Meeting of the Centre for Topological Data Analysis on 13-15 September 2021

photo of Materials Innovation Factory

Overall plan on 13-17 September 2021 (online)

  • The meeting was planned in a hybrid form at the MIF, but will be only on Zoom because of too strict Covid-related rules.
  • To register and receive a Zoom link, please e-mail your full name and affiliation to Dr Vitaliy Kurlin.
  • Monday afternoon 13th - Wednesday noon 15th September : annual meeting of the Centre for TDA.
  • Wednesday afternoon 15th - Friday noon 17th September : second annual conference MACSMIN 2021.
  • Friday afternoon 17th September (2-4.30pm) : meeting of the Scientific Advisory Board (SAB) of the centre.
  • The meeting is joint with the UK network Applied Algebraic Topology funded by the London Mathematical Society. We have a modest budget to cover some travel and accommodation, please e-mail Vitaliy Kurlin if you are interested.

Daily schedule : Monday 13th, Tuesday 14th, Wednesday 15th, SAB on Friday afternoon 17th

  • Monday afternoon 13th September 2021 (all UK times)
    • 14.00-14.40 Ran Levi. An application of neighbourhoods in directed graphs in the classification of binary dynamics. Video
      Abstract. A binary state on a graph means an assignment of binary values to its vertices. For example, if one encodes a network of spiking neurons as a directed graph, then the spikes produced by the neurons at an instant of time is a binary state on the encoding graph. Allowing time to vary and recording the spiking patterns of the neurons in the network produces an example of a binary dynamics on the encoding graph, namely a one-parameter family of binary states on it. The central object of study in this talk is the neighbourhood of a vertex v in a graph G, namely the subgraph of G that is generated by v and all its direct neighbours in G. We present a topological/graph theoretic method for extracting information out of binary dynamics on a graph, based on a selection of a relatively small number of vertices and their neighbourhoods. As a test case we demonstrate an application of the method to binary dynamics that arises from sample activity on the Blue Brain Project reconstruction of cortical tissue of a rat.
    • 14.50-15.30 Gesine Reinert. Credit risk prediction using TDA. Abstract (docx). Video
    • 15.40-16.20 Florian Pausinger. Long shortest vectors in low dimensional lattices. Abstract (pdf). Video
  • Tuesday morning 14th September 2021
    • 9.30-10.10 Michael Farber. Topology and automated decision making. Video
      Abstract. I will discuss the problem of decision making when an algorithm has to make choice out of a continuum of possibilities. Such situation arises in robotics, in the problem of designing motion planning algorithms. Tools of algebraic topology and cohomology theory play a crucial role. In my talk I will discuss some old and new mathematical results inspired by this problem. In particular I will describe joint work with D. Cohen and S. Weinberger on parametrised motion planning algorithms. No prior knowledge of the subject will be assumed.
    • 10.20-11.00 Omer Bobrowski. Persistent cycle registration and bootstrap. Video
      Abstract. The motivating question for this talk is the following: Suppose that we are given a persistence diagram computed from random data. Pick a feature (cycle) of interest in the diagram. Can we determine whether this feature represents a "statistically significant" phenomenon, or was it generated purely by chance? We propose a novel approach to answer this question using a bootstrap-like method. The key idea is that if we resample the data, we expect features that represent essential phenomena in the underlying system to appear frequently in the resamples. The main ingredient in this framework is a new "cycle-registration" method that allows us to match persistent cycles that appear in two different samples. This is joint work with Yohai Reani (Technion).
    • 11.10-11.50 Vidit Nanda. Conormal spaces and Whitney stratifications
      I will describe a new algorithm for computing Whitney stratifications of complex projective varieties, their flags, and algebraic maps between them. This is joint work with Martin Helmer. Video
  • Tuesday afternoon 14th September 2021
    • 14.00-14.40 Ginestra Bianconi. The topological Dirac operator and the dynamics of topological signals. Video
      Topological signals associated not only to nodes but also to links and to the higher dimensional simplices of simplicial complexes are attracting increasing interest in signal processing, machine learning and network science. Typically, topological signals of a given dimension are investigated and filtered using the corresponding higher-order Laplacian. In this talk I will cover notable spectral properties of higher-order Laplacians and I will reveal how these properties affect higher-order diffusion and higher-order synchronization dynamics. Moreover, I will introduce the topological Dirac operator that can be used to process simultaneously topological signals of different dimensions. I will discuss the main spectral properties of the Dirac operator defined on networks, simplicial complexes and multiplex networks, and their relation to higher-order Laplacians. Finally I will show how the higher-order Laplacians can be used to define a higher-order Kuramoto model and how the Dirac operator allows to define the topological synchronization of locally coupled topological signals defined on nodes and links of a network.
