Viernes 18 de noviembre de 2016,

• Sesión conjunta con el Seminario de Computación de CIMAT-Guanajuato.
  Salón Diego Bricio Hernández, CIMAT, Guanajuato.
  12.30-13.30. "Computational Techniques for Persistent Homology".
  Clément Maria, University of Queensland, Australia. 
  Resumen

  Persistent homology is a method that studies the evolution of the topology of the level sets of a function. It has found many applications in practice, and hence requires efficient methods to compute it. In this talk, we introduce diverse algorithms and data structures involved at the different stages of the computation. We introduce the "simplex tree" data structure to represent simplicial complexes---which are combinatorial representations of the level sets---in high-dimensions, and the "compressed annotation matrix", which encodes and allows efficient updates of the cohomology groups---i.e., the topological information---of the simplicial complexes.
  
  This is joint work with Jean-Daniel Boissonnat and Tamal K. Dey.

 Viernes 2 de septiembre de 2016,

• Salón K-201 (antes usos múltiples del nivel H), CIMAT, Guanajuato.
  12.00-12.50. "Modelos estándar de 2-variedades estratificadas".
  José Carlos Gómez Larrañaga​, CIMAT. 
  Resumen

  Se definirá lo que son las 2-variedades estratificadas (2-VA’s) y como aparecen en el estudio de la categoría de Lusternik-Schnirelmann (cat L-S). Estos espacios topológicos se proponen como posibles modelos a usarse en el análisis topológico de datos. Luego de mencionarse algunos algoritmos que determinan si estos objetos tienen alguna propiedad algebraica-topológica especifica tal como ser simplemente conexo u homotópicamente equivalente a una esfera de dimensión dos, se inicia el estudio de encontrar modelos estándar de 2-VA´s trivalentes. Se indicará como el encontrar estos modelos serviría para calcular la categoría de L-S de estos objetos. Si el tiempo lo permite se mencionará el trabajo de Scoville et al. donde proponen una versión discreta de la Cat L-S para complejos simpliciales. (Trabajo conjunto con F. J. González Acuña y W. Heil).

• Salón K-201 (antes usos múltiples del nivel H), CIMAT, Guanajuato.
  13.00-13.50. "Bases de Markov en el modelo de regresión de Poisson".
  Abraham Martin del Campo​, CONACYT-CIMAT.
  Resumen

  El teorema fundamental de la estadística algebraica, establece la conexión entre el álgebra y la estadística, a través de las bases de Markov. En esta charla, daremos una introducción a las bases de Markov, y explicaremos cómo se pueden utilizar para hacer una prueba de bondad de ajuste para el model de Poisson, el cual estamos implementando en R.

• CIMAT Guanajuato, Auditorio J.A. Canavati

12:00 hrs. Topological, Geometric and Combinatorial Properties of Random Polyominoes
Érika Berenice Roldán Roa
Examen para la Obtención del grado de Doctora en Ciencias con Orientación en Probabilidad y Estadística

Resumen: We study two models of random shapes in this thesis: the Eden Cell Growth Model (EGM) and uniform and percolation distributed polyominoes (also known as lattice-based animals). These models have long been of interest in mathematical physics, probability, and statistical mechanics. Both structures have interesting topological, combinatorial, and geometrical properties. However, until now, tools from stochastic topology and topological data analysis have not been used to study these properties. By introducing these methods and techniques, we established and proved new results that increase the understanding of topological, geometric, and combinatoric properties of these random models.

First, we study the maximum number of holes (the rank of the first homology group) that a polyomino with a given number of tiles can have. We prove a tight bound for the asymptotic behavior and give an exact formula for an infinite sequence of natural numbers for this maximum number of holes. Our second set of contributions of this thesis are about the rate of growth of the expectation of the number of holes in a polyomino with uniform and percolation distributions. We prove the existence of linear bounds for the expected number of holes of a polyomino with respect to both the uniform and percolation distributions. Furthermore, we exhibit particular constants for the upper and lower bounds in the uniform distribution case. Finally, we characterize how the rank of the first homology group, of the stochastic process defined by the EGM, changes in time. This allowed us to design and implement a new algorithm that computes the persistence homology associated to this stochastic process at each time and that keeps track of geometric features of the homology of the process as the area and location of the holes. We present and analyze the results of the computational experiments obtained with this algorithm. We also state conjectures based on these experiments about the asymptotic behavior of the number of holes, the locations of these holes, their associated area, and other geometric and topological properties of this stochastic process.

Lunes 14 de noviembre de 2016, 

• Sesión conjunta con el Seminario de Estadística de CIMAT-Guanajuato. 
  Salón Diego Bricio Hernández. CIMAT, Guanajuato.
  13.00-13.50. "Zigzag Persistent Homology: Theory and Algorithms".
  Clément Maria, University of Queensland, Australia. 
  Resumen

  Persistent homology is a method that studies the evolution of the topology of the level sets of a function. It restricts to the case where the sets grow monotically with regards to inclusion. This is quite restrictive in practice. Zigzag persistent homology is a powerful generalisation that allows the level sets to both grow and shrink. In this talk, we introduce and motivate zigzag persistence, and focus on algorithms to compute it. Specifically, we formalise zigzag persistence within the field of quiver theory, and introduce new transformation theorems---called diamond---to track the evolution of the decomposition of quiver representations under local modifications. We deduce an algorithm from these results.
  
