07-05

30. Kolloquium aus Berlin mit Live-Übertragung nach München

Programm:

14.15      Peter Schröder (Caltech)@TUB
"Schrödinger's Smoke"

15.30      Yuri Suris (TU Berlin)@TUB
"News about confocal quadrics"


Peter Schröder: Schrödinger's Smoke

We describe a new approach for the purely Eulerian simulation of incompressible fluids. In it, the fluid state is represented by a C2 C2-valued wave function evolving under the Schrödinger equation subject to incompressibility constraints. The underlying dynamical system is Hamiltonian and governed by the kinetic energy of the fluid together with an energy of Landau-Lifshitz type. The latter ensures that dynamics due to thin vortical structures, all important for visual simulation, are faithfully reproduced. This enables robust simulation of intricate phenomena such as vortical wakes and interacting vortex filaments, even on modestly sized grids. Our implementation uses a simple splitting method for time integration, employing the FFT for Schrödinger evolution as well as constraint projection. Using a standard penalty method we also allow arbitrary obstacles. The resulting algorithm is simple, unconditionally stable, and efficient. In particular it does not require any Lagrangian techniques for advection or to counteract the loss of vorticity. We demonstrate its use in a variety of scenarios, compare it with experiments, and evaluate it against benchmark tests. Joint work with Albert Chern, Felix Knöppel, Ulrich Pinkall and Steffen Weißmann. For more details search "Schrödinger's Smoke" on YouTube.

Yuri Suris: News about confocal quadrics

I will report about some recent developments, both in geometry and dynamics, related to confocal quadrics. First, I will provide a new approach to billiards in confocal quadrics and their integrability, based on the pluri-Lagrangian structure. Second, I will describe a novel construction of discrete confocal coordinate systems, based on a joint work with Bobenko, Schief and Techter.



16-06-14

SFB Seminar München

Programm:

14.15      Sara Krause-Solberg (TUM)
"Non-Negative Dimensionality Reduction in Signal Separation"


Sara Krause-Solberg: Non-Negative Dimensionality Reduction in Signal Separation

In this talk, we discuss the application of (non-negative) dimensionality reduction methods in signal separation. In single-channel separation, the decomposition techniques as e.g. non-negative matrix factorization (NNMF) or independent component analysis (ICA) are typically applied to time-frequency data of the mixed signal obtained by a signal transform.

Starting from this classical separation procedure in the time-frequency domain, we considered an additional preprocessing step, in which the dimension of the data is reduced in order to facilitate the computation. Depending on the separation methods, different properties of the dimensionality reduction technique are required. We focused on the non-negativity of the low-dimensional data or - since the time-frequency data is non-negative - rather on the non-negativity preservation beyond the reduction step, which is mandatory for the application of NNMF.

Finally, we discuss the application of the developed non-negative dimensionality reduction techniques to signal separation. We present some numerical results when using our non-negative PCA (NNPCA) and compare its performance with other versions of PCA and different separation techniques, namely NNMF and ICA.



16-05-31

29. Kolloquium aus München / Berlin mit Live-Übertragung

Programm:

14.15      Sergei Tabachnikov (Pennsylvania State University)@TUB
"Tire track geometry and the filament equation: results and conjectures"

15.30      Gary Froyland (UNSW)@TUM
"Dynamic isoperimetry and Lagrangian coherent structures"


Sergei Tabachnikov: Tire track geometry and the filament equation: results and conjectures

The simplest model of a bicycle is a segment of fixed length that can move, in n-dimensional Euclidean space, so that the velocity of the rear end is always aligned with the segment (the rear wheel is fixed on the frame). The rear wheel track and a choice of direction uniquely determine the front wheel track; changing the direction to the opposite, yields another front track. The two track are related by the bicycle (Darboux) transformation which defines a discrete time dynamical system on the space of curves. I shall discuss the symplectic, and in dimension 3, bi-symplectic, nature of this transformation and, in dimension 3, its relation with the filament equation. An interesting problem is to describe the curves that are in the bicycle correspondence with themselves (in this case, given the front and rear tracks, one cannot tell which way the bicycle went). In dimension two, such curves yield solutions to Ulam's problem: is the round ball the only body that floats in equilibrium in all positions? I shall discuss F.~Wegner's results on this problem and relate them with the planar filament equation. Open problems and conjectures will be emphasized.


Gary Froyland: Dynamic isoperimetry and Lagrangian coherent structures

The study of transport and mixing processes in dynamical systems is important for the analysis of mathematical models of physical systems. I will describe a novel, direct geometric method to identify subsets of phase space that remain strongly coherent over a finite time duration. The method is based on a dynamic extension of classical (static) isoperimetric problems; the latter are concerned with identifying submanifolds with the smallest boundary size relative to their volume. I will introduce dynamic isoperimetric problems; the study of sets with small boundary size relative to volume as they are evolved by a general dynamical system. I will state dynamic versions of the fundamental (static) isoperimetric (in)equalities; a dynamic Federer-Fleming theorem and a dynamic Cheeger inequality. I will also introduce a dynamic Laplace operator and describe a computational method to identify coherent sets based on eigenfunctions of the dynamic Laplacian. Our results include formal mathematical statements concerning geometric properties of finite-time coherent sets, whose boundaries can be regarded as Lagrangian coherent structures. The computational advantages of this approach are a well-separated spectrum for the dynamic Laplacian, and flexibility in appropriate numerical approximation methods. Finally, we demonstrate that the dynamic Laplace operator can be realised as a zero-diffusion limit of a recent probabilistic transfer operator method for finding coherent sets, based on small diffusion.



16-05-24

28. Kolloquium aus München/Berlin mit Live-Übertragung

Programm:

14.15      Mark Iwen (Michigan State University)@TUM
"An Introduction to Distance Preserving Projections of Smooth Manifolds"

15.30      Ludwig Gauckler (TU Berlin)@TUB
"Numerical long-time energy conservation for the nonlinear Schrödinger equation"


Mark Iwen: An Introduction to Distance Preserving Projections of Smooth Manifolds

Manifold-based image models are assumed in many engineering applications involving imaging and image classification. In the setting of image classification, in particular, proposed designs for small and cheap cameras motivate compressive imaging applications involving manifolds. Interesting mathematics results when one considers that the problem one needs to solve in this setting ultimately involves questions concerning how well one can embed a low-dimensional smooth sub-manifold of high-dimensional Euclidean space into a much lower dimensional space without knowing any of its detailed structure. We will motivate this problem and discuss how one might accomplish this seemingly difficult task using random projections. Little if any prerequisites will be assumed.



Ludwig Gauckler: Numerics of nonlinear Schrödinger equations on long time intervals

In the talk, the long-time behaviour of numerical methods for Hamiltonian differential equations is discussed, in particular the near-conservation of energy by symplectic numerical methods on long time intervals. In the case of Hamiltonian ordinary differential equations, this can be rigorously shown by a backward error analysis. After an introduction to this classical result, the difficulties in extending such a result to Hamiltonian partial differential equations like the nonlinear Schrödinger equation are described. Finally, a recent result on long-time near-conservation of energy by the (symplectic) split-step Fourier method applied to the (Hamiltonian) nonlinear Schrödinger equation is presented.


16-04-19

27. Kolloquium aus München mit Live-Übertragung nach Berlin

Programm:

14.15      Marian Mrozek (Jageillonian University, Kraków)@TUM
"On a discretization of Conley theory for flows in the setting of discrete Morse theory"

15.30      Magnus Bakke Botnan (TU München) @TUM
"Stability and Approximations in Topological Data Analysis"


Marian Mrozek: On a discretization of Conley theory for flows in the setting of discrete Morse theory

Conley theory studies qualitative features of dynamical systems by means of topological invariants of isolated invariant sets. Since the invariants are stable under perturbations and computable from a finite sample, they provide a tool for rigorous, qualitative numerical analysis of dynamical systems.

