Research
Research
(In reverse chronological order)
Noncommutative Model Selection and the Data-Driven Estimation of Real Cohomology Groups (with Araceli Guzmán-Tristán and Eduardo Velázquez-Richards), arxiv.org:2411.19894, 2024, submitted
Noncommutative Model Selection for Data Clustering and Dimension Reduction Using Relative von Neumann Entropy (with Araceli Guzmán-Tristán), arxiv.org:2411.19902, submitted
A New Construction of the Vietoris-Rips Complex, arXiv:2301.07191, submitted
Semi-coarse Spaces, Homotopy and Homology (with Jonathan Treviño), Advances in Applied Mathematics, volume 167 (2025)
Cofibration and Model Category Structures for Discrete and Continuous Homotopy, arXiv:2209.13510, submitted
Grothendieck Topologies and Sheaves on Čech Closure Spaces, arXiv:2109.13867, submitted
Kunneth Theorems for Vietoris-Rips Homology (with Alejandra Trujillo), Acta Mathematica Hungarica, volume 166, pages 239-253 (2022)
Vietoris-Rips Homology Theory for Semi-Uniform Spaces, arXiv:2008.05739, submitted
A Topological Approach to Spectral Clustering, Foundations of Data Science, Volume 3, Number 1, 2021
Čech Closure Spaces: A Unified Framework for Discrete and Continuous Homotopy, Topology and Its Applications, Volume 296, Number 1, pp 1-41, 2021
Coisotropic Hofer-Zehnder capacities and non-squeezing for relative embeddings (with Samuel Lisi), Journal of Symplectic Geometry, Volume 18, Number 3, pp. 819-865, 2020
Lagrangian Blow-ups, blow-downs, and applications to real packing, Journal of Symplectic Geometry, Volume 12, Number 4, pp. 725-789, 2014
Topología aplicada: nuevos fundamentos teóricos, hacia nuevas prácticas, Seminario de Estudiantes, CIMAT, Guanajuato, México, May, 2025
Resumen: En esta plática, primero daremos un resumen de la teoría de homotopía discreta del punto de vista de los espacios de cerradura y pseudotopológicos, y algunos de sus invariantes. En una segunda parte, presentamos algunas nuevas propuestas para la estimación de la cohomología real de una variedad desde una muestra.
Topología algebraica dónde no se debe, Muestrario Matemático, CIMAT, Guanajuato, México, May, 2025
Resumen: En esta plática, primero daremos un resumen de la teoría de homotopía discreta del punto de vista de los espacios de cerradura y pseudotopológicos, y algunos de sus invariantes. En una segunda parte, presentamos algunas nuevas propuestas para la estimación de la cohomología real de una variedad desde una muestra.
El grupo fundamental, discreto y continuo, Escuela Temática de Geometría Diferencial y Topología, CIMAT, Guanajuato, México, May, 2025
Resumen: Damos una introducción a las propiedades del grupo fundamental en complejos CW, y mostramos como se puede extender esta construcción hacía espacios metricos finitos con una escala destacada como uno se encuentra en el análisis topologico de datos.
Towards non-commutative inference for topological data analysis, Seminario de Matrices Aleatorias y Probabilidad No Conmutativa, CIMAT, Guanajuato, México, May 20th, 2025
Resumen: Despite its suggestive name, current methods in topological data analysis typically involve the computation and analysis of one or more decidedly non-topological metric invariants such as persistent homology or Euler characteristic curves. In this work, we return to the question of how to estimate genuinely topological invariants of a closed manifold from a set of sample points, and we present several newly developed methods which conjecturally compute the real cohomology groups of a metric-measure space with high probability, and which show good performance in numerical examples. Our strategy involves choosing an operator semi-group acting on R^n, where n is the number of sample points, which best reflects the properties of the heat semigroup of the target manifold, and we highlight several points of contact with random matrices and non-commutative probability which motivate current and future work. This work is joint with Araceli Guzmán-Tristan and Eduardo Velázquez-Richards
Sheaf theory for data and combinatorics, CIMA, Benemérito Universidad Autonóma de Puebla, Puebla, México, September 2nd, 2024
Abstract: In this talk, we will describe how to construct sheaf theory on graphs in order to facilitate its use in data analysis and combinatorics. The approach we take is by generalizing sheaf theory on topological spaces to categories which contain both reflexive graphs and topological spaces as subcategories. There are a number of choices of such categories, and we will describe several of them, the advantages and disadvantages of working in each, as well as the construction of a Grothendieck topos on Cech closure spaces, the simplest of these categories.
Homotopy and sheaf theory for data and combinatorics, University of Florida Geometry and Topology Seminar, University of Florida, Gainsville, FL, USA, May 2024
Abstract: In this talk, we'll give a survey of the current state of a program to 'discretize' algebraic topology - and in particular homotopy and sheaf theory - in order to facilitate their use in data analysis and combinatorics. The approach we take is by generalizing homotopy and sheaf theory on topological spaces to categories which contain both reflexive graphs and topological spaces as subcategories. There are a number of choices of such categories, and we will describe several of them, the advantages and disadvantages of working in each, and, time permitting, demonstrate their use in answering several questions in topological data analysis whose solutions are either overly technical or have been unattainable without them.
