CIMAT Applied Geometry and Topology Seminar

CIMAT Applied Geometry and Topology Seminar

Spring 2023

• 10am, February 14th , 2023 (Mexico City Time) - Pierre Tarrago, Université Pierre y Marie Curie

Title: Quantum cohomology of Grassmannians and unitary Brownian motion

Abstract: The quantum cohomology ring of Grassmannians is a deformation of the original cohomology ring which counts certain holomorphic maps from the complex projective space to the Grassmannian variety. In this talk, I will explain how a probabilistic reinterpretation of the quantum cohomology ring allows to get asymptotic enumerative results on the number of such holomorphic maps in terms of the kernel of the Brownian motion on unitary groups. The talk is based on a joint work with Jeremie Guilhot and Cedric Lecouvey and should remain introductory.

• 10am, February 28th, 2023 (Mexico City Time) - Peter Bubenik, University of Florida

Title: Persistent homology using filtered closure spaces

Abstract: We develop persistent homology in the setting of filtered (Cech) closure spaces. Examples of filtered closure spaces include filtered topological spaces, metric spaces, weighted graphs, and weighted directed graphs. We use various products and intervals for closure spaces to obtain six homotopy theories, six cubical singular homology theories and three simplicial singular homology theories. Applied to filtered closure spaces, these homology theories produce persistence modules. We extend the definition of Gromov-Hausdorff distance to filtered closure spaces and use it to prove that these persistence modules and their persistence diagrams are stable. We also extend the definitions Vietoris-Rips and Cech complexes to give functors on closure spaces and prove that their persistent homology is stable. The Vietoris-Rips functor has a left adjoint which we call the star functor; in contrast the Cech functor does not have a left or right adjoint. This is joint work with Nikola Milicevic.

• 10am, March 7th, 2023 (Mexico City Time) - Ximena Fernández, Durham University

Title: Morse theory for group presentations and applications to the persistent fundamental group

Abstract: Discrete Morse theory is a combinatorial tool to simplify the structure of a given (regular) CW-complex up to homotopy equivalence, in terms of the critical cells of discrete Morse functions. In this talk, I will present a refinement of this theory that guarantees not only a homotopy equivalence with the Morse CW-complex, but also a Whitehead's simple homotopy equivalence. Moreover, it provides an explicit description of the attaching maps of the critical cells in the simplified complex and bounds on the dimension of the complexes involved in the deformation. This result provides the suitable theoretical framework for the study of different problems in combinatorial group theory and topological data analysis. I will show an application of this technique that allows to prove that some potential counterexamples to the Andrews-Curtis conjecture do satisfy the conjecture. Moreover, the method can also be extended to filtrations of CW-complexes, providing an efficient algorithm for the computation of the persistent fundamental group of point clouds in terms of group presentations.

The talk is based on the article: Fernandez, X. Morse theory for group presentations. Transactions of the AMS (2023, accepted) arXiv:1912.00115 

• March 21st, Mario Fuentes Rumí, University of Málaga/CIMAT Mérida

Title: Modelos de Lie de fibraciones clasificantes.

Abstract: El espacio topológico $B\text{aut}(X)$ es doblemente interesante: resulta ser el espacio clasificante de aquellas fibraciones cuya fibra es  el espacio $X$, y, por otro lado, nos proporciona información homotópica de $\text{aut}(X)$ el espacio de aquellos automorfismos de $X$ que son equivalencias homotópicas. Su homotopía racional es bien conocida en el contexto clásico: es decir, con $X$ simplemente conexo y tomando el recubridor universal de $B\text{aut}(X)$. Gracias a los recientemente desarrollados modelos de Lie, podemos dar el salto al mundo no simplemente conexo. En esta charla se presentan algunos resultados sobre la homotopía racional de $B\text{aut}(X)$ en un contexto no simplemente conexo.

