At an individual level, research has been carried out through visits between the project participants. As a collective effort, we have organized 19 workshops and conferences. More than 200 joint publications were produced as a direct result of these actions. Given the volume of results obtained, we are forced to mention only some of them.
WP1
How many different plane networks can be defined on n sites in the plane, provided that each link must be a straight line segment? We gave a new lower bound on this number.
We have improved the upper bound on the minimum number of pairs of links that must cross in complete straight line network. This is an important problem in Combinatorial Geometry. This was made possible with new software developed by us. The software has been made publicly available.
WP2
We have been working in combinatorial optimization problems on stochastic elements. We have studied the algorithmic problem of finding the maximum enclosing box on points in the line or the plane, where each point has assigned a weight and a probability of being present.
WP3
The convex hull of a point set is the minimum polygon that encloses the point set. A variant called the rectilinear convex hull requires that the polygon has vertical or horizontal sides. We obtained an algorithm for maintaining the rectilinear convex hull while the point set is rotated.
We gave an algorithm for finding the angle of rotation of a polygon so that it encloses as many points as possible from a given set of points. Applications of polygon placement problems include global localization of mobile robots, pattern matching, and geometric tolerance.
We computed the region (called the kernel) of a polygon from which every other point of the polygon can be seen, subject to restrictions on this visibility. This has applications in surveillance scenarios.
WP4
We designed algorithms to assign flight plans for drones equipped with cameras, in order to film a prescribed set of scenes. The real life application of this algorithm is for sport events; in particular for bicycle races; in which the center of attention is constantly in motion. A key feature of the algorithms is that they use algorithms for computing flows in networks to find the flight plans.
Using algorithms for network partitioning, we designed an algorithm to classify flamenco melodies.
Using algorithms for shorted paths in directed networks, we designed an algorithm to compute melodic templates for flamenco songs.
WP5
Dissemination of results has followed several lines. First, several academic talks have spread results of the project to a specialized audience in universities, workshops, and conferences. Second, dissemination talks covering results of the project were included in activities like the “Pint of Science” festival, the MATRIX conference, or a satellite activity of the Meeting of the Spanish Royal Mathematical Society. Third, dissemination articles have been written at the Mapping Ignorance journal (ISSN 2529-8992). Fourth, and final, dissemination articles on newspapers (like ABC or The Conversation) have helped to reach a broad public explaining mathematics to some extent related to the project.
To promote gender equality, we organized a summer school for women in mathematics. The main objective is to encourage female students to continue with graduate studies in mathematics. Various advanced courses and talks have been scheduled. The event took place the city of Zacatecas, Mexico in 2019.
