Applications of Computational Algebraic Topology

In this note a course given at the "UN Encuentro de Matemáticas 2016" held in Bogotá (Colombia) is described. The course was divided into four parts, devoted, respectively, to basic denitions in Algebraic Topology, to the algorithmic aspects of Topology, to the Kenzo computer system and, finally, to the applications of Computational Algebraic Topology. In the first talk the basic concepts of Algebraic Topology were presented, stressing the importance of replacing the topological space notion by other combinatorial analogues, easing the introduction of homological techniques. In the second talk, we reviewed several algorithms devised to compute homological objects. In the third talk, we described the Kenzo program, using its new version accessible via Internet. In the last talk, persistent homology was introduced and an actual application of topology to image processing was presented, in the frame of a project related to the fight against neurodegenerative diseases, as Alzheimer.