In this talk, after a brief introduction to Conley theory, I will present its recent extension to combinatorial multivector fields, a generalization of the concept introduced by R. Forman in his discrete (combinatorial) Morse theory. I will also show some numerical examples indicating potential applications in the study of sampled dynamical systems



Magnus Bakke Botnan: Stability and Approximations in Topological Data Analysis

Persistent homology assigns a topological descriptor to a real-valued function defined on a topological space. This descriptor is called a persistence diagram and comprises a collection of intervals summarizing the dimension and 'size' of the homological features of the associated sublevel filtration. Two immediate questions arise: 1) is the descriptor stable with respect to perturbation of the function? 2) can we approximate the descriptor? By introducing the language of interleavings we shall see how positive answers have been given to both these questions. From here we shall discuss similar questions related to other, albeit related, topological descriptors.



16-02-02

26. Kolloquium aus München mit Live-Übertragung nach Berlin

Programm:

14.15      Helmut Pottmann (TU Wien & John v. Neumann Guest Professor )@TUM
"Polyhedral Patterns"

15.30      Daniel Karrasch (TU München)@TUM
"The flux integral revisited: the Lagrangian perspective"


Helmut Pottmann: Polyhedral Patterns

We study the design and optimization of polyhedral patterns, which are patterns of planar polygonal faces on freeform surfaces. Working with polyhedral patterns is desirable in architectural geometry and industrial design. However, the classical tiling patterns on the plane must take on various shapes in order to faithfully and feasibly approximate curved surfaces. We define and analyze the deformations these tiles must undertake to account for curvature, and discover the symmetries that remain invariant under such deformations. We propose a novel method to regularize polyhedral patterns while maintaining these symmetries into a plethora of aesthetic and feasible patterns. Finally, we raise the question of faithful ap- proximations of smooth surfaces by polyhedral surfaces and present some initial results in this direction.



Daniel Karrasch: The flux integral revisited: the Lagrangian perspective.

Advective transport of scalar quantities through surfaces is of fundamental importance in many scientific applications. From the Eulerian perspective of the surface it can be quantified by the well-known flux integral. The recent development of highly accurate semi-Lagrangian methods for solving scalar conservation laws and of Lagrangian approaches to coherent structures in turbulent (geophysical) fluid flows necessitate a new approach to transport, i.e. accumulated (over time) flux, from the (Lagrangian) material perspective.

In my talk, I present a Lagrangian framework for calculating transport of conserved quantities through a given (hyper-)surface in n-dimensional, fully aperiodic, volume-preserving flows.


2015


15-12-08

25. Kolloquium aus München mit Live-Übertragung nach Berlin

Programm:

14.15      Daniel Karrasch (TU München)@München CANCELED
"The flux integral revisited: the Lagrangian perspective"

15.30      Nicolai Reshetikhin (UC Berkeley)@Berlin
"On limit shapes and their integrability"


Daniel Karrasch: The flux integral revisited: the Lagrangian perspective.

Cancelled due to illness. Will be repeated on February 2nd, 2016

Nicolai Reshetikhin: On limit shapes and their integrability

The talk will start with the introduction into the limit shape phenomenon which is the emergence of a deterministic structure from a random one for large systems. Then the focus will be on the 6-vertex model and on dimer models and it will be argued that corresponding limit shapes are given by integrable PDE's.



15-09-22

23. Kolloquium aus München mit Live-Übertragung nach Berlin

Programm:

14.15      David Krieg (TU Freiberg)
"Domain Filling Circle Packings - Existence and Uniqueness"

15.30      Gero Friesecke (TU München)
"Molecular structures generated by discrete symmetries, X-ray diffraction, and structure identification"


David Krieg: Domain Filling Circle Packings - Existence and Uniqueness"

Circle packings are discrete models of analytic functions which mimic and approximate their classical counterparts. For example, William Thurston conjectured and Rodin and Sullivan proved that discrete conformal mappings modelled by circle packings converge to the classical mappings if the packings are appropriately refined.

In the standard approach to discrete conformal mapping the domain packing is constructed by "cookie cutting", which results in poor approximation of the domain near its boundary. To overcome this deficiency, different algorithms have been proposed which allow one to construct domain filling packings, with boundary circles touching the boundary of the given domain. Though these algorithms work well in practice, some questions concerning existence and uniqueness of domain filling packings remained open. Oded Schramm proved very general results for domain filling (not only circle) packings, but in particular his uniqueness statements require assumptions on the boundary (being ``decent''), which seem not to be completely natural. For more general domains Schramm predicted the appearance of degenerate circles with radius zero; but he gives no explicit criteria when this may happen.

In the talk I present existence and uniqueness statements for circle packings filling arbitrary bounded, simply connected domains. As normalization three boundary points (prime ends if the domain is not Jordan) are associated with three boundary circles, which have to be touched in a generalized sense. This leads to a discrete version of Carathéodory's Theorem, which provides the existence and uniqueness of discrete conformal mappings under weak assumptions. Afterwards a generalization of circle packings is introduced (circle agglomerations), degeneracy is taken into account, and other normalizations are discussed.

Gero Friesecke: Molecular structures generated by discrete symmetries, X-ray diffraction, and structure identification

The scattering of incoming plane waves at crystals, i.e., periodic structures, results in discrete structure- identifying diffraction patterns. Much of what is known from experiment about the atomic-scale structure of matter is found using this method. Mathematically, the remarkable diffraction patterns have been explained by the multi-dimensional Poisson summation formula. But why - really - does all this work, i.e. how do you get from some continuous object with discrete symmetry (the electron density of a crystal, a continuous periodic function on $\R^3$) to a discrete object on some other space (the X-ray diffraction pattern) and back?

A closer examination (joint work with Dominik Juestel, TUM, and Richard James, University of Minnesota) reveals that, in fact, the hypothesis of periodicity is not fundamentally what is being used, but rather a group structure. The key point is that the structure is invariant under a DISCRETE subgroup of the Euclidean group of rotations and translations, and the incoming radiation is equivariant under a related CONTINUOUS subgroup. This suggests the possiblity of novel X-ray methods, via the replacement of the pair (plane waves, crystals) by (other solutions to Maxwell's equations, 'objective structures'). The details have been worked out for helical structures and lead, surprisingly, to the recently discovered beams with orbital angular momentum (OAM). Objective structures are defined mathematically as orbits of a finite number of points under a discrete Euclidean group. Many important structures in biology and nanoscience are of this form, including buckyballs and many fullerenes, the parts of many viruses, actin, carbon nanotubes (all chiralities).

The talk will informally present a mathematical picture of all the objects involved, not assuming any background in the underlying physics.



15-07-14

22. Kolloquium aus München mit Live-Schaltung nach Berlin

Programm:

14.15      Mike Lesnick (IMA, University of Minneapolis & Columbia University, NY)
"Interactive Visualization of 2-D Persistent Homology"

15.30      Norbert Peyerimhoff (Durham University, UK)
"Cheeger constants and frustration indices for magnetic Laplacians on graphs and manifolds"


Mike Lesnick: Interactive Visualization of 2-D Persistent Homology

Abstract: In topological data analysis, we study data by associating to the data a filtered topological space, whose structure we can then examine using persistent homology. However, in many settings, such as in the study of point cloud data with noise or non-uniformities in density, a single filtered space is not a rich enough invariant to encode the interesting structure of our data. This motivates the study of multidimensional persistence, which associates to the data a topological space simultaneously equipped with two or more filtrations. The homological invariants of these “multifiltered spaces,” while much richer than their 1-D counterparts, are also far more complicated. As such, adapting the usual 1-D persistent homology methodology for data analysis to the multi-D setting requires new ideas. In this talk, I’ll introduce multi-D persistent homology and discuss joint work with Matthew Wright on the development of a practical tool for the interactive visualization of 2-D persistence.