Sheaves for data and graphs through closure spaces, University of Minnesota Topology Seminar, online, March 2024
Abstract: We present a new approach to sheaf theory for data sets by constructing a Grothendieck topology associated to a Cech closure space. A particularly attractive aspect of this theory is that it applies to many of the major classes of interest to applications: directed and undirected graphs, finite simplicial complexes, and metric spaces decorated with a privileged scale, and on topological spaces, the resulting sheaf cohomology is isomorphic to the usual one. In this talk, we will introduce Cech closure spaces and discuss the the construction and its basic properties.
Recent Developments in the Algebraic Topology of Mesoscopic Spaces, Joint Mathematics Meetings, San Francisco, USA, January 2024
Abstract: We will discuss recent work on the development of algebraic topology for mesoscopic spaces, or metric spaces decorated with a preferred scale, which we argue are the natural spaces of interest for applied topology. We will describe how doing homotopy theory for these spaces requires working in a category more general than topological or uniform spaces, and we will discuss relationships between this work and discrete homotopy theory.
Sheaves on data through closure spaces, Northeastern University Topology Seminar, Northeastern University, Boston, MA, USA, November 2023
Abstract: We present a new approach to sheaf theory for data sets by constructing a Grothendieck topology associated to a closure space. Two particularly attractive aspects of this theory are that, first, it applies unchanged to directed graphs, and, second, the higher-dimensional sheaf cohomology of graphs may be non-trivial, making it a good candidate method for discretizing certain sheaves on manifolds using point clouds.
Algebraic Topology of Mesoscopic Spaces, AATRN Seminar, online, March 29th, 2023. Video
Abstract: In this talk, we introduce the notion of a mesoscopic space: a metric space decorated with a privileged scale, and we survey recent developments in the algebraic topology of such spaces. Our approach begins with the observation that mesoscopic spaces, together with reflexive graphs and topological spaces, all naturally induce pseudotopological spaces (and in particular Cech closure spaces), and we describe how homotopy and homology may be studied in these categories. We then show that, by adapting this perspective to semi-uniform spaces, we are able to construct a version of the metric cohomology of Hausmann at arbitrary scales, which allows us to prove results on homotopy invariance and excision for Vietoris-Rips (co)homology.
An Overview of Hodge Theory, Smooth and Discrete, Banff Workshop: Applications of Hodge Theory on Networks, online, January 31st, 2023. Video
Abstract: I will give an overview of the basic ideas of Hodge theory and the Hodge decomposition for smooth manifolds and simplicial complexes, including several prominent example applications.
Applied Topology from the Classical Point of View, GEOTOP-A Seminar. online, November 21st, 2021. Video
Abstract: We generalize several basic notions in algebraic topology to categories which contain both topological spaces classically treated by classical homotopy theory as well as more discrete and combinatorial spaces of interest in applications, such as graphs and point clouds. The advantage of doing so is that there are now non-trivial 'continuous' maps from paracompact Hausdorff spaces to finite spaces (given the appropriate structure), and one may then compare the resulting topological invariants on each side functorially. We find that there are a number of possible such categories, each with its own particular homotopy theory and associated homologies, and, additionally, that there is a generalization of the coarse category which allows finite sets to be non-trivial (i.e. not 'coarsely' equivalent to a point). We will give an overview of these theories and several applications, show how they are related to familiar objects in applied topology, such as the Vietoris-Rips homology, and discuss the advantages and disadvantages of each. We finish by describing a recent construction of sheaf theory in the category of Cech closure spaces, a strict generalization of the category of topological spaces
Algebraic Topology in the Mesoscopic Regime, IMSI Workshop: Topological Data Analysis, online, April 27th, 2021. Video
Abstract: There have been a number of attempts to extend the realm of application of algebraic topological tools to discrete spaces such as graphs, digital images, and point clouds, which one more typically encounters in computer science and data analysis. In each of these theories, one of two strategies has typically been taken. In topological data analysis, one usually replaces the original space with one or more topological spaces that one hopes will retain the relevant topological information in the original set. In various approaches to discrete or digital topology, we find instead different attempts to develop algebraic topology from scratch for some class of discrete objects of interest, proceeding largely by analogy with classical algebraic topology. In this work, we propose a third option: we generalize algebraic topology to categories which contain both the topological spaces classically treated by classical homotopy theory, but which also include as objects the more discrete and combinatorial spaces of interest in applications. The advantage here is that there are now non-trivial ‘continuous maps’ from classical topological spaces to the discrete spaces (given the appropriate structure), and one may then compare the resulting topological invariants on each side functorially. We find that there are a number of possible such categories, each with its own particular homotopy theory and associated homologies, and, additionally, that there is a generalization of the coarse category which allows finite sets to be non-trivial (i.e. not ‘coarsely’ equivalent to a point). We will give an overview of these theories and several applications, discussing the advantages and disadvantages of each.