• Special Session: Wednesday March 29th, 4:15pm, CIMAT/DEMAT Coloquium (in person), Samuel Lisi, University of Mississippi

Title: Symplectic capacities: embeddings and dynamics

Abstract:  This talk will give an introduction to symplectic capacities, a class of numerical invariants that relate dynamical properties of Hamiltonian systems to the geometry of symplectic manifolds. Symplectic manifolds are the natural setting in which to consider Hamiltonian dynamical systems. Indeed, they very naturally generalize cotangent bundles (phase space). Unlike Riemannian manifolds, symplectic manifolds don't have local invariants. They do, instead, admit invariants that obstruct symplectic embedding. Remarkably, these embedding obstructions and dynamical properties of Hamiltonian systems are intimately connected by means of symplectic capacities. (Most of the talk will be an overview of the field, but I will also discuss some joint work with Antonio Rieser.)

• April 25th, Eduardo Velázquez, UNAM/CIMAT

Título: Aspectos Computacionales del Flujo de Curvatura Media

Resumen: El Flujo de curvatura Media (FCM) es un flujo geométrico que consiste en deformar una subvariedad a lo largo de su haz normal con una rapidez igual a su vector de curvatura media. Este flujo se ha utilizado en diversas aplicaciones, por ejemplo, el modelo del crecimiento de cristales, propagación de incendios, aproximación de contornos, superficies mínimas, entre otras. En esta plática presentaremos técnicas que podemos utilizar para resolver numéricamente problemas de FCM y algunas de sus variantes; en particular presentaremos un método que permite tanto parametrizar contornos en imágenes, como aproximar la evolución por FCM de la curva perfil de una subvariedad lagrangiana equivariante.

• May 9th, Luigi Caputi, University of Torino

• May 16th, Julián Candela, CIMAT

• May 23rd, Jonathan Treviño, CIMAT

• May 30th, Teresa Hoekstra, CINVESTAV/CIMAT

Previous Semesters:

Fall 2022

• Monday, September 26th, 11am (Mexico City Time) - Bruno Benedetti, University of Miami

• Monday, October 3rd, 11am (Mexico City Time) - Omer Bobrowski, Queen Mary University

• Monday, October 10th, 11am (Mexico City Time) - Anibal Medina, Max Planck Institute Bonn

• Monday, October 17th, 11am (Mexico City Time) - Nikola Milicevic, Penn State University

• November 7th, 11am (Mexico City Time) - Henry Adams, Colorado State University

• November 14th - part of the VIIIth Mexican Workshop on GTDA

Spring 2022

• Thursday, February 10th, 16:00 (Mexico City Time, UTC -6) - Araceli Guzmán, CIMAT Guanajuato

Título: De Nudos y Acumulamientos

Resumen: En esta plática abordaré dos problemas en distintas ramas de la topología que describen mi trabajo actual de investigación, así como los posibles caminos para conducirla en un futuro. El primer problema reside en el  área de Teoría de Nudos, es un trabajo conjunto con el Dr. Mario Eudave y el Dr. Enrique Ramírez. Consiste en encontrar familias de nudos no casifibrados, una propiedad que nos habla de la descomposición en asas del exterior de un nudo. Construiré una familia de nudos no casifibrados y que además son primos, propiedad análoga al concepto de número primo en los enteros. 

El segundo problema, en el área Análisis Topológico de Datos, lo trabajo con el Dr. Antonio Rieser. En  éste describiré un algoritmo que parte de un conjunto de datos tomados de una distribución uniforme en un espacio métrico y que elige una gráfica que mejor representa sus acumulamientos. Esta elección se consigue usando entropía cuántica relativa en los operadores de calor de cierta familia de gráficas.

• Thursday, February 17th, 15:00 (Mexico City Time, UTC -6) - José Ángel Frías García, CIMAT Guanajuato

Título: Invariantes polinomiales de gráficas y el estudio de las 3-variedades.

Resumen: El estudio de las 3-variedades es un problema central en Topología de dimensiones bajas. Una herramienta muy útil para entender estas variedades es mediante descomposiciones de Heegaard. A partir de un diagrama de Heegaard para una 3-variedad, se puede construir una gráfica listón en la superficie de Heegaard correspondiente. A dicha gráfica listón se le pueden asociar invariantes polinomiales como pueden ser los polinomios de Tutte, Penrose o listón.      