Norbert Peyerimhoff: Cheeger constants and frustration indices for magnetic Laplacians on graphs and manifolds

Abstract: Due to Mark Kac' famous question "Can one hear the shape of a drum?" in 1966, the study of spectra of Laplacians became a very fashionable research area with many beautiful results. A fundamental fact in this area is Cheeger's inequality, which relates an isoperimetric constant to the first non-trivial eigenvalue of the Laplace operator. Remarkably, this result is not only fundamental in the setting of Riemannian manifolds: its combinatorial counterpart is of central importance for the development of efficient and well connected networks or, more theoretically formulated, in the construction of expander graph families. It is natural to try to extend spectral results from classical Laplacians to more general operators like magnetic Laplacians. The magnetic potential leads to a specific first order modification of the classical Laplacian and requires to change from a real to a complex-valued Hilbert space. However, self-adjointness guarantees that the spectrum of a magnetic Laplacian is still real. In this talk we introduce these operators on graphs and on Riemannian manifolds, and we will discuss in detail how the classical Cheeger inequality needs to be modified. The classical isoperimetric constant has to be replaced by a new invariant, balancing between isoperimetric properties of subsets and an contribution of the additional magnetic potential. The contribution of the magnetic potential is measured by the so-called frustration index. If time permits, I will also explain how our results can be applied for the construction of a particular spectral clustering algorithm for partially oriented



15-05-19

20. Kolloquium aus München mit Live-Schaltung nach Berlin

Programm:

14.15      Matteo Novaga (Università di Pisa & John von Neumann Gastprofessor@TUM)
"Confined elastic curves"

15.30      Lucia de Luca (TU München)
"Dynamics of Discrete Screw Dislocations in SC, BCC, FCC and HCP crystals"


Matteo Novaga: Confined elastic curves

Abstract: I will consider curves confined in a prescribed container, minimizing their bending or elastic energy. In particular, I will discuss the existence and regularity of the optimal curves, and describe their shape in some particular cases.

Lucia de Luca: Dynamics of Discrete Screw Dislocations in SC, BCC, FCC and HCP crystals

Abstract: Dislocations are line defects in the periodic structure of crystals and their motion represents the main mechanism of plastic deformation in metals. We discuss a variational approach for modeling the dynamics of screw dislocations in some specific crystal structures, i.e., Simple Cubic (SC), Body Centered Cubic (BCC), Face Centered Cubic (FCC), and Hexagonal Close Packed (HCP). More precisely, using a discrete (in space and in time) variational scheme, we study the motion of a configuration of dislocations toward low energy con gurations. We deduce an effective fully overdamped dynamics that follows the maximal dissipation criterion introduced by Cermelli & Gurtin and predicts motion along the glide directions of the crystal. The results are fruit of a joint work with Roberto Alicandro (University of Cassino), Adriana Garroni and Marcello Ponsiglione ("La Sapienza", University of Rome)


15-04-21

19. Kolloquium aus Berlin mit Live-Schaltung nach München

Programm:

14.15      Sebastian Heller (Universität Tübingen)
"Navigating the space of symmetric CMC surfaces"

15.30      Kristin Shaw (TU Berlin)
"Tropical (p,q)-homology and algebraic cycles"


Sebastian Heller: Navigating the space of symmetric CMC surfaces

Abstract: We consider compact surfaces of constant mean curvature (CMC) in 3-dimensional space forms. While the only CMC spheres are round spheres and CMC tori can be explicitly parametrized via integrable systems methods, only very little is known about higher genus CMC surfaces. In this talk we first give a brief introduction to the spectral curve approach to CMC tori due to Hitchin. In general the constant mean curvature condition of a surface can be translated into the flatness condition of an associated family of SL(2,ℂ)-connections. For tori, the abelian fundamental group allows to reduce flat SL(2,ℂ)-connections to flat line bundle connections, and the associated family can be parametrized in terms of certain algebraic geometric objects - the spectral data - from which the conformal immersion can be reconstructed. Under the assumption of certain discrete symmetries, irreducible connections on higher genus surfaces can also be parametrized by flat line bundle connections. This enables us to define a generalization of the spectral curve theory for higher genus CMC surfaces. Due to the non-abelian nature, it is hard to construct spectral data for higher genus CMC surfaces explicitly. In a recent preprint (joint work with L. Heller and N. Schmitt) we have introduced a flow on the spectral data from CMC tori towards higher genus CMC surfaces. We explain how this flow can be used to construct spectral data for higher genus CMC surfaces and to study the moduli space of symmetric CMC surfaces of higher genus.



Kristin Shaw: Tropical (p,q)-homology and algebraic cycles

Abstract: I will define tropical (p,q)-homology as introduced by Itenberg, Katzarkov, Mikhalkin and Zharkov (IKMZ). In fact, despite their name these homology groups can be defined for any polyhedral space not necessarily having a tropical structure. However, in the tropical setting these groups are particularly interesting since it follows from (IKMZ) that these homology groups can give Hodge numbers of complex projective varieties. The second part of this talk I will describe how one can use these groups to study "tropical algebraic cycles". Here I will focus on two main examples. Firstly tropical hypersurfaces in ℝ3, which are polyhedral complexes dual to subdivisions of lattice polytopes. Secondly, tropical Abelian varieties, which are real tori equipped with an integral affine structure.




15-03-10

Workshop on the occasion of the 60th Birthday of Prof. Ulrich Pinkall: Geometry of smooth and discrete surfaces
Workshop aus Berlin mit Live-Schaltung nach München

Programm:

10.00 - 17:00       The full workshop-programm can be accessed under: DGD Website

15-02-10

18. Kolloquium aus Berlin mit Live-Schaltung nach München

Programm:

14.15      Sina Ober-Blöbaum (Universität Paderborn/FU Berlin)
"Discrete variational mechanics in structure-preserving integration and optimal control"

15.30      Alexander Mielke (WIAS Berlin)
"The Chemical Master Equation as a discretization of the Fokker-Planck and Liouville equation for chemical reactions"


Sina Ober-Blöbaum: Discrete variational mechanics in structure-preserving integration and optimal control

Abstract: Discrete variational mechanics plays a fundamental role in constructing and analyzing structure-preserving numerical methods for the simulation and optimization of mechanical systems. Based on a discrete variational principle that approximates the continuous one, time-stepping schemes denoted as variational integrators can be derived that are structure preserving, i.e. they are symplectic-momentum conserving and exhibit good energy behavior, meaning that no artificial dissipation is present and the energy error stays bounded over longterm simulations. After a brief introduction to variational integrators and an overview of their different fields of applications, we derive two different kinds of high order variational integrators based on different dimensions of the underlying approximation space. While the first well-known integrator is equivalent to a symplectic partitioned Runge-Kutta method, the second integrator, denoted as symplectic Galerkin integrator, yields a method which in general, cannot be written as a standard symplectic Runge-Kutta scheme. In the second part of the talk, we use these integrators for the discretization of optimal control problems. For the numerical solution of optimal control problems, direct methods are based on a discretization of the underlying differential equations which serve as equality constraints for the resulting finite dimensional nonlinear optimization problem. By analyzing the adjoint systems of the optimal control problem and its discretized counterpart, we prove that for these particular integrators dualization and discretization commute. This property guarantees that the convergence rates are preserved for the adjoint system which is also referred to as the Covector Mapping Principle.



