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Position and direction
Introduces the concepts of position and direction, and describes how they are used to indicate an object's position in space and how they are useful when reading a map.
Book
Gromov’s Theory of Multicomplexes with Applications to Bounded Cohomology and Simplicial Volume
The simplicial volume is a homotopy invariant of manifolds introduced by Gromov in his pioneering paper
The first aim of this paper is to lay the foundation of the theory of
multicomplexes. After setting the main definitions, we construct the singular multicomplex
In the second part of this work we apply the theory of multicomplexes to the study of the bounded
cohomology of topological spaces. Our constructions and arguments culminate in the complete proofs of Gromov’s Mapping Theorem (which
implies in particular that the bounded cohomology of a space only depends on its fundamental group) and of Gromov’s Vanishing Theorem,
which ensures the vanishing of the simplicial volume of closed manifolds admitting an amenable cover of small multiplicity.
The
third and last part of the paper is devoted to the study of locally finite chains on non-compact spaces, hence to the simplicial volume
of open manifolds. We expand some ideas of Gromov to provide detailed proofs of a criterion for the vanishing and a criterion for the
finiteness of the simplicial volume of open manifolds. As a by-product of these results, we prove a criterion for the
eBook
Tracking Topology in Casual Dynamical Triangulations: A Toy Model in Dimension 1+1
This work presents an approach to track the spacetime topology in casual dynamical triangulations. The focus will be on a basic demonstration of the validity of the model through simulations in dimension 1+1. We will evaluate this model using standard tools, such as the spectral dimension, and will also present a calculation of the expectation value of a topological invariant, the Euler Characteristic. We will also briefly mention the recently completed dimension 3 case, in which we implemented an algorithm to find the fundamental group in this framework.
Journal Article
Presentation ECOGyT 2024
ECOGyT 2024, the First Colombian Meeting on Geometry and Topology, was held at the National University of Colombia, Bogotá campus, from July 22 to August 2, 2024.
Journal Article
Goodwillie Approximations to Higher Categories
We construct a Goodwillie tower of categories which interpolates between the category of pointed spaces and the category of spectra.
This tower of categories refines the Goodwillie tower of the identity functor in a precise sense. More generally, we construct such a
tower for a large class of
eBook
