SFB-Seminar München

  • Next Occurrence: 20.04.2018, 10:15 - 11:45
  • Nina Otter: Magnitude meets persistence. Homology theories for filtered simplicial sets
  • Type: Seminar
  • Location: TU München

Room: 02.06.011 at Zentrum Mathematik, Boltzmannstraße 3, 85748 Garching
Time: Tuesday 14:15



  • 14:15 - 15:15 (@TUM) Decoding of neural data using cohomological learning, Erik Rybakken (NTNU, Trondheim)
  • We introduce a novel data-driven approach to discover and decode features in the neural code coming from large population neural recordings with minimal assumptions, using cohomological learning. We apply our approach to neural recordings of mice moving freely in a box, where we find a circular feature. We then observe that the decoded value corresponds well to the head direction of the mouse. Thus we capture head direction cells and decode the head direction from the neural population activity without having to process the behaviour of the mouse.


  • 10:15 - 11:45 (Room 03.03.011) Magnitude meets persistence. Homology theories for filtered simplicial sets, Nina Otter (Oxford University, England)
  • The Euler characteristic is an invariant of a topological space that in a precise sense captures its canonical notion of size, akin to the cardinality of a set. The Euler characteristic is closely related to the homology of a space, as it can be expressed as the alternating sum of its betti numbers, whenever the sum is well-defined. Thus, one says that homology categorifies the Euler characteristic. In his work on the generalisation of cardinality-like invariants, Leinster introduced the magnitude of a metric space, a real number that gives the “effective number of points” of the space. Recently, Leinster and Shulman introduced a homology theory for metric spaces, called magnitude homology, which categorifies the magnitude of a space. In this talk I will introduce magnitude and magnitude homology, give an answer to these questions and show that they are intertwined: it is the blurred version of magnitude homology that is related to persistent homology.


  • 14:00 - 15:00 The Matroid of Barcodes: Combinatorial Foundations in TDA, Greg Henselman (Princeton University)
  • Topological data analysis (TDA) is a robust field of mathematical data science specializing in complex, noisy, and high-dimensional data. While the elements of modern TDA have existed since the mid-1980’s, applications over the past decade have seen a dramatic increase in systems analysis, engineering, medicine, and the sciences. Two of the primary challenges in this field regard modeling and computation: what do topological features mean, and are they computable? While these questions remain open for some of the simplest structures considered in TDA — homological persistence modules and their indecomposable submodules — in the past two decades researchers have made great progress in algorithms, modeling, and mathematical foundations through diverse connections with other fields of mathematics. This talk will give a first perspective on the idea of matroid theory as a framework for unifying and relating some of these seemingly disparate connections (e.g. with quiver theory, classification, and algebraic stability), and some questions that the fields of matroid theory and TDA may mutually pose to one another. No expertise in homological persistence or general matroid theory will be assumed, though prior exposure to the definition of a matroid and/or persistence module may be helpful.


  • 14:15 - 15:15 Non-Negative Dimensionality Reduction in Signal Separation, Sara Krause-Solberg (TU München)
  • 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.


  • 14:15 - 15:30 (with Live-Broadcast to the TU Berlin) Strong Equivalence of the Interleaving and Functional Distortion Distances for Reeb Graphs, Elizabeth Munch (University at Albany, NY)
  • The Reeb graph is a construction on a topological space with a real valued function which encodes information about the changing connected components of level sets of the function. It has found many uses in the applied setting, particularly in computer graphics and topological data analysis where the Reeb graph acts as a signature for the function/space pair. As with any method where we want to study data, it is necessary to understand the behavior of the signature in the presence of noise. Thus, several metrics have recently been defined for Reeb graphs; these include the interleaving distance and the functional distortion distance. We will discuss the definitions of each of these metrics as well as recent work showing that they are strongly equivalent.


  • 14:15 - 15:15 A tour through modern methods in multiple time scale dynamics, Christian Kühn (TU Wien)
  • 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.


  • 14:15 - 15:15 Variational Integrators in Plasma Physics, Michael Kraus (IPP)
  • 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.


  • 14:15 - 16:30 Optimal topological simplification of discrete functions on surfaces, Carsten Lange (TU München)
  • 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.


  • 14:15 - 15:15 (Thursday) Topological chaos, braiding and bifurcation of almost-cyclic sets, Shane Ross (Virginia Tech)
  • 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.
  • 15:30 - 16:30 (Thursday) Transfer operator based numerical analysis of time-dependent transport, Kathrin Padberg-Gehle (TU Dresden)
  • 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.


  • 16:15 - 17:15 The Real Dynamics of Bieberbach’s Example, Sandra Hayes (Queen College, City University of New York)
  • 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.


  • 16:15 - 17:15 Visual Exploration of Complex Functions, Elias Wegert (TU Freiberg)
  • 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. 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, meromor- phic 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 re- sults (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.


  • 16:15 - 17:15 Gradient flows and Ricci curvature in discrete analysis, Jan Maas (Universität Bonn)
  • 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).


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


  • 16:15 - 17:15 Discrete variational principles in atomistic mechanics, Gero Friesecke