Alexander Mielke: The Chemical Master Equation as a discretization of the Fokker-Planck and Liouville equation for chemical reactions

Abstract: We discuss the reaction kinetics according to a finite set of mass-action type reactions under the additional assumption of detailed balance. This nonlinear ODE system has an entropic gradient structure. The associated Chemical Master Equation is the Markov chain counting the number of particles for each species, where a reaction corresponds to a jump in the discrete state space NI0. This Markov process has again a gradient structure on the set of probability measures. Scaling the number of particles by the total volume V , we show evolutionary -convergence for V → ∞ of the discrete Markov process to the continuous Liouville equation which is a gradient flow with respect to a Wasserstein-type metric. We discuss the role of the Fokker-Planck equation as a singularly perturbed Liouville equation with better approximation properties. This is joint work with Jan Maas, IST Wien.


15-01-27

17. Kolloquium aus München mit Live-Schaltung nach Berlin

Programm:

14.15      Roman Schubert (Bristol University, UK)
"What is the semiclassical limit of non-Hermitian time evolution?"

15.30      Michael La Croix (MIT Massachusetts)
"Combinatorial Models for Random Matrices with Gaussian Entries"


Roman Schubert: What is the semiclassical limit of non-Hermitian time evolution?

Abstract: It is well known that in the semiclassical limit (or the high frequency limit) solutions to many PDE's in physics and other sciences are driven by a Hamiltonian flow, i.e., by solutions to a system of ODE's. Examples include Maxwell's equations and the Schroedinger equation. In this setting the propagation of waves can be described geometrically as the propagation of Lagrangian submanifolds of a symplectic manifold by the Hamiltonian flow.

We will consider the case that the Hamiltonian flow is generated by a complex valued Hamilton function, which corresponds to a system with loss or gain and a Schroedinger equation where the Hamilton operator is non-Hermitian. We derive a new class of ODE's which govern the semiclassical limit, and which combine a Hamiltonian vectorfield, generated by the real part of the Hamilton function, with a gradient vectorfield, generated by the imaginary part. It turns out that the propagation can again be described in terms of symplectic geometry, but this time it includes a complex structure which is generated by the dynamics along the rays. We will give an overview of these new geometric structures emerging from the semiclassical limit. This is joint work with Eva Maria Graefe.



Michael La Croix: Combinatorial Models for Random Matrices with Gaussian Entries

Abstract: Random matrix theory studies the statistics of functions defined on matrices with random entries. Typical questions involve understanding how the spectrum of such a matrix depends on the size of the matrix. Universality principles show that it is often sufficient to consider matrices constructed from independent Gaussian entries. For several such models, expectations of symmetric functions of eigenvalues are polynomial in the size of the matrix. With appropriate scaling and choice of basis, these polynomials appear with non-negative integer coefficients, and can be given combinatorial interpretations as generating series for combinatorial maps, equivalence classes of graphs embedded in surfaces, considered up to homeomorphism. The analytic problem of evaluating high-dimensional integrals is thus discretized as a counting problem, which in this case can be studied using algebraic combinatorics, by encoding the same maps in terms of permutation factorizations.

The equivalence between the discreet and continuous approaches is a consequence of the duality relating the representation theories of the finite groups and Lie groups acting on the permutation and matrix models, but in some ways the equivalence is more general than either of the models. Parallel theories for matrices with complex entries (combinatorialized by orientable maps) and matrices with real entries (combinatorialized by non-oriented maps) can be combined into a single parametrized theory that continues to make sense even when the parameter is evaluated outside of its natural domain.




15-01-13

16. Kolloquium aus München mit Live-Schaltung nach Berlin

Programm:

14.15      Carsten Lange (TU München)
"Signed graphs, nested set complexes and spines"

15.30      Herbert Edelsbrunner (IST Austria)
"Approximation and Convergence of the Intrinsic Volume"


Carsten Lange: Signed graphs, nested set complexes and spines

Abstract: Fulton and MacPherson used nested set structures to describe compactifications of configuration spaces in 1994 and examples for related nested set complexes can be constructed for every finite graph. These examples, called graph associahedra, have a simple combinatorial description, can be realized by convex polyhedra and yield instances of generalized permutahedra. All necessary objects will be introduced in the first part of my presentation. In the second part, I present a generalized construction of nested set complexes for finite graphs with an addditional vertex labeling by + and - which exhibits a larger class of generalized permutahedra. A key ingredient of this construction are spines which are used to describe a generalization of nested sets.

Herbert Edelsbrunner: Approximation and Convergence of the Intrinsic Volume

Abstract: We study the computation of intrinsic volumes of a solid body from a sequence of binary images of progressively finer resolution. While the intrinsic volumes of the binary images do not necessarily converge to the correct value, we show that the formula can be rigged to give the correct limit for the first intrinsic volume, which in R^3 relates to the total mean curvature of the boundary of the body. Work with Florian Pausinger.





2014


14-12-16

15. Kolloquium aus Berlin mit Live-Schaltung nach München

Programm:

14.15      Dierk Schleicher (Jacobs University Bremen)
"Hausdorff dimension in transcendental dynamics"

15.30      Vassili Gelfreich (University of Warwick)
"A portrait of Arnold Diffusion"


Dierk Schleicher: Hausdorff dimension in transcendental dynamics

Abstract: Transcendental dynamics, the iteration of transcendental entire functions from C to C, has become a very active field of research especially in the last decade (even though foundations were laid by Pierre Fatou about one hundred years ago). We describe some of the basic questions and properties, including the interesting dynamics near essential singularities. The dynamics also has some interesting and surprising dynamical properties: for instance, in a most natural way we obtain a decomposition of the complex plane into two disjoint sets E and R such that every point in E is connected to infinity by its own curve in R so that all curves are disjoint from each other and from E — so that one might think that R is much bigger than E. However, R has Hausdorff dimension 1, while E is the complement of R and thus has full planar measure.

Vassili Gelfreich: A portrait of Arnold Diffusion

Abstract: Over the past decades, a large amount of research in geometry processing has been dedicated to computational tools for processing single geometries. From a more global perspective, one might ask to study surfaces as points in a space of shapes. Over the past years, concepts from Riemannian manifolds have been applied to design and investigate shape spaces, with applications to shape morphing and modeling, computational anatomy, as well as shape statistics. Studying shape space from the point of view of Riemannian geometry enables transfer of important geometric concepts, such as geodesics or curvatures, from classical geometry to these (usually) infinite-dimensional spaces of shapes. In my talk I will highlight some of the recent developments and open questions in understanding shape space from a Riemannian perspective, with a particular down-to-earth focus on how one can relate Riemannian metrics to elastic energies.




14-11-18

14. Kolloquium aus München mit Live-Schaltung nach Berlin

Programm:

14.15      Ulrich Bauer (TU München)
"The Morse theory of Čech and Delaunay filtrations"

15.30      Max Wardetzky (Uni Göttingen)
"From Shapes to Shape Spaces"


Ulrich Bauer: "The Morse theory of Čech and Delaunay filtrations".

Abstract: Given a finite set of points in ℝⁿ and a positive radius, we consider the Čech, Delaunay–Čech, Delaunay (alpha shape), and wrap complexes as examples of a generalized discrete Morse theory. We prove that the four complexes are simple-homotopy equivalent by a sequence of simplicial collapses, and the same is true for their weighted versions. Our results have applications in topological data analysis and in the reconstruction of shapes from sampled data. Joint work with Herbert Edelsbrunner.

