Topology Session

Join Zoom Meeting

https://vc-cudi.zoom.us/j/87174027537
Meeting ID: 871 7402 7537

Guest Speakers

  • Alberto Verjovsky - Institute of Mathematics, UNAM
  • Aldo Guzmán-Sáenz - Thomas J. Watson Research Center, IBM
  • Alex Adem - University of British Columbia
  • Gunnar Carlsson - Stanford University
  • Herbert Edelsbrunner - Institute of Science and Technology Austria
  • José Seade - Institute of Mathematics, UNAM
  • Kathryn Hess - École Polytechnique Fédérale de Lausanne
  • Luis Paris - University of Burgundy
  • Michael Farber - Queen Mary University of London
  • Rita Jiménez - Institute of Mathematics, UNAM
  • Robert Ghrist - University of Pennsylvania
  • Santiago López de Medrano - Institute of Mathematics, UNAM

Program

  • Gunnar Carlsson
    Monday, September 13th, 2021
    10:00 am (CDT)

    Topological Methods for Deep Learning

    Abstract: Deep learning is a very powerful methodology used in a number of situations. One is the study of data sets of images, where it can produce very accurate using a particular class of computation graphs called convolutional neural networks. On the other hand, topological data analysis has produced a good understanding for the statistics of natural images. In this talk, we will show how to bridge this gap, and also produce ways to understand the inner workings of convolutional neural networks. We will also show how to use topology to inform the constructions of specialized architectures that increase the generalization power as well as lessen the number of data points needed to train.

  • Aldo Guzmán-Sáenz
    Friday, September 17th, 2021
    10:00 am (CDT)

    Análisis Topológico de Datos en el contexto de Genómica Computacional

    Abstract: La biología computacional, cuyo objetivo es obtener información sobre sistemas biológicos mediante el uso de herramientas de cómputo, ya desde hace algún tiempo utiliza herramientas como el Análisis Topológico de Datos (ATD). En esta plática mencionaremos un par de aplicaciones de estas herramientas: ATD junto a aprendizaje de máquinas en una tarea de clasificación de Enfermedad de Parkinson, y ATD aplicado en la identificación de organismos presentes en una muestra metagenómica.

  • Robert Ghrist
    Monday, September 20th, 2021
    10:00 am (CDT)

    Opinion Dynamics on Sheaves

    Abstract: There is a long history of networked dynamical systems that models the spread of opinions over social networks, with the graph Laplacian playing a lead role. One of the difficulties in modelling opinion dynamics is the presence of polarization: not everyone comes to consensus. This talk will describe joint work with Jakob Hansen introducing a new model for opinion dynamics using sheaves of vector spaces over social networks. The graph Laplacian is enriched to a Hodge Laplacian, and the resulting dynamics on discourse sheaves can lead to some very interesting and perhaps more realistic outcomes. Additional work with Hand Riess extending the theory will be hinted at.

  • Alberto Verjovsky
    Friday, September 24th, 2021
    10:00 am (CDT)

    Compact 3-dimensional geometric solenoidal manifolds

    Abstract: We present several results about solenoidal manifolds motivated by results by Dennis Sullivan in [1] with commentaries developed in [2] and on a work in progress with Dennis Sullivan. Solenoidal manifolds of dimension n are topological spaces which are locally homeomorphic to the product of a Cantor set with an open subset of ℝn. Geometric 3-dimensional solenoidal manifolds are the analog of geometric 3-manifolds in the sense of Thurston. We will give some results related to 3-dimensional geometric solenoidal manifolds.
    References
    [1] D. Sullivan, Solenoidal manifolds, J. Singul. 9 (2014), 203–205.
    [2] A. Verjovsky, Commentaries on the paper “Solenoidal manifolds” by Dennis Sullivan. J. Singul. 9 (2014), 245–251.

  • Herbert Edelsbrunner
    Monday, September 27th, 2021
    10:00 am (CDT)

    Distortion, on the Average and in Expectation

    Abstract: We generalize the concept of the Voronoi path of a line to more general shapes and compute the distortion constant, which describes how it changes volume on the average. Although initially asked for a Poisson point process, the distortion is a characteristic property of the space rather than the point process. In other words, the constant ratio of the perimeter of a circle and its pixelation---and the analogous ratios for spheres in three and higher dimensions---hold for all smoothly embedded shapes on average.
    This is joint work with Anton Nikitenko.

  • José Seade
    Friday, October 1st, 2021
    10:00 am (CDT)

    Sobre la topología de una función holomorfa cerca de un punto crítico

    Abstract: Hablaré sobre trabajo reciente con Marcelo Aguilar y Aurelio Menegón. Estudiamos funciones holomorfas f:(Cn+1,0) → (C,0) con punto crítico no aislado en el orígen, y la manera como los niveles no-críticos de la función degeneran al nivel crítico.

