Past Applied Topology Seminar

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. 

The TMD algorithm (Kanari et al. 2018) computes the barcode of a neuron (tree) with respect to the radial or path distance from the soma (root). We are interested in the inverse problem: how to understand the space of trees that are represented by the same barcode. Our tool to study this spaces is the stochastic TNS algorithm (Kanari et al. 2020) which generates trees from a given barcode in a biologically meaningful way. 

I will present some theoretical results on the space of trees that have the same barcode, as well as the effect of adding noise to the barcode. In addition, I will provide a more combinatorial perspective on the space of barcodes, expressed in terms of the symmetric group. I will illustrate these results with experiments based on the TNS.

This is joint work with L. Kanari and K. Hess. 

One of the main concerns in social network science is the study of positions and roles. By "position" social scientists usually mean a collection of actors who have similar ties to other actors, while a "role" is a specific pattern of ties among actors or positions. Since the 1970s a lot of research has been done to develop these concepts in a rigorous way. An open question in the field is whether it is possible to perform role and positional analysis simultaneously. In joint work in progress with Mason Porter we explore this question by proposing a framework that relies on the principle of functoriality in category theory. In this talk I will introduce role and positional analysis, present some well-studied examples from social network science, and what new insights this framework might give us.

Contagion maps are a family of maps that map nodes of a network to points in a high-dimensional space, based on the activations times in a threshold contagion on the network. A point cloud that is the image of such a map reflects both the structure underlying the network and the spreading behaviour of the contagion on it. Intuitively, such a point cloud exhibits features of the network's underlying structure if the contagion spreads along that structure, an observation which suggests contagion maps as a viable manifold-learning technique. We test contagion maps as a manifold-learning tool on several different data sets, and compare its performance to that of Isomap, one of the most well-known manifold-learning algorithms. We find that, under certain conditions, contagion maps are able to reliably detect underlying manifold structure in noisy data, when Isomap is prone to noise-induced error. This consolidates contagion maps as a technique for manifold learning. 

Synchronization is a collective phenomenon that pervades the natural systems from neurons to fireflies. In a network, synchronization of the dynamical variables associated to the nodes occurs when nodes are coupled to their neighbours as captured by the Kuramoto model. However many complex systems include also higher-order interactions among more than two nodes and sustain dynamical signals that might be related to higher-order simplices such as nodes of triangles. These dynamical topological signals include for instance fluxes which are dynamical variables associated to links.

In this talk I present a new topological approach [1] to synchronization on simplicial complexes. Here the theory of synchronization is combined with topology (specifically Hodge theory) for formulating the higher-order Kuramoto model that uses the higher-order Laplacians and provides the main synchronization route for topological signals. I will show that the dynamics defined on links can be projected to a dynamics defined on nodes and triangles that undergo a synchronization transition and I will discuss how this procedure can be immediately generalized for topological signals of higher dimension. Interestingly I will show that when the model includes an adaptive coupling of the two projected dynamics, the transition becomes explosive, i.e. synchronization emerges abruptly.

This model can be applied to study synchronization of topological signals in the brain and in biological transport networks as it proposes a new set of topological transformations that can reveal collective synchronization phenomena that could go unnoticed otherwise.

[1] Millán, A.P., Torres, J.J. and Bianconi, G., 2019. Explosive higher-order Kuramoto dynamics on simplicial complexes. Physical Review Letters (in press) arXiv preprint arXiv:1912.04405.

Eigenvalue-eigenvector pairs of combinatorial graph Laplacians are extensively used in graph theory and network analysis. It is well known that the spectrum of the Laplacian L of a given graph G encodes aspects of the geometry of G  - the multiplicity of the eigenvalue 0 counts the number of connected components while the second smallest eigenvalue (called the Fiedler eigenvalue) quantifies the well-connectedness of G . In network analysis, one uses Laplacian eigenvectors associated with small eigenvalues to perform spectral clustering. In graph signal processing, graph Fourier transforms are defined in terms of an orthonormal eigenbasis of L. Eigenvectors of L also play a central role in graph neural networks.

Motivated by this we study eigenvalue-eigenvector pairs of Laplacians of random graphs and their potential use in TDA. I will present simulation results on what persistent homology barcodes of Bernoulli random graphs G(n, p) look like when we use Laplacian eigenvectors as filter functions. Also, I will discuss the conjectures made from the simulations as well as the challenges that arise when trying to prove them. This is work in progress.
 

The last fifty years of dynamical systems theory have established that dynamical systems can exhibit extremely complex behavior with respect to both the system variables (chaos theory) and parameters (bifurcation theory). Such complex behavior found in theoretical work must be reconciled with the capabilities of the current technologies available for applications. For example, in the case of modelling biological phenomena, measurements may be of limited precision, parameters are rarely known exactly and nonlinearities often cannot be derived from first principles. 