Max Wardetzky: "From Shapes to Shape Spaces"

Abstract: Over the past decades, a large amount of research in geometry processing has been dedicated to computational tools for processing single geometries. From a more global perspective, one might ask to study surfaces as points in a space of shapes. Over the past years, concepts from Riemannian manifolds have been applied to design and investigate shape spaces, with applications to shape morphing and modeling, computational anatomy, as well as shape statistics. Studying shape space from the point of view of Riemannian geometry enables transfer of important geometric concepts, such as geodesics or curvatures, from classical geometry to these (usually) infinite-dimensional spaces of shapes. In my talk I will highlight some of the recent developments and open questions in understanding shape space from a Riemannian perspective, with a particular down-to-earth focus on how one can relate Riemannian metrics to elastic energies.




14-10-21

13. Kolloquium aus Berlin und München mit Live-Schaltung

Programm:

14.15      Maurice de Gosson (Universität Wien)
"The Symplectic Camel and the Uncertainty Principle"

15.30      Matthias Beck (San Francisco State University)
"Integer partitions from a geometric viewpoint"


Maurice de Gosson: "The Symplectic Camel and the Uncertainty Principle".

Abstract: Gromov’s discovery in 1985 of the symplectic non-squeezing theorem, dubbed “the principle of the symplectic camel”by Arnol’d, can be viewed as a classical version of the uncertainty principle of quantum mechan- ics. We will show that a derived notion, the symplectic capacity of subsets of phase space, allows a topological reformulation of the uncertainty princi- ple. Using recent results by Artstein-Avidan, R. Karasev, and Y. Ostrover we propose a new notion of indeterminacy. We also discuss the relationship between the notion of symplectic capacity and Hardy’s uncertainty principle about the localization of a function and its Fourier transform, which can be reformulated in terms of the symplectic capacity of the covariance ellipsoid of the Wigner transform of that function.

Matthias Beck: "Integer partitions from a geometric viewpoint".

Abstract: The study of partitions and compositions (i.e., ordered partitions) of integers goes back centuries and has applications in various areas within and outside of mathematics. Partition analysis is full of beautiful--and sometimes surprising--identities, starting with Euler's classic theorem that the number of partitions of an integer k into odd parts equals the number of partitions of k into distinct parts. Motivated by work of G. Andrews et al from the last 1 1/2 decades, we will show how one can shed light on certain classes of partition identities by interpreting partitions as integer points in polyhedra. Our approach yields both "short" proofs of known results and new theorems. This is joint work with Ben Braun, Ira Gessel, Nguyen Le, Sunyoung Lee, and Carla Savage




14-06-24

12. Kolloquium aus Berlin und München mit Live-Schaltung

Programm:

14.15      Wolfgang Schief (UNSW Sydney)
"Universal aspects of geometric and algebraic integrability in Pluecker and Lie geometry"

15.30     Tomoki Ohsawa (University of Michigan-Dearborn, USA)
"The Geometry and Dynamics of Semiclassical Wave Packets"


Wolfgang Schief: "Universal aspects of geometric and algebraic integrability in Pluecker and Lie geometry".

Abstract: We discuss a novel geometric interpretation of a "universal" system of difference equations in terms of line complexes in a three-dimensional complex projective space. This system of algebraic equations is integrable in the sense of multi-dimensional consistency and admits a variety of canonical reductions. These are associated with particular classes of integrable line complexes such as linear and quadric line complexes. In the former case, one may reinterpret the line complexes in terms of Lie circle geometry. In fact, another reduction of the universal system leads to integrability in Lie sphere geometry and a generalisation to quaternionic line complexes is available which corresponds to Lie 4-sphere geometry. Connections with "master" equations such as the discrete BKP, CKP and Darboux equations are presented

Tomoki Ohsawa: “The Geometry and Dynamics of Semiclassical Wave Packets"

Abstract: I will talk about the geometry and dynamics of semiclassical wave packets, which provide a description of the transition regime between quantum and classical mechanics. It is well known that both classical and quantum mechanical systems are described as Hamiltonian systems: finite-dimensional one for the former, and infinite-dimensional for the latter with respect to the corresponding symplectic geometric structures. I will show how to exploit such geometric structures to formulate semiclassical dynamics from the Hamiltonian/symplectic point of view. The Hamiltonian/symplectic formulation reveals the role of symmetry, conservation laws, and reduction in semiclassical wave packet dynamics.




14-05-27

11. Kolloquium aus München und Berlin mit Live-Schaltung

Programm:

14.15      Hartmut Monien (Universität Bonn)
"Solving problems in algebraic geometry using applied mathematics"

15.30     Peter Schröder (Caltech)
"Smoke Rings from Smoke"


Hartmut Monien:"Solving problems in algebraic geometry using applied mathematics"

Algebraic geometry is concerned with finding the locus of zeros of sets of polynomial equations in a commutative ring. In my talk I will show how some hard questions in that field can in fact be answered by explicitly solving a 2D partial differential equation on a Riemann surface. The monodromy group of the Riemann surface is closely related to the "dessin d'enfant" of Grothendiek which allows to gain insight into the absolute Galois group. We will present some interesting non-congruence subgroups with interesting Galois groups.

Peter Schröder: “ Smoke Rings from Smoke”

We give an algorithm which extracts vortex filaments ("smoke rings") from a given 3D velocity field. Given a filament strength h>0, an optimal number of vortex filaments, together with their extent and placement, is given by the zero set of a complex valued function over the domain. This function is the global minimizer of a quadratic energy based on a Schr{ödinger operator. Computationally this amounts to finding the eigenvector belonging to the smallest eigenvalue of a Laplacian type sparse matrix. Turning traditional vector field representations of flows, for example, on a regular grid, into a corresponding set of vortex filaments is useful for visualization, analysis of measured flows, hybrid simulation methods, and sparse representations. To demonstrate our method we give examples from each of these.

Joint work with Ulrich Pinkall and Steffen Weißmann


14-05-06

SFB Seminar
14.15     Christian Kühn (TU Wien)
"A tour through modern methods in multiple time scale dynamics"


An overview of multiple time scale systems and singular perturbation problems will be given. It is the goal of this talk to show the breadth of this field and outline some of its major techniques and applications developed within the last 15 years. First, I am going to briefly introduce the background for the geometric viewpoint for normally hyperbolic systems covering Fenichel theory and the notion of canards. Next, switching between fast and slow systems will be considered using the Exchange Lemma. When normal hyperbolicity is lost the blow-up method for desingularization is employed. To conclude we briefly illustrate the general problem of multiscale dynamics near instability in the context of stochastic fast-slow systems. Throughout the talk, specialized numerical methods will be used to illustrate the dynamics.




14-04-29

10. Kolloquium aus München nach Berlin per Live-Schaltung

Programm:

14.15      Caroline Lasser (TU München)
"The multivariate Hermite-Laguerre connection"

15.30      Mauro Maggioni (Duke University, Durham, USA)
"Geometric Methods for the Approximation of High-dimensional Dynamical Systems"

Caroline Lasser: "The multivariate Hermite-Laguerre connection"


The Hermite polynomials have been born in the 19th century and live in many mathematical fields: in numerical analysis, in quantum theory, in combinatorics, in probability. The same can be said about Laguerre polynomials. Our talk will review the connection between the two and lift this connection to the multivariate situation such that the symplectic geometry of classical phase space comes into play. Our results are joint work with S. Troppmann.

