Research
Research
(In reverse chronological order)
Vietoris-Rips Homology Theory for Semi-Uniform Spaces, arXiv:2008.05739, submitted
Grothendieck Topologies and Sheaves on Čech Closure Spaces, arXiv:2109.13867v2, submitted
Noncommutative Model Selection and the Data-Driven Estimation of Real Cohomology Groups (with Araceli Guzmán-Tristán and Eduardo Velázquez-Richards), arxiv.org:2411.19894, 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
Kunneth Theorems for Vietoris-Rips Homology (with Alejandra Trujillo), Acta Mathematica Hungarica, volume 166, pages 239-253 (2022)
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
2025
Pseudotopological Foundations of Topological Data Analysis, The Geometric Realization of AATRN, Institute for Mathematical and Statistical Innovation, University of Chicago, Chicago, Illinois, August 21st, 2025, Constancia
De los Operadores de Calor al Análisis de Datos (una plática en cinco actos), Taller Mexicano de Topología, IMATE UNAM, Ciudad de México, México, June 17th 2025, Constancia
Towards non-commutative inference for topological data analysis, Seminario de Matrices Aleatorias y Probabilidad No Conmutativa, CIMAT, Guanajuato, México, May 20th, 2025
Abstract: 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
Topología aplicada: nuevos fundamentos teóricos, hacia nuevas prácticas, Seminario de Estudiantes, CIMAT, Guanajuato, México, May 16th, 2025, Constancia
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 9th, 2025, Constancia
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.
2024
Sheaf theory for data and combinatorics, CIMA, Benemérito Universidad Autonóma de Puebla, Puebla, México, September 2nd, 2024, Constancia
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 Homology for Graphs and Data Through Closure Spaces, Journée de Géometrie y Topologie, Mediterranean Institute for the Mathematical Sciences, MedTech, Tunis, Tunisia, July 18th, 202, Constancia
Homotopy and sheaf theory for data and combinatorics, University of Florida Geometry and Topology Seminar, University of Florida, Gainsville, FL, USA, May 2024, Constancia
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, Constancia
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, Constancia
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.
2023
Sheaves on data through closure spaces, Northeastern University Topology Seminar, Northeastern University, Boston, MA, USA, November 2023, Constancia
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, Constancia
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.
2022
Discrete algebraic topology for metric spaces and topological data analysis, University of Central Florida Colloquium (online), Orlando, Florida, January 28th, 2022
2021
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
Applied Topology from the Classical Point of View, Annual Meeting of the Society for Industrial and Applied Mathematics Mexico Section, Online, June 21-23, 2021
Algebraic Topology in the Mesoscopic Regime, IMSI Workshop: Topological Data Analysis, Online, April 27th, 2021. Video, Constancia
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.
2020
Más Allá de la Topología, 53 Congreso Nacional de la Sociedad Matemática Mexicana, Puebla, Mexico, Online, October 22nd, 2020
The Vietoris-Rips Complex Through the Looking Glass, Joint Mathematics Meeting, Denver, CO, USA, January 15-18, 2020
2019
Cech Closure Spaces: A Unified Framework for Discrete and Continuous Homotopy, Topological and Geometric Data Analysis Seminar, Ohio State University, Columbus, Ohio, July 25th, 2019, Constancia
2018
Homotopy and the Vietoris-Rips homology, 5to Escuela de Análisis Topológico de Datos y Temas Relacionados, CIMAT Guanajuato, Guanajuato, México, November 19-23, 2018
Homotopy and homologies for point clouds, 10th Conference on Geometric and Topological Methods in Computer Science, Oaxaca, México, September 10-14, 2018
Everything old is new again: Cech closure spaces and the foundations of topological data analysis, CMO workshop on Multiparameter Persistent Homology, CMO Oaxaca, México, August 2018
Homotopy theory for point clouds, 15th Annual Workshop on Topology and Dynamical Systems, University of Nipissing, Canada, May, 2018
2017
Homotopy theory for data and simplicial complexes, 47th John H. Barrett Memorial Lectures, University of Tennessee, United States, May 2017
An introduction to algebraic topology, stochastic topology, and topological data analysis (2 hours), Workshop on Homotopy Probability Theory, University of Saarbrucken, Saarbrucken, Germany, September, 2017
Homotopy and homology on point clouds and combinatorial objects (4 hours), Winter School on Algebraic Topology, CIMAT Mérida, Mérida, México, December, 2017
2016
Homotopy Theory for Data and Simplicial Complexes, Computational Mathematics Seminar, Jagiellonian University, Krakov, Poland, June 2nd, 2016
Homotopy Theory for Data and Simplicial Complexes, Stochastic and Applied Topology Seminar, Technion, Haifa, Israel, June, 2016