The contrast between the richness of dynamical systems and the imprecise nature of available modeling tools suggests that we should not take models too seriously. Stating this a bit more formally, it suggests that extracting features which are robust over a range of parameter values is more important than an understanding of the fine structure at some particular parameter.

The goal of this talk is to present a high-level introduction/overview of computational Conley-Morse theory, a rigorous computational approach for understanding the global dynamics of complex systems.  This introduction will wander through dynamical systems theory, algebraic topology, combinatorics and end in game theory.

We present a recipe for constructing filter functions on graphs with parameters that can optimised by gradient descent. This recipe, based on graph Laplacians and spectral wavelet signatures, do not require additional data to be defined on vertices. This allows any graph to be assigned a customised filter function for persistent homology computations and data science applications, such as graph classification. We show experimental evidence that this recipe has desirable properties for optimisation and machine learning pipelines that factors through persistent homology. 

In the last years complex networks tools contributed to provide insights on the structure of research, through the study of collaboration, citation and co-occurrence networks. The network approach focuses on pairwise relationships, often compressing multidimensional data structures and inevitably losing information. In this paper we propose for the first time a simplicial complex approach to word co-occurrences, providing a natural framework for the study of higher-order relations in the space of scientific knowledge. Using topological methods we explore the conceptual landscape of mathematical research, focusing on homological holes, regions with low connectivity in the simplicial structure. We find that homological holes are ubiquitous, which suggests that they capture some essential feature of research practice in mathematics. Holes die when a subset of their concepts appear in the same article, hence their death may be a sign of the creation of new knowledge, as we show with some examples. We find a positive relation between the dimension of a hole and the time it takes to be closed: larger holes may represent potential for important advances in the field because they separate conceptually distant areas. We also show that authors' conceptual entropy is positively related with their contribution to homological holes, suggesting that polymaths tend to be on the frontier of research.

[[{"fid":"51386","view_mode":"default","fields":{"format":"default"},"type":"media","attributes":{"class":"file media-element file-default"},"link_text":"Barbara Mahler"}]]

We will discuss the algebraicity of two quantities central to the computation of persistent homology. We will also connect persistent homology and algebraic optimization. Namely, we will express the degree corresponding to the distance variable of the offset hypersurface in terms of the Euclidean distance degree of the starting variety, obtaining a new way to compute these degrees. Finally, we will describe the non-properness locus of the offset construction and use this to describe the set of points that are topologically interesting (the medial axis and center points of the bounded components of the complement of the variety) and relevant to the computation of persistent homology.

Persistent homology is an algebraic tool for quantifying topological features of shapes and functions, which has recently found wide applications in data and shape analysis. In the first and introductory part of this talk I recall the underlying ideas and basic concepts of this very active field of research. In the second part, I plan to sketch a concrete application of this concept to digital image processing. 

This talk focuses on algebraic and combinatorial-topological problems motivated by neuroscience. Neural codes allow the brain to represent, process, and store information about the world. Combinatorial codes, comprised of binary patterns of neural activity, encode information via the collective behavior of populations of neurons. A code is called convex if its codewords correspond to regions defined by an arrangement of convex open sets in Euclidean space. Convex codes have been observed experimentally in many brain areas, including sensory cortices and the hippocampus,where neurons exhibit convex receptive fields. What makes a neural code convex? That is, how can we tell from the intrinsic structure of a code if there exists a corresponding arrangement of convex open sets?

This talk describes how to use tools from combinatorics and commutative algebra to uncover a variety of signatures of convex and non-convex codes.

This talk is based on joint works with Aaron Chen and Florian Frick, and with Carina Curto, Elizabeth Gross, Jack Jeffries, Katie Morrison, Mohamed Omar, Zvi Rosen, and Nora Youngs.

How can we adapt the Topological Data Analysis (TDA) pipeline to use several filter functions at the same time? Two orthogonal approaches can be considered: (1) running the standard 1-parameter pipeline and doing statistics on the resulting barcodes; (2) running a multi-parameter version of the pipeline, still to be defined. In this talk I will present two recent contributions, one for each approach. The first contribution considers intrinsic compact metric spaces and shows that the so-called Persistent Homology Transform (PHT) is injective over a dense subset of those. When specialized to metric graphs, our analysis yields a stronger result, namely that the PHT is injective over a subset of full measurem which allows for sufficient statistics. The second contribution investigates the bi-parameter version of the TDA pipeline and shows a decomposition result "à la Crawley-Boevey" for a subcategory of the 2-parameter persistence modules called "exact modules". This result has an impact on the study of interlevel-sets persistence and on that of sheaves of vector spaces on the real line. 