Mauro Maggioni: “Geometric Methods for the Approximation of High-dimensional Dynamical Systems”


We discuss a novel statistical learning framework for performing model reduction and modeling of stochastic high-dimensional dynamical systems. We consider two complementary settings. In the first one, we are given long trajectories of the system and we discuss new techniques for estimating, in a robust fashion, an effective number of degrees of freedom of the system, which may vary in the state space of the system, and a local scale where the dynamics is well-approximated by a reduced dynamics with a small number of degrees of freedom. We then use these ideas to produce an approximation to the generator of the system and obtain reaction coordinates for the system that capture the large time behavior of the dynamics. We present various examples from molecular dynamics illustrating these ideas. This is joint work with C. Clementi, M. Rohrdanz and W. Zheng. In the second setting we only have access to a (large number of expensive) simulators that can return short simulations of high-dimensional stochastic system, and introduce a novel statistical learning framework for learning automatically a family of local approximations to the system, that can be (automatically) pieced together to form a fast global reduced model for the system, called ATLAS. ATLAS is guaranteed to be accurate (in the sense of producing stochastic paths whose distribution is close to that of paths generated by the original system) not only at small time scales, but also at large time scales, under suitable assumptions on the dynamics. We discuss applications to homogenization of rough diffusions in low and high dimensions, as well as relatively simple systems with separations of time scales, and deterministic chaotic systems in high-dimensions, that are well-approximated by stochastic differential equations. This is joint work with M. Crosskey.


14-04-15

SFB Seminar
14.15     Michael Kraus (IPP Garching)
"Variational Integrators in Plasma Physics"

Variational integrators provide a systematic way to derive geometric numerical methods for Lagrangian dynamical systems, which preserve a discrete multisymplectic form as well as momenta associated to symmetries of the Lagrangian via Noether’s theorem. An inevitable prerequisite for the derivation of variational integrators is the existence of a variational formulation for the considered dynamical system. Even though a large class of systems fulfills this requirement, there are many interesting examples which do not belong to this class, e.g., equations of advection-diffusion type like they are often found in fluid dynamics or plasma physics. We propose the application of the variational integrator method to so called adjoint Lagrangians, which formally allow us to embed any dynamical system into a Lagrangian system by doubling the number of variables. Thereby we are able to derive variational integrators for arbitrary systems, extending the applicability of the method significantly. A discrete version of the Noether theorem for adjoint Lagrangians yields the discrete momenta preserved by the resulting numerical schemes. The presented method therefore provides a systematic way to construct numerical schemes which respect certain conservation laws of a given system. The basics of variational integrators for field theories are presented including the discrete Noether theorem. The theory is then applied to several prototypical systems from plasma physics like the Vlasov-Poisson system and ideal magnetohydrodynamics. Numerical examples confirm the good theoretical properties.




14-04-08

SFB Seminar - Live-Übertragung aus Berlin
14.15     Johannes Wallner (TU Graz)
"Formfinding with statics for polyhedral meshes"

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

14-01-07

8. Kolloquium aus München nach Berlin per Live-Schaltung

Programm:

14.15      Marco Cicalese (TU München)
"Variational methods for lattice systems"

15.30      Massimo Fornasier (TU München)
"Mean Field Sparse Optimal Control"

Marco Cicalese: "Variational methods for lattice systems"


I will give a concise introduction to the variational analysis of the micro-to-macro limits of energy driven lattice systems. To address the problem I will review a general scheme based on the notion of Gamma-convergence. Then I will present several examples from materials science devoting special attention to the continuum limit of some simple network models entailing multiple scales where new effects of microscopic origin add up to the usual macroscopic description and give rise to complex energies.

Massimo Fornasier: “Mean Field Sparse Optimal Control”


We present the concept of mean-field optimal control which is the rigorous limit process connecting finite dimensional optimal control problems with ODE constraints modeling multi-agent interactions to an infinite dimensional optimal control problem with a constraint given by a PDE of Vlasov-type, governing the dynamics of the probability distribution of interacting agents. While in the classical mean-field theory one studies the behavior of a large number of small individuals freely interacting with each other, by simplifying the effect of all the other individuals on any given individual by a single averaged effect, we address the situation where the individuals are actually influenced also by an external policy maker, and we propagate its effect for the number N of individuals going to infinity. On the one hand, from a modeling point of view, we take into account also that the policy maker is constrained to act according to optimal strategies promoting its most parsimonious interaction with the group of individuals. This is realized by considering cost functionals including L^1-norm terms penalizing a broadly distributed control of the group, while promoting its sparsity. On the other hand, from the analysis point of view, and for the sake of generality, we consider broader classes of convex control penalizations. In order to develop this new concept of limit rigorously, we need to carefully combine the classical concept of mean-field limit, connecting the finite dimensional system of ODE describing the dynamics of each individual of the group to the PDE describing the dynamics of the respective probability distribution, with the well-known concept of Gamma-convergence to show that optimal strategies for the finite dimensional problems converge to optimal strategies of the infinite dimensional problem.



2013


13-12-10

7. Kolloquium per Live-Schaltung (Berlin und München)

Programm:

14.15     Bernd Sturmfels (UC Berkeley and MPI Bonn)
"The Euclidean Distance Degree"

15.30     Johannes Wallner (TU Graz)
"Cell Packing Structures"

Bernd Sturmfels: "The Euclidean Distance Degree"


The nearest point map of a real algebraic variety with respect to Euclidean distance is an algebraic function. The Euclidean distance degree is the number of critical points of this optimization problem. We focus on varieties seen in engineering applications, and we discuss exact computational methods. Our running example is the Eckart-Young Theorem which states that the nearest point map for low rank matrices is given by the singular value decomposition. This is joint work with Jan Draisma, Emil Horobet, Giorgio Ottaviani, Rekha Thomas.

Johannes Wallner: "Cell Packing Structures"


We give an overview of architectural structures which are either composed of polyhedral cells or closely related to them. In particular we discuss so-called support structures of polyhedral cell packings, which are mostly relevant if they are derived from quadrilateral or hexagonal meshes. There are interesting connections between discrete differential geometry on the one hand, and applications on the other hand. Such applications range from load-bearing structures to shading and lighting systems. On a higher level, we illustrate the interplay between geometry, optimization, statics, and manufacturing, with the overall aim of combining form, function and fabrication into novel integrated design tools. This is joint work with H. Pottmann et al.


13-12-03

SFB Seminar
14.15     Carsten Lange (TU München)
"Optimal topological simplification of discrete functions on surfaces"

Given a function f on a surface and a tolerance d>0, a fundamental problem is the construction of a perturbed function g such that N(f-g) is at most d with respect to some norm N and g has the minimum number of critical points. I will present a solution to this problem (with respect to the supremum norm) and describe how homological noise of persistence at most 2d can be completely removed from an input function on a discrete surface. The solution obtained is not unique and a convex polyhedron of possible solutions is identified. As a consequence, the method can be complemented to construct a solution that satisfies an additional constraint. The construction relies on a connection between discrete Morse theory and persistence homology. A brief introduction to both subjects will be included.


13-11-21 DONNERSTAG 14:15

SFB Seminar

Programm:

14.15     Kathrin Padberg-Gehle, TU Dresden
"Transfer operator based numerical analysis of time-dependent transport"

15.30     Shane Ross, Virginia Tech
"Topological chaos, braiding and bifurcation of almost-cyclic sets"

Kathrin Padberg-Gehle

"Transfer operator based numerical analysis of time-dependent transport"
Numerical methods involving transfer operators have only recently been recognized as powerful tools for analyzing and quantifying transport processes in time-dependent systems. This talk discusses several different constructions that allow us to extract coherent structures and dynamic transport barriers in nonautonomous dynamical systems. Moreover, we will explore in example systems how diffusion as well as the finite-time duration of the computations influence the structures of interest.

