instituto de matemáticas universidad de sevilla
Antonio de Castro Brzezicki
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Top-models for understanding degrees of connectivity of semi-continuous objects
Seminario IMUS
Fecha: 10.04.2014 De 17.30 a 18.30
Lugar: Sala de reuniones del IMUS, Edificio Celestino Mutis
Autor:

Topology is an area of mathematics that provides us strong qualitative results about the degree of connectivity of the objects. Algebraic Topology works for generating algebraic tools (topological invariants) trying to “measure” the local and global topological complexity of a continuous, semi-continuous or a discrete object. Within a subdivided,semi-continuous, cellular or discrete context and endowed with an arithmetic structure, linear “algebrizations" of topological spaces has been a very productive technique for developing algorithmic techniques for computing topological invariants. In this way, homology is the first topological invariant computable in polynomial time with regards the number of elemental “bricks" of a subdivided object. To progress in the difficult problems of computability and complexity of more advanced topological invariants than homology, is a priority for Computational Algebraic Topology (CAT, for short). In fact, one of the main objectives of CAT is to design fast and efficient data representation structures and algorithms for computing algebraic and numerical topological invariants (Euler characteristics, Betti numbers, homology groups, A(infty)-coalgebra structure of homology, cohomology operations, homotopy groups, ....) for an object and its evolution. The power of Topology in technological contexts stems from its capacity of representing and extracting high-level abstract "cognition" from the shapes, the local-to-global nature of the topological invariants, and the robustness under perturbation of the topological methods and, to a lesser degree, from its classifying and discriminating role of "separating classes of patterns”. We deal with here an gentle introduction to CAT, focussing on the representational approach of topology within the setting of Digital Imagery and Computer-Aided-Design. A collateral goal in order to understand these topological models will be the visualization of highly complex algebraic topological invariants on them


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