Applied Topology Seminar

Ollivier Ricci curvature is a notion originated from Riemannian Geometry and suitable for applying on different settings from smooth manifolds to discrete structures such as (directed) hypergraphs. In the past few years, alongside Forman Ricci curvature, this curvature as an edge based measure, has become a popular and powerful tool for network analysis. This notion is defined based on optimal transport problem (Wasserstein distance) between sets of probability measures supported on data points and can nicely detect some important features such as clustering and sparsity in their structures. After introducing this notion for (directed) hypergraphs and mentioning some of its properties, as one of the main recent applications, I will present the result of implementation of this tool for the analysis of chemical reaction networks. 

In this talk I will present an efficient algorithm to produce a provably dense sample of a smooth compact algebraic variety. The procedure is partly based on computing bottlenecks of the variety. Using geometric information such as the bottlenecks and the local reach we also provide bounds on the density of the sample needed in order to guarantee that the homology of the variety can be recovered from the sample.

In this talk, I will give a brief introduction to level-set methods for image analysis. I will then describe an application of level-sets to the construction of filtrations for persistent homology computations. I will present several case studies with various spatial data sets using this construction, including applications to voting, analyzing urban street patterns, and spiderwebs. I will conclude by discussing the types of data which I might imagine such methods to be suitable for analyzing and suggesting a few potential future applications of level-set based computations.

In this talk, I will present the theory of combinatorial multivector fields for finite topological spaces, the main subject of my thesis. The idea of combinatorial vector fields came from Forman and emerged naturally from discrete Morse theory. Lately, Mrozek generalized it to the multivector fields theory for Lefschetz complexes. In our work, we simplified and extended it to the finite topological spaces settings. We developed a combinatorial counterpart for dynamical objects, such as isolated invariant sets, isolating neighbourhoods, Conley index, limit sets, and Morse decomposition. We proved the additivity property of the Conley index and the Morse inequalities. Furthermore, we applied persistence homology to study the evolution and the stability of Morse decomposition. In the last part of the talk, I will show numerical results and potential future directions from a data-analysis perspective. 

Research on robot manipulation has focused, in recent years, on grasping everyday objects, with target objects almost exclusively rigid items. Non–rigid objects, as textile ones, pose many additional challenges with respect to rigid object manipulation. In this seminar we will present how we can employ topology to study the ``state'' of a rectangular textile using the configuration space of $n$ points on the plane. Using a CW-decomposition of such space, we can define for any mesh associated with a rectangular textile a vector in an euclidean space with as many dimensions as the number of regions we have defined. This allows us to study the distribution of such points on the cloth and define meaningful states for detection and manipulation planning of textiles. We will explain how such regions can be defined and computationally how we can assign to any mesh the corresponding region. If time permits, we will also explain how the CW-structure allows us to define more than just euclidean distance between such mesh-distributions.

High dimensionality of biological data is a crucial element that is in need of different methods to unravel their complexity. The current and rich biomedical material that hospitals generate every other day related to cancer detection can benefit from these new techniques. This is the case of diseases such as Acute Lymphoblastic Leukaemia (ALL), one of the most common cancers in childhood. Its diagnosis is based on high-dimensional flow cytometry tumour data that includes immunophenotypic expressions. Not only the intensity of these markers is meaningful for clinicians, but also the shape of the points clouds generated, being then fundamental to find leukaemic clones. Thus, the mathematics of shape recognition in high dimensions can turn itself as a critical tool for this kind of data. This is why we resort to the use of tools from Topological Data Analysis such as Persistence Homology.

Given that ALL relapse incidence is of almost 20% of its patients, we provide a methodology to shed some light on the shape of flow cytometry data, for both relapsed and non-relapsed patients. This is done so by combining the strength of topological data analysis with the versatility of machine learning techniques. The results obtained show us topological differences between both patient sets, such as the amount of connected components and 1-dimensional loops. By means of the so-called persistence images, and for specially selected immunophenotypic markers, a classification of both cohorts is obtained, highlighting the need of new methods to provide better prognosis. 

This talk is motivated by the following question: "how can one reconstruct the geometry of a state space given a collection of observed time series?" A well-studied technique for metric fusion is Similarity Network Fusion (SNF), which works by mixing random walks. However, SNF behaves poorly in the presence of correlated noise, and always reconstructs an intrinsic metric. We propose a new methodology based on delay embeddings, together with a simple orthogonalization scheme that uses the tangency data contained in delay vectors. This method shows promising results for some synthetic and real-world data. The authors suspect that there is a theorem or two hiding in the background -- wild speculation by audience members is encouraged. 

Persistence theory provides useful tools to extract information from real-world data sets, and profits of techniques from different mathematical disciplines, such as Morse theory and quiver representation. In this seminar, I am going to present a new approach for studying persistence theory using model categories. I will briefly introduce model categories and then describe how to define a model structure on the category of the tame parametrised chain complexes, which are chain complexes that evolve in time. Using this model structure, we can define new invariants for tame parametrised chain complexes, which are in perfect accordance with the standard barcode when restricting to persistence modules. I will illustrate with some examples why such an approach can be useful in topological data analysis and what new insight on standard persistence can give us. 

Distinguishing classes of surfaces in $\mathbb{R}^n$ is a task which arises in many situations. There are many characteristics we can use to solve this classification problem. The Persistent Homology Transform allows us to look at shapes in $\mathbb{R}^n$ from $S^{n-1}$ directions simultaneously, and is a useful tool for surface classification. Using the Julia package DiscretePersistentHomologyTransform, we will look at some example curves in $\mathbb{R}^2$ and examine distinguishing features.