Por otro lado, técnicas de Machine Learning y Análisis de Datos están tomando relevancia en diversas áreas de las Matemáticas en las que se cuenta con cantidades masivas de información. En el caso particular de Topología en dimensiones bajas, R. Sazdanovic y colaboradores han usado Análisis de Datos para calcular la dimensionalidad del conjunto de polinomios de Jones para todos los nudos con hasta 17 cruces, así como para establecer relaciones del polinomio de Jones con la signatura de los nudos. Más recientemente, A. Juhász y M. Lackenby junto con un equipo de DeepMind utilizaron Machine Learning para establecer relaciones entre invariantes hiperbólicos de nudos y la signatura.

En el presente proyecto, nos planteamos el estudio de bases de datos de polinomios asociados a gráficas listón de diagramas de Heegaard para establecer relaciones con invariantes clásicos como el grupo fundamental. Una pregunta interesante es si dichos polinomios tienen relación con otros invariantes modernos en 3-variedades como la Homología de Heegaard-Floer. 

• Thursday, February 24th, 16:00 (Mexico City Time, UTC -6) - Hannah Alpert, Auburn University

Title: Homology crowding in configuration spaces of disks

Abstract: Configuration spaces of disks in a region of the plane vary according to the radius of the disks, and their topological invariants such as homology also vary.  Realizing a given homology class means coordinating the motion of several disks, and if there is not enough space for the disks to move, the homology class vanishes.  We explore how clusters of orbiting disks can get too crowded, some topological conjectures that describe this behavior, and some progress toward those conjectures.

• Thursday, March 10th, 11:00 (Mexico City Time, UTC -6) - Erika Roldán, Technische Universtiät München and EPFL Lausanne

Title: Topology of random 2-dimensional cubical complexes 

Abstract: We study a natural model of random 2-dimensional cubical complexes which are subcomplexes of an n-dimensional cube, and where every possible square (2-face) is included independently with probability p. Our main result exhibits a sharp threshold p=1/2 for homology vanishing as the dimension n goes to infinity. This is a 2-dimensional analogue of the Burtin and Erdős-Spencer theorems characterizing the connectivity threshold for random graphs on the 1-skeleton of the n-dimensional cube. Our main result can also be seen as a cubical counterpart to the Linial-Meshulam theorem for random 2-dimensional simplicial complexes. However, the models exhibit strikingly different behaviors. We show that if p > 1 - sqrt(1/2) (approx 0.2929), then with high probability the fundamental group is a free group with one generator for every maximal 1-dimensional face. As a corollary, homology vanishing and simple connectivity have the same threshold. This is joint work with Matthew Kahle and Elliot Paquette.

• Tuesday, March 29th, 11:00 (Mexico City Time, UTC -6) - Nikola Milicevic, Penn State University

Title: Homology, Homotopy, and Persistent Homology using filtered Čech’s closure

spaces

Abstract: We develop persistent homology in the setting of filtered Čech’s closure spaces. Examples of filtered closure spaces include filtered topological spaces, metric spaces, weighted graphs, and weighted directed graphs. We use various products and intervals for closure spaces to obtain six homotopy theories, six cubical singular homology theories and

three simplicial singular homology theories. Applied to filtered closure spaces, these homology theories produce persistence modules. We extend the definition of Gromov-Hausdorff

distance to filtered closure spaces and use it to prove that our persistence modules and their persistence diagrams are stable. We also extend the definitions Vietoris-Rips and Čech complexes to closure spaces and prove that their persistent homology is stable.