Shane Ross

"Topological chaos, braiding and bifurcation of almost-cyclic sets"
In certain two-dimensional time-dependent flows, the braiding of periodic orbits provides a way to analyze chaos in the system through application of the Thurston-Nielsen classification theorem. Periodic orbits generated by the dynamics can behave as physical obstructions that 'stir' the surrounding domain and serve as the basis for this topological analysis. We provide evidence that, even for the case where periodic orbits are absent, almost-cyclic sets can be used. These are individual components of almost-invariant sets identified using a transfer operator approach which act as stirrers or 'ghost rods' around which the surrounding fluid appears to be stretched and folded. We discuss the bifurcation of the almost-cyclic sets as a system parameter is varied, which results in a sequence of topologically distinct braids. We show that, for Stokes’ flow in a lid-driven cavity, these various braids give good lower bounds on the topological entropy over the respective parameter regimes in which they exist. Hence, we develop a connection between set-oriented statistical methods and topological methods, which provides an additional analysis tool in the study of complex flows.


13-11-12

6. Kolloquium aus München

Programm:

14.15     Tim N. Hoffmann (TU München)
"On a surface theory for quadrilateral nets"

Tim N. Hoffmann

"On a surface theory for quadrilateral nets"
will report on recent work on a discrete version of surface theory for quadrilateral nets. Our approach aims to generalize the known integrable cases into a more general framework. There are many well working examples of integrable discretizations of special surface classes as well as well working discrete definitions of fundamental forms, curvatures, shape operator and similar fundamental objects of surface theory for but so far little effort has been made to formulate a general framework that covers the integrable cases with their fundamental properties and still works on a broader class of nets. This is joint work with Andrew O. Sageman-Furnas (Furnas) and Max Wardetzky.


13-10-15

5. Kolloquium per Live-Schaltung von Berlin nach München

Programm:

14.15     Wolfgang K. Schief (UNSW Sydney)
"Bianchi (hyper-)cubes and a geometric unification of the Hirota and Miwa equations"

15.30     David Cimasoni (University of Geneva)
"The critical temperature for the Ising model on biperiodic graphs"

Wolfgang K. Schief

"Bianchi (hyper-)cubes and a geometric unification of the Hirota and Miwa equations"
We present a geometric and algebraic way of unifying two discrete master equations of inte- grable system theory, namely the dKP (Hirota) and dBKP (Miwa) equations. We demonstrate that so-called Cox lattices encapsulate Bianchi (hyper-)cubes associated with either simultaneous so- lutions of a novel 14-point and the dBKP equations or solutions of the dKP equation, depending on whether the Cox lattices are generic or degenerate.

David Cimasoni

"The critical temperature for the Ising model on biperiodic graphs"
The Ising model is one of the most studied models in statistical physics. It is one of the simplest models to exhibit a ”phase transition”, that is, a sharp change of behavior when some parameter (here, the temperature) crosses some critical value. In this talk, I will start with a gentle introduc- tion to the Ising model and its phase transition. Then, I will explain how to determine the critical temperature of the Ising model on any biperiodic planar weighted graph (or equivalently, on any finite weighted graph embedded in the torus). Although this result lies in the realm of statistical physics, the statement is formulated in homological terms, and the proof uses several geometric tools (Kramers-Wannier duality on surfaces, Harnack curves, . . . )


13-08-09   !  FREITAG, 10.15  !    Videotalk from TU Berlin

Igor Pak (UCLA)  "How to prove Steinitz's theorem"
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.

13-07-23

Sandra Hayes (City College, City University of New York)  "The Real Dynamics of Bieberbach’s Example"
Bieberbach constructed in 1933 domains in $\mathbb{C}^2$ which were biholomorphic to $\mathbb{C}^2$ but omitted an open set. The existence of such domains was unexpected, because the analogous statement for the one-dimensional complex plane is false. The special domains Bieberbach considered are given as basins of attraction of a cubic Henon map. This classical method of the construction is one of the first applications of dynamical systems to complex analysis. In this talk the boundaries of the real sections of Bieberbach’s domains will be shown to be smooth. The real Julia sets of Bieberbach’s map will be calculated explicitly and illustrated with computer generated graphics.

13-07-02

4. Kolloquium per Live-Schaltung von München nach Berlin

Programm:

16.15    Prof. Dr. Ileana Streinu (Smith College, Northampton, MA.,USA & SFB TRR109 Gastprofessorin an der TUM)
          “Rigidity of origami surfaces”

17.15   Kaffeepause

17.30    Dr. Ciprian S. Borcea (Rider University, New Jersey, USA)
          "Rigidity and flexibility of periodic frameworks"

Prof. Dr. Ileana Streinu (Smith College, Northampton, USA)

“Rigidity of origami surfaces”

Cauchy's famous rigidity theorem for 3D convex polyhedra has been extended in various directions by Dehn, Weyl, A.D.Alexandrov, Gluck and Connelly. These results imply that a disk-like polyhedral surface with simplicial faces is, generically, flexible, if the boundary has at least 4 vertices. What about surfaces with rigid but not necessarily simplicial faces? A natural, albeit extreme family is given by flat-faced origamis.
Around 1995, Robert Lang, a well-known origamist, proposed a method for designing a crease pattern on a flat piece of paper such that it has an isometric flat-folded realization with an underlying, predetermined metric tree structure. Important mathematical properties of this algorithm remain elusive to this day.
In this talk I will show that Lang's beautiful method leads, often but not always, to a crease pattern that cannot be continuously deformed to the desired flat-folded shape if its faces are to be kept rigid. Most surprisingly, sometimes the initial crease pattern is simply rigid: the (real) configuration space of such a structure may be disconnected, with one of the components being an isolated point.
Joint work with my PhD student John Bowers, who has also implemented a computer program to visualize the research.

Dr. Ciprian S. Borcea (Rider University, New Jersey, USA)

“Rigidity and flexibility of periodic frameworks”

A d-periodic bar-and-joint framework is an abstraction (and generalization to arbitrary dimension d) of an atom-and-bond crystal structure. We present a general introduction to the deformation theory of this type of frameworks. Questions of generic rigidity highlight the role of sparsity conditions on the underlying quotient graph.
Joint work with Dr. Ileana Streinu, Smith College.



13-06-04

3. Kolloquium per Live-Schaltung von Berlin nach München

Programm:

16.15    Prof. Dr. Julian Pfeifle (Universitat Politècnica de Catalunya)
          "Minimum-cardinality triangulations of polytopes and manifolds"

17.15   Kaffeepause

17.30    Prof. Dr. Martin Rumpf (Univerität Bonn)
          "Variational Time Discretization of Geodesic Calculus in Shape Space"

Prof. Dr. Julian Pfeifle
"Minimum-cardinality triangulations of polytopes and manifolds"


Triangulations are important in both discrete and numerical mathematics, but different properties are studied in each of these areas. On the discrete side, attention tends to focus on structural and combinatorial properties, such as the ``shape'' of the set of all triangulations of a fixed object, or the (minimal) number of simplices in any one of them. In this talk, I will (briefly) survey some of the principal results in this area, and report on recent progress in finding triangulations of minimal cardinality of some interesting polytopes and topological manifolds derived from them. Some of these results exploit the new capabilities of the software package ``polymake'' for exact and efficient calculations in quadratic extension rings of the rationals.