  • Kathryn Hess
    Monday, October 4th, 2021
    10:00 am (CDT)

    Trees, barcodes, and neuron morphologies

    Abstract: Motivated by the desire to automate classification of neuron morphologies, we designed a topological signature, the Topological Morphology Descriptor (TMD), that assigns a barcode to any geometric tree (i.e, any finite binary tree embedded in ℜ3). We showed that the TMD effectively determines the reliability of clusterings of random and neuronal trees. Moreover, using the TMD we performed an objective, stable classification of pyramidal cells in the rat neocortex, based only on the shape of their dendrites.

    We have also reverse-engineered the TMD, in order to digitally synthesize dendrites, to compensate for the dearth of available biological reconstructions. The algorithm we developed, called Topological Neuron Synthesis (TNS), stochastically generates a geometric tree from a barcode, in a biologically grounded manner. The synthesized neurons are statistically indistinguishable from real neurons of the same type, in terms of morpho-electrical properties and connectivity. We synthesized networks of structurally altered neurons, revealing principles linking branching properties to the structure of large-scale networks.

    In this talk I will provide an overview of the TMD and the TNS and then describe the results of our theoretical and computational analysis of their behavior and properties, in which symmetric groups play a key role.

    This talk is based on joint work with Adélie Garin and Lida Kanari, as well as with Justin Curry, Jordan Desha, and Brendan Mallery, building on earlier collaborations led by Lida Kanari.

  • Santiago López de Medrano
    Friday, October 8th, 2021
    10:00 am (CDT)

    Intersections of concentric ellipsoids: topology, geometry and implications

    Abstract: I will present the problem of describing the topology of the intersection of several concentric ellipsoids in ℝn y and that of other quadrics. We will briefly present the main topological results and their proofs, as well as the difficulties to solve the problem in full generality.

    We will also review the relation between this theory and other branches of Mathematics, as well as some possible applications.

  • Alejandro Adem
    Monday, October 11th, 2021
    10:00 am (CDT)

    Minimal Euler Characteristics for Even-Dimensional Manifolds with Finite Fundamental Group

    Abstract: In this talk we will discuss estimates for the minimal Euler characteristic of even dimensional manifolds with a given finite fundamental group and a highly connected universal cover. In the particular case when χ(M) = 2 we have the related problem of determining which finite groups can be the fundamental group of a rational homology 2n-sphere. As an application we obtain new restrictions for non-abelian finite groups arising as fundamental groups of rational homology 4–spheres.
    This is joint work with Ian Hambleton.

  • Rita Jiménez Rolland
    Friday, October 15th, 2021
    10:00 am (CDT)

    Homología y cohomología de mapping class groups

    Abstract: El grupo modular de una superficie, `mapping class group' en inglés, es el grupo de clases de isotopía de homeomorfismos de una superficie topológica. En esta charla describiremos algunas de las propiedades de este grupo y ciertos contextos en los que aparece. Nuestro objetivo es presentar un panorama de los resultados conocidos (y algunas preguntas abiertas) sobre la homología y cohomología de mapping class groups de superficies compactas orientables. Si el tiempo lo permite, hablaremos sobre nuestra contribución en este tema.

  • Michael Farber
    Monday, October 18th, 2021
    10:00 am (CDT)

    Autonomous robot motion and topology

    Abstract: In the autonomous regime a robot gets high level description of tasks and implements them without further human intervention. I will discuss topological questions which arise when programming an autonomous robot. I will make emphasise on some recent developments: topological complexity of groups, parametrised motion planning and Rationality Conjecture.

  • Luis Paris
    Friday, October 22nd, 2021
    10:00 am (CDT)

    Grupos de trenzas virtuales

    Abstract: Los grupos de trenzas virtuales fueron introducidos por Kauffman en 1999 en su artículo seminal sobre la teoría de los nudos y enlaces virtuales. Juegan el mismo papel para los enlaces virtuales que los grupos de trenzas clásicos para los enlaces clásicos. Esta charla será una presentación general sobre los grupos de trenzas virtuales para un público no especializado. Comenzaremos con la definición de trenzas virtuales en términos de diagramas, como hizo Kauffman en su artículo seminal, la versión virtual del "Teorema de Markov", y sus (posibles) aplicaciones a los nudos y enlaces virtuales. En un segundo tiempo presentaremos una interpretación topológica, debida a Cisneros de la Cruz, de las trenzas virtuales en términos de arcos en una superficie espesada. En una última parte abordaremos algunos aspectos algebraicos de estos grupos (presentación, solución al problema de la palabra, centro, torsión, etc.).