• Tuesday, April 5th, 11:00 (Mexico City Time, UTC -5) - Ginestra Bianconi, Queen Mary University London

Title: The topological Dirac operator and the dynamics of topological signals 

Abstract: Topological signals associated not only to nodes but also to links and to the higher dimensional simplices of simplicial complexes are attracting increasing interest in signal processing, machine learning and network science. Typically, topological signals of a given dimension are investigated and filtered using the corresponding higher-order Laplacian. In this talk I will cover notable spectral properties of higher-order Laplacians and I will reveal how these properties affect higher-order diffusion and higher-order synchronization dynamics. Moreover, I will introduce the topological Dirac operator that can be used to process simultaneously topological signals of different dimensions. I will discuss the main spectral properties of the Dirac operator defined on networks, simplicial complexes and multiplex networks, and their relation to higher-order Laplacians. Finally I will show how the Dirac operator allows to define topological synchronization of locally coupled topological signals defined on nodes and links of a network. 

• Tuesday, April 26th, 11:00 (Mexico City Time, UTC -5) - Parvaneh Joharinad, Max Planck Institute for the Mathematical Sciences, Leipzig

Title: Curvature of data

Abstract: The aim of this presentation is to briefly present generalized curvatures and see which kind of relations between data points each of them evaluate and what kind of information they reveal. While in topological data analysis the objective is to extract qualitative features, the shape of data, geometric data analysis mainly deals with quantitative features of data. For instance, the prominent scheme of manifold learning is applied to find the comparatively low dimensional Riemannian manifold on which the data set fits best. It then raises the question of whether one can anticipate some geometric properties from initial model before finding this manifold structure.The most important quantitative measures that in a good extent reveal the geometry of a Riemannian manifold are its (sectional and Ricci) curvatures. Therefore, we wish to see how one can determine the curvature of data and how does it help to derive the salient structural features of a data set. The talk is based on joint work with Jürgen Jost

• Tuesday, May 3rd, 11:00 (Mexico City Time, UTC -5) - Matthew Cordes, ETH 

Title: Geometric approximate group theory

Abstract: An approximate group is a subset of a group that is "almost closed" under multiplication. Finite approximate subgroups play a major role in additive combinatorics. Recently Breuillard, Green and Tao established a structure theorem concerning finite approximate subgroups and used this theory to reprove Gromov’s polynomial growth theorem. Certain infinite approximate groups also are also classic well studied objects. For instance, nice approximate subgroups of R^n are models for mathematical quasi-crystals. Recently, Björklund and Hartnick have begun a program investigating infinite approximate lattices in locally compact second countable groups using geometric and measurable structures. In the talk I will introduce infinite approximate groups and present a new approach to studying these groups geometric group theoretic methods. This is joint work with Tobias Hartnick and Vera Tonić.

• Tuesday, May 24th, 11:00 (Mexico City Time, UTC -5) - Baris Coskunuzer, University of Texas at Dallas

Title: Geometric Approaches on Persistent Homology

Abstract: Persistent Homology is one of the most important techniques used in Topological Data Analysis. In the first part of the talk, we give an introduction to the subject. In the second half, we study the persistent homology output via geometric topology tools. In particular, we give a geometric description of the term “persistence”. The talk will be non-technical, and accessible to graduate students. This is joint work with Henry Adams.

• Tuesday, May 31st, 11:00 (Mexico City Time, UTC -5) - Jesús Rodriguez, CONACYT-CIMAT

Título: Análisis Topológico de Datos en el Lenguaje Natural

Resumen: Usamos homología persistente en el espacio LSA de documentos matemáticos de ArXiv para tratar de encontrar posibles huecos de conocimiento. Utilizamos la distancia Bottleneck para comparar los resultados entre diferentes categorías y para comparar la estructura local de diferentes documentos. Presentaremos nuestros avances y los objetivos futuros de este proyecto.

• Tuesday, June 7th, 11:00 (Mexico City Time, UTC -5) - Hans Riess, University of Pennsylvania

Title: A Sheaf Laplacian for Lattice-Valued Sheaves

Abstract: In this talk, we develop a discrete Hodge theory for cellular sheaves taking data in the category of complete lattices and Galois connections. The key development is the Tarski Laplacian, an endomorphism on the cochain complex whose fixed points agree with global sections in degree zero. After laying the foundation for the basic theory, we, then, provide an application to multi-agent semantics.