Prof. Dr. Martin Rumpf
"Variational Time Discretization of Geodesic Calculus in Shape Space"


The talk will introduce a time discrete geometric calculus on the space of shapes with applications in geometry processing and computer vision. The discretization is based on a suitable local approximation of the squared distance, which can be efficiently computed. The approach covers shape morphing and the robust distance evaluation between shapes based on the computation of discrete geodesic paths, shape extrapolation via a discrete exponential map, and natural transfer of geometric details along shape paths using discrete parallel transport. Furthermore, it can be used for the statistical analysis of time indexed shape data in terms of discrete geodesic regression.
The talk will describe how concepts from Riemannian manifold theory are combined with application dependent models of physical dissipation. Furthermore, a rigorous consistency and convergence analysis will be outlined. Applications will be presented in the shape space of viscous fluidic objects and the space of viscous thin shells.


13-05-07

2. Kolloquium mit Live-Schaltung von München nach Berlin

Programm:

16.15   Prof. Jesus de Loera (Univ. of California, Davis / John von Neumann Gastprofessor, TU München)
          "Linearizing Hilbert Nullstellensatz and the Orientability of Matroids"

17.15   Kaffeepause

17.30   Prof. Jürgen Richter-Gebert, TU München
          "Complex matroids: rigidity and syzygies"

Prof. Jesus De Loera

Univ. of California, Davis & John von Neumann Gastprofessor an der TU München

"Linearizing Hilbert Nullstellensatz and the Orientability of Matroids"

Systems of multivariate polynomial equations can be used to model the combinatorial problems. In this way, a problem is feasible (e.g. a graph is 3-colorable, Hamiltonian, etc.) if and only if a certain system of polynomial equations has a solution over an algebraically closed field. Such modeling has being used to prove non-trivial combinatorial results via polynomials (e.g. work by Alon, Tarsi, Karolyi,etc). But the polynomial method is not just for proving theorems but a rather exciting method to compute with combinatorial objects. In this talk we introduce the audience to this new idea. We show that for combinatorial feasibility problems, Hilbert's Nullstellensatz gives a sequence of linear algebra problems, over an algebraically closed field, that eventually decides feasibility. We call this method the Nullstellensatz-Linear Algebra approach or NulLA method for short.

Matroids and oriented matroids play an important role in discrete geometry and questions about orientability and realizability of matroids give rise to highly structured systems of polynomial equations with connections to classical mathematics. In the second part of the talk we connect the study of matroids to the NulLA method. We present systems of polynomial equations that correspond to a matroid M and each of these systems has a zero solution if and only if M is orientable. In this case Hilbert's Nullstellensatz gives us that M is non-orientable if and only if certain certificate to the given polynomials system exists.

Since Richter-Gebert showed that determining if a matroid is orientable is NP-complete, thus determining if these systems have solutions is also NP-complete. However, we also show that these systems of equations and the corresponding linear-algebra relaxations have rather rich structure. For example, it turns out the associated polynomial systems corresponding to M is linear if M is a binary matroid and thus one can determined if binary matroids are orientable much more easily.

This talk is based on joint work with J. Lee, J. Miller, and S. Margulies.

Prof. Jürgen Richter-Gebert

TU München, Lehrstuhl für Geometrie und Visualisierung

„Complex matroids: rigidity and syzygies“

The talk focuses on the interrelation of phirotopes and chirotopes -- the latter forming an abstraction of the signature information of a real vector configuration, the first forming an abstraction of phase information of a complex vector configuration.

We will see that in contrast to the real case, the complex phirotope in general already encodes the geometric location of the vectors of the configuration.

Thus phirotopes are by far more rigid than chirotopes. Within the realm of phirotopes those related to real chirotopes form in a sense a singular situation. This singularity is the reason that chirotopes have a by far richer realization theory than phirotopes.

As a consequence of the rigidity of phirotopes, explicit algebraic relations must exist among the data of a phirotope. We will extract these relations and interpret them as (slightly surprising) results in elementary geometry.


13-04-16

1. Kolloquium per Live-Schaltung aus Berlin

Programm:

16.15    Vladimir Fock (Univ. Strasbourg)
          "Discrete and continuous integrable systems on cluster varieties"

17.15   Kaffeepause

17.30    Yuri B. Suris (TU Berlin)
          "Variational formulation of commuting Hamiltonian flows"

Vladimir Fock, Univ. Strasbourg: "Discrete and continuous integrable systems on cluster varieties"

A.B. Goncharov and R. Kenyon defined a class of integrable systems on cluster varieties enumerated by convex polygons on a plane with integral vertices. Every such system has a family of commuting continuous flows enumerated by integral points inside the polygons and discrete flows enumerated by integral points on the boundary and represented by algebraic transformations. We will present our study of their construction using elementary diagramatic technique introduced by Dylan Thurston. We will also show the relation of this construction with affine Lie groups and show that the integrable systems coincide with well known ones on such groups (for example the relativistic Toda chain). However, we will try to show that construction of Goncharov and Kenyon gives a rather new point of view on integrable systems simplifying the constructions and admitting several generalisations.

Yuri B. Suris, TU Berlin: "Variational formulation of commuting Hamiltonian flows"

We give a complete solution of the following problem: one looks for a function of several variables (interpreted as multi-time) which delivers critical points to the action functionals obtained by integrating a Lagrangian 1-form along any smooth curve in the multi-time. We derive the corresponding multi-time Euler-Lagrange equations and show that, under the multi-time Legendre transformation, they are equivalent to a system of commuting Hamiltonian flows. Involutivity of the Hamiltonian functions turns out to be characteristic for systems of commuting symplectic maps.



13-03-26

Elias Wegert (TU Bergakademie Freiberg)  "Visual Exploration of Complex Functions using Phase Plots"
During the last years it became quite popular to visualize complex (analytic) functions as images. The talk gives an introduction to “phase plots” (or “phase portraits”), which depict a function f directly on its domain by color-coding the argument of f.
Testfoto
The picture shows a phase plot of the Riemann zeta function. Phase portraits are like fingerprints: though part of the information (the modulus) is neglected, meromorphic functions are (almost) uniquely determined by their phase plot – and the first part of the lecture will explain how properties of a function can be recovered. In the second part we investigate the phase plots of some special functions and illustrate several known results (theorems of Jentzsch and Szego ̈, universality of the Riemann zeta function). Finally we give a few examples which demonstrate that phase plots and related “phase diagrams” are useful tools for exploring complex functions in teaching and research.

13-03-19

Martin von Gagern (Doktorand, M10, TUM)  "Hyperbolization of Ornaments"
By changing the combinatorics of the underlying symmetry group, images of Euclidean ornaments can be transported into the hyperbolic plane. It can be seen that conformality is an important requirement when mapping the actual artistic content. Discrete conformal maps make such computations feasible, leading to new pictures derived from existing works of art.

13-02-05

Jan Maas (Bonn University) "Gradient flows and Ricci curvature in discrete analysis"
Since the seminal work of Jordan, Kinderlehrer and Otto, it is known that the heat flow on $R^n$ can be regarded as the gradient flow of the entropy in the Wasserstein space of probability measures. Meanwhile this interpretation has been extended to very general classes of metric measure spaces, but it seems to break down if the underlying space is discrete. In this talk we shall present a new metric on the space of probability measures on a discrete space, based on a discrete Benamou-Brenier formula. This metric defines a Riemannian structure on the space of probability measures and it allows to prove a discrete version of the JKO-theorem. This naturally leads to a notion of Ricci curvature based on convexity of the entropy in the spirit of Lott-Sturm-Villani. We shall discuss how this is related to functional inequalities and present discrete analogues of results from Bakry-Emery and Otto-Villani. This is joint work with Matthias Erbar (Bonn).

13-01-22

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

13-01-15

Gero Friesecke (TUM) "Discrete variational principles in atomistic mechanics"



 

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