    • 14.50-15.30 Jacek Brodzki. Topological characteristics of time-evolution of physical systems. Video
      In this talk I will describe structural heterogeneity, a new topological characteristic for semi-ordered materials that captures their degree of organisation and tracks their time-evolution. This new invariant allows us to detect the phase transitions between ordered and disordered states. I will present applications of this method in a specific physical system, but its flexibility will allow it to become a useful tool in the study of the dynamics of semi-ordered physical systems.
    • 15.40-16.20 David Mendez. A directed persistent homology theory for dissimilarity functions.
      Abstract. Persistent homology is one of the most successful tools in Topological Data Analysis, having been applied in numerous scientific domains such as medicine, neuroscience, robotics, and many others. However, a fundamental limitation of persistent homology is its inability to incorporate directionality. In this talk we will introduce a theory of persistent homology for directed simplicial complexes which detects directed cycles in odd dimensions. To do so, we introduce a homology theory with coefficients in semirings for these complexes: by explicitly removing additive inverses, we can detect directed cycles algebraically. We will also exhibit some of the features of this persistent homology theory, including its stability and how the obtained persistent diagrams relate to those obtained from persistent homology with ring coefficients. We will end the talk by highlighting some of the computational challenges towards the effective computation of the directed persistent diagram of a point cloud.
  • Wednesday morning 15th September 2021 :
    • 9.30-10.10 Heather Harrington. Algebraic systems biology: topology links models and biological data. Video
      Abstract. Signalling pathways in molecular biology can be modelled by polynomial dynamical systems. I will present models describing biological systems involved in development and cancer. I will overview approaches to analyse these models with data using computational geometry, topology, statistics and dynamics. These methods can provide new insights to better understand model parameter values and biological insights, such as how changes at the molecular scale (e.g. molecular mutations) result in kinetic differences and observed phenotypic changes (e.g. mutations in fruit fly wings).
    • 10.20-11.00 Jeffrey Giansiracusa. Persistent homology and topological phase transitions. Video
      Abstract. A phase transition is when a statistical system undergoes an abrupt change as a parameter such as temperature passes across a critical value. Some phase transitions are relatively easy to understand, such as the transition in the Ising model from a magnetised state to a disordered state. Some are more challenging and involve topological objects such as vortices or knots, as in the famous Berezinskii-Kosterlitz-Thouless transition in the 2D XY-model; these are called topological phase transitions. And some phase transitions are not yet understood but are expected to be topological, such as the transition in QCD (the quantum field theory that describes the strong nuclear force inside atomic nuclei) from quarks weakly interacting at high energies to confinement at low energies where we only ever see quarks bound into pairs and triples. In this talk I will report on work with Nick Sale and Biagio Lucini to use persistent homology to detect and measure properties of topological phase transitions without a priori knowledge of the details of the topological objects involved.
    • 11.10-11.50 Lisbeth Fajstrup. TDA and amorphous materials – persistent homology taking non-persistent classes seriously.
      Abstract. When using TDA to study data, the short lived features are often disregarded as noise; they are not persistent. If the points at the center of growing balls are actual atoms, points near the diagonal in the persistence diagram are in fact information. We accumulate the information in the Accumulated Persistence Function, APF, and use this, together with representatives of the persistent homology classes to extract characteristics of e.g. sodium silicate glasses. I will explain the method and some of the results obtained so far. It is an ongoing project.
  • From 14.00 on Wednesday to 12.00 on Friday 17th September : MACSMIN 2021.
  • Friday afternoon 17th September 2021 : online SAB between 14.00 - 16.30.

Organisers : Dr Vitaliy Kurlin's group

Dr Nicola Kirkham (Oxford) kindly helps with the organization. Main contact for enquiries : e-mail Dr Vitaliy Kurlin.