This is joint work with Elchanan Solomon on the one hand, with Jérémy Cochoy on the other hand.

Steady state chemical reaction models can be thought of as algebraic varieties whose properties are determined by the network structure. In experimental set-ups we often encounter the problem of noisy data points for which we want to find the corresponding steady state predicted by the model. Depending on the network there may be many such points and the number of which is given by the euclidean distance degree (ED degree). In this talk I show how certain properties of networks relate to the ED degree and how the runtime of numerical algebraic geometry computations scales with the ED degree.

Single parameter persistent homology has proven to be a useful data analytic tool and single parameter persistence modules enjoy a concise description as a barcode, a complete invariant. [Bubenik, 2012] derived a topological summary closely related to the barcode called the persistence landscape which is amenable to statistical analysis and machine learning techniques.

The theory of multidimensional persistence modules is presented in [Carlsson and Zomorodian, 2009] and unlike the single parameter case where one may associate a barcode to a module, there is not an analogous complete discrete invariant in the multiparameter setting. We propose an incomplete invariant derived from the rank invariant associated to a multiparameter persistence module, which generalises the single parameter persistence landscape in [Bubenik, 2012] and satisfies similar stability properties with respect to the interleaving distance. Our invariant naturally lies in a Banach Space and so is naturally endowed with a distance function, it is also well suited to statistical analysis since there is a uniquely defined mean associated to multiple landscapes. We shall present computational examples in the 2-parameter case using the RIVET software presented in [Lesnick and Wright, 2015].

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.

The aim of applied topology is to use and develop topological methods for applied mathematics, science and engineering. One of the main tools is persistent homology, an adaptation of classical homology, which assigns a barcode, i.e., a collection of intervals, to a finite metric space. Because of the nature of the invariant, barcodes are not well adapted for use by practitioners in machine learning tasks. We can circumvent this problem by assigning numerical quantities to barcodes, and these outputs can then be used as input to standard algorithms. I will explain how we can use tropical-like functions to coordinatize the space of persistence barcodes. These coordinates are stable with respect to the bottleneck and Wasserstein distances. I will also show how they can be used in practice.

Phase transitions and critical phenomena are ubiquitous in Nature. They permeate physics, chemistry, biology and complex systems in general, and are characterized by the role of correlations and fluctuations of many degrees of freedom. From a mathematical viewpoint, in the vicinity of a critical point, thermodynamic quantities exhibit singularities and scaling properties. Theoretical attempts to describe classical phase transitions using tools from differential topology and Morse theory provided strong arguments pointing that a phase transition may emerge as a consequence of topological changes in the configuration space around the critical point.

On the other hand, much work was done concerning the topology of networks which spontaneously emerge in complex systems, as is the case of the genome, brain, and social networks, most of these built intrinsically based on measurements of the correlations among the constituents of the system.

We aim to transpose the topological methodology previously applied in n-dimensional manifolds, to describe phenomena that emerge from correlations in a complex system, in which case Hamiltonian models are hard to invoke. The main idea is to embed the network onto an n-dimensional manifold and to study the equivalent to level sets of the network according to a filtration parameter, which can be the probability for a random graph or even correlations from fMRI measurements as height function in the context of Morse theory.  By doing so, we were able to find topological phase transitions either in random networks and fMRI brain networks.  Moreover, we could identify high-dimensional structures, in corroboration with the recent finding from the blue brain project, where neurons could form structures up to eleven dimensions.The efficiency and generality of our methodology are illustrated for a random graph, where its Euler characteristic can be computed analytically, and for brain networks available in the human connectome project.  Our results give strong arguments that the Euler characteristic, together with the distributions of the high dimensional cliques have potential use as topological biomarkers to classify brain Networks. The above ideas may pave the way to describe topological phase transitions in complex systems emerging from correlation data.

I will present Vidit Nanda's paper "Local homology and stratification" (https://arxiv.org/abs/1707.00354), and briefly explain how in my master thesis I am applying ideas from the paper to study word embedding problems.

Abstract of the paper:  We outline an algorithm to recover the canonical (or, coarsest) stratification of a given regular CW complex into cohomology manifolds, each of which is a union of cells. The construction proceeds by iteratively localizing the poset of cells about a family of subposets; these subposets are in turn determined by a collection of cosheaves which capture variations in cohomology of cellular neighborhoods across the underlying complex. The result is a finite sequence of categories whose colimit recovers the canonical strata via (isomorphism classes of) its objects. The entire process is amenable to efficient distributed computation.
 

This is the continuation of a discussion of  Ezra Miller's paper https://arxiv.org/abs/1709.08155.

This is the first part of a discussion of  Ezra Miller's paper https://arxiv.org/abs/1709.08155.