  This is joint work with Steve Oudot

Miércoles 14 de diciembre de 2016,  

• Salón G001 (Antes salón 1, junto al Auditorio Canavati). CIMAT, Guanajuato.
  13.00-14.00, A classification of trivalent simply connected 2-stratifols.
   Wolfgang Heil, Florida State University.
  Resumen

  Trivalent 2-stratifolds are a generalization of 2-manifolds in that there are disjoint simple closed curves where three sheets meet. We present a classification of simply connected 2-stratifolds in terms of their associated labeled graphs. This is joint work with Jose Carlos Gomez-Larrañaga and Francisco Gonzalez-Acuña.

Miércoles 7 de diciembre de 2016,  

• Salón Diego Bricio Hernández, CIMAT, Guanajuato.
  13.00-14.00, Analysis of cancer genomic data using computational algebraic topology.
  Javier Arsuaga, University of California, Davis.
  Resumen

  Genomic technologies measure thousands of molecular signals with the goal of understanding essential biological processes. In cancer these molecular signals have been used to characterize disease subtypes, cancer pathways as well as subsets of patients with specific prognostic factors. This large amount of information however is so complex that new mathematical methods are required for further analyses. Computational homology provides such a method. We have developed a new homology based supervised method that identifies significant copy number changes in the tumor genome. This method associates a set of point clouds to any given profile and uses β0 of the surfaces to detect frequent copy number changes and β1 to further analyze the structure of the copy number changes. We applied this method to a set of breast cancer patients with known molecular subtype. The analysis using β0 confirmed previously reported copy number changes and found three new significant changes in the basal subtype: 1p, 2p and 14q. The analysis using β1 identified multiple co-occurring amplifications. I will discuss those related with the ERBB2/HER2 subtype (17q12, 17q21.2 and 17q21.33). The talk will end discussing possible extensions of this approach.   

• Salón G101, CIMAT Guanajuato

Sesión conjunta del seminario de Estadística y las Sesiones ATD

13:00 - 14:00. Nonparametric Estimation of Probability Density Functions of Random Persistence Diagrams
Vasileios Maroulas, University of Tennessee at Knoxville 

Resumen: We introduce a nonparametric way to estimate the global probability density function for a random persistence diagram. Precisely, a kernel density function centered at a given persistence diagram and a given bandwidth is constructed. Our approach encapsulates the number of topological features and considers the appearance or disappearance of features near the diagonal in a stable fashion. In particular, the structure of our kernel individually tracks long persistence features, while considering features near the diagonal as a collective unit. The choice to describe short persistence features as a group saves computational time and reduces scaling while simultaneously retaining accuracy. Indeed, we prove that the associated kernel density estimate converges to the true distribution as the number of persistence diagrams increases and the bandwidth shrinks accordingly. We also establish the convergence of the mean absolute deviation estimate, defined according to the bottleneck metric. Lastly, examples of kernel density estimation are presented for typical underlying datasets. 

• Auditorio J.A. Canavati (G002), CIMAT Guanajuato

5a Escuela de Análisis Topológico de Datos y Temas Relacionados
atd2018.eventos.cimat.mx

Viernes 10 de octubre, 12-14 horas, Salón de usos múltiples del Nivel 6 (nuevo edificio). 

• Nociones de inferencia estadística necesarias para el estudio de TDA, Miguel Nakamura, CIMAT.
• Nociones de probabilidad necesarias para el estudio de TDA y topología estocástica, Víctor Pérez Abreu, CIMAT.

Viernes 13 de marzo en Morelia, Michoacán. Sesión conjunta con el Seminario Interinstitucional Centro Norte de México en Combinatoria y Probabilidad.

La sesión consta de dos conferencias: 

11-12 horas.
El método probabilista en acción.
Leonardo Martínez, IMATE, UNAM-Querétaro.

12-13 horas.
Propiedades de conexidad de gráficas aleatorias. Octavio Arizmendi. CIMAT, Guanajuato.

Salón 7 del Centro de Ciencias Matemáticas UNAM en Morelia www.matmor.unam.mx.

El semestre agosto-diciembre 2014 estuvo dedicado a conocer el tema de TDA, así como algunas de sus aplicaciones. En particular, varios de alumnos de licenciatura que realizaron proyectos de verano en TDA, harán presentaciones de sus proyectos. El objetivo es contar con mayores elementos y nociones de topología algebraica, inferencia estadística y probabilidad para participar en la Escuela de Análisis Topológico de Datos y Topología Estocástica, evento internacional a celebrarse del 19 al 23 de enero de 2015.

Las sesiones serán en los siguientes viernes del semestre agosto-diciembre 2014.