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Todos los participantes están invitados a presentar una contribución (tamaño A1) a la sesión de pósteres del Encuentro.

Pósteres XXII Encuentr de Topología

1er premio: Marithania Silvero Casanova (Universidad de Sevilla)

A geometric realization of extreme Khovanov homology

Khovanov homology groups are algebraic invariants of oriented links, which categorify the Jones polynomial. Given a diagram D of an oriented link L, the homology group Hi,j(L) is known to be trivial for j< jmin, where jmin depends on the diagram and is known as the hypotetical extreme degree of D. In this work we show that the hypotetical extreme Khovanov homology groups, Hi,jmin(L), are precisely the cohomology groups of the independence complex of a specific graph constructed from the diagram. Using this, we provide examples of links whose extreme Khovanov homology groups are nontrivial for as many values of i as desired. These are particular examples of H-thick links.

(Trabajo conjunto con Pedro M. González Manchón y Juan González-Meneses)

2o premio: Ricard Riba (Universitat Autònoma de Barcelona)

Invariantes de esferas de homologia racional

Los invariantes de esferas de homologia racional se pueden construir a partir de un cociclo de un cierto subgrupo del mapping class group formado por el producto de potencias p-éssimas de twists de Dehn i elementos del grupo de Torelli. En particular, se puede usar esta construccion para intentar demostrar la conjetura de Perron.

María García Monera (Universitat Politècnica de València)

Critical points for the Tangent map. Surfaces in R4

In this work we give the definition of r-critical points of a smooth map between manifolds and we apply this definition to the study of the 1-critical points of the tangent map Ω :TMRk+n, when M is an immersed surface immersed in R4, TM is the tangent bundle of M and Ω (m,y) = y if y ∈ Tm M .

Carlos Andrés Giraldo Hernández (Universidad Autónoma de Barcelona)

Haces Fibrados y Fibraciones Minimales

Dada una EI-categoría C con un número finito de objetos, mostramos como clasificar diagramas de fibraciones indexados por C que comparten el mismo espacio base. Empleando métodos similares, estudiamos sobre que categorías es posible clasificar diagramas de fibraciones cuyo espacio base es un diagrama no necesariamente constante.

Luis Javier Hernández Paricio (Universidad de La Rioja)

Toroidal graphics for discrete semi-flows on P1+(R) x P1+(R)

In this work we study discrete semi-flows induced by the iteration of functions of rational type defined on a product of two augmented real projective lines. In this way we obtain a technique which solves the problem that appears if one want to compute the values of a rational iteration when the denominators are zero or close to zero. These models allow us to design algorithms and implementations to visualize the basins of fixed points either as subsets of a toroidal surface or as subsets of a square which represents a torus module some identifications.

(Trabajo conjunto con J. I. Extremiana Aldana y M. T. Rivas Rodríguez)

Pablo Simón Isaza Peñaloza (Universidad de Zaragoza)

Topología de la fibra de Milnor de singularidades de la forma zn-xa yb=0

Nuestro objetivo general es describir la topología de la fibra de milnor de una singularidad de hipersuperficie. Para el caso de una curva algebraica la topología es conocida. Para el caso de una singularidad aislada de dimensión mayor, existen resultados parciales. Sin embargo, muy poco se sabe en el caso de singularidades no aisladas de dimensión mayor. Nosotros estudiamos este último caso para singularidades con ecuaciones del tipo z- xa y= 0. Mediante la construcción de complejos celulares damos una descripción topológica de la fibra y de la fibración de Milnor. Luego, utilizamos esta descripción para calcular la homología de la fibra.

Irma Pallarés Torres (Universitat de València)

Ae-codimension of a map germ and the vanishing topology

We will prove some results for smooth or holomorphic map germs (R or C, respectively) : (K,S) → (K2,0), where R or C. The Ae-codimension of a germ f is an A-invariant, it permits us classify different types of singularities. We will prove a first theorem that relates the Ae-codimension of f with the codimension in R3 of the set of viewpoints for a type of singularity X, View(X). Later, we will talk about the vanishing topology and we will define the image Milnor number of f, μI(f). We will prove a second result that says us that the Ae-codimension of f is less than or equal to μI(f), and with equality if and only if f is a quasihomogeneous germ. These two results permit us prove a simple formula for calculating the codimension in R3 of View(X) using the number of branches and the maximum number of nodes in a neighbourhood of the singularity. In higher dimensions, the second theorem is true for germs : (K,S) → (K2,0) and : (K2,S) → (K3,0), however it is conjectured for higher dimensions.

María Teresa Rivas Rodríguez (Universidad de La Rioja)

Dyamics of Bivariate Rational Functions on the Real Projective Plane

We use topological models to obtain algorithms and implementations for the study of the dynamics induced by the iteration of bivariate rational functions defined on a real augmented projective plane. As an application of our techniques, we can plot the basins of attraction of fixed points in the following geometric models: hemisphere, hemicube, Moebius band, square and disk. We can also give local graphical representations on any rectangle of the plane.

(Trabajo conjunto con J. I. Extremiana Aldana y L. J. Hernández Paricio )

David Rochera Plata (Universitat de València)

On some examples of rational offset curves

Given a planar curve α, its offset curve at a distance d is defined as the envelope of a family of congruent circles of radius d centered on the points of α. In general, offset curves are not rational even though the progenitor curve is rational. It is clear that a sufficient condition for the offset to be rational is to have the original curve a Pythagorean hodograph. Moreover, it is known the implicit equation of the offset as an algebraic curve when the original curve is rational. With that algebraic basis, examples of rational curves with rational offsets are studied, we give the implicit equation for them and some singularities are analyzed.

José Ignacio Royo Prieto (Universidad del País Vasco UPV/EHU)

Fibrados foliados gruesos (Thick Foliated Bundles)

Las hojas de la foliación de un fibrado foliado clásico son transversas a las fibras del fibrado. Proponemos el concepto de fibrado foliado grueso, en el cual se permite que las fibras del fibrado tengan una estructura foliada más rica.

Este tipo de estructura aparece en los entornos de los estratos de las foliaciones Riemannianas singulares, y lo utilizamos para relacionar la minimalidad de las foliaciones de cada estrato.

(Trabajo conjunto con M.Saralegi y R.Wolak)




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