The Heat Operator and Data Clustering, University of Colima Mathematics Seminar, Colima, Mexico, May 13th, 2016
Topology and Geometry in Data Analysis, University of Colima Mathematics Student Seminar, Colima, Mexico, May 12th, 2016
Everything you ever wanted to know about homotopy but were afraid to ask, part III, ATD Seminar, CIMAT Guanajuato, April 29th, 2016
Everything you ever wanted to know about homotopy but were afraid to ask, part II, ATD Seminar, CIMAT Guanajuato, April 8th, 2016
Everything you ever wanted to know about homotopy but were afraid to ask, part I, ATD Seminar, CIMAT Guanajuato, March 4th, 2016
2015
Homotopy theory for data sets, Segunda escuela/conferencia de análisis topológico de datos, Querétaro, Mexico, December, 2015
Homotopy theory for data sets, Toposys Workshop, Jagiellonian University, Krakow, Poland, September, 2015
A Topological Approach to Spectral Clustering, DyToComp 2015, Dynamics, Topology, and Computations, Bedlewo, Poland, June, 2015
A Topological Approach to Data Clustering, AUS-ISMS 2015, 2nd International Conference on Mathematics and Statistics, Sharjah, United Arab Emirates, April, 2015
A Topological Approach to Data Clustering, Stochastic and Applied Topology Seminar, Technion, June, 2015
From the Heat Operator to Data Clustering, Differential Geometry Seminar, Centro de Investigación en Matemáticas, Guanajuato, Mexico, May, 2015
Approximating the Laplace-Beltrami operator, Graph Theory Seminar, Centro de Investigación en Matemáticas, Guanajuato, Mexico, May, 2015
2014
A Topological Approach to Data Clustering, Unit of Computational Medicine, Karolinska Institute, Stockholm, Sweden, November, 2014
Diffusion-based approaches to manifold learning, Toposys Workshop, IST Vienna, Vienna, Austria, September, 2014
Coisotropic Hofer-Zehnder Capacities, Non-squeezing for Relative Embeddings, and Energy-Capacity inequalitites, Geometry Seminar, ENS-Lyon, Lyon, France, March, 2014
Coisotropic Hofer-Zehnder capacities, non-squeezing for relative embeddings, and energy-capacity inequalitites, Warwick-Seoul National University Symplectic Geometry Workshop, Warwick, England, February, 2014
Coisotropic Hofer-Zehnder capacities and non-squeezing for relative embeddings, Symplectix Seminar, Institut Henri Poincar\'e, Paris, France, January, 2014
2013
Hofer-Zehnder capacities for Lagrangian submanifolds, Universit\'e Paul Sabatier (Toulouse III), Institut des Math\'ematiques Geometry Seminar, Toulouse, France, June 2013
Relative symplectic packing in 4-manifolds, Bar-Ilan University, Department of Mathematics Emmy Noether Seminar, Ramat Gan, Israel, May 2013
2011
Relative blow-ups and real packing in symplectic 4-manifolds University of Nantes, Department of Mathematics Topology Seminar, Nantes, France, September 2011
Relative packing in real symplectic 4-manifold, University of Haifa, Department of Mathematics Topology and Geometry Seminar, Haifa, Israel, January 2012
2010
Symplectic and relative packing in real symplectic 4-manifolds, International Mathematical Union winter meeting, Tel Aviv University, Tel Aviv, Israel, December 2010
Blow-ups and blow-downs of Lagrangian submanifolds and real symplectic packing, TAU-Technion Seminar on Geometry and Dynamics, Tel Aviv University, Tel Aviv, Israel, November 2010
Lagrangian Submanifolds, blow-ups, and real packing, GESTA 2010, Symplectic Geometry with Algebraic Techniques, Lisbon, Portugal, June 2010
Posters
2014
A topological approach to spectral clustering, Discrete, Computational, and Algebraic Topology Workshop, University of Copenhagen, Copenhagen, Denmark, November, 2014
Scientific Visits
Institute of Mathematics, UNAM, June 18th-20th, 2025
Mathematics Department, University of Florida, May 12th-18th, 2024
Member, Complementary Program, Simons Laufer Mathematical Sciences Institute, Berkeley, California, January 8th - July 5th, 2024 (Sabbatical Spring Semester)
Member, Long Program: Math+Neuroscience: Strenthening the Interplay Between Theory and Mathematics, ICERM, Brown University, Providence, Rhode Island, September 6th - December 8th, 2023 (Sabbatical Fall Semester)
Member, MSRI program: Higher Categories and Categorification, part 2, Instituto de Matemáticas Unidad Cuernavaca, UNAM, Cuernavaca, México, June 1st-30th, 2022
Jagiellonian University, Krakov, Poland, May 22nd - June 5th, 2016
Université de Nantes, Nantes, France, February 20th - 28th, March 9th - 11th, 2014
Université de Nantes, Nantes, France December 8th - 14th, 2013
IMA General Membership, Institute for Mathematics and Its Applications (IMA), Minneapolis, Minnesota, USA, October 1st - November 2nd, 2013
Université de Nantes, Nantes, France, July 15th - August 1st, 2013, ESF CAST short visit grant
Université de Nantes, Nantes, France, January 31st - March 14th, 2013, ESF CAST short visit grant
Université de Nantes, Nantes, France, October 10th - November 6th, 2012
Université Libre de Bruxelles, Brussels, Belgium, August 12th - 19th, 2012, ESF CAST short visit grant
Université de Nantes, Nantes, France, September 3rd-10th, 2011