Doctoral School Mathematics and Scientific Computing

Abstract: Uncertainty in the input data is an important issue in optimisation. While robust optimisation provides solutions that are feasible in any realisation of the input data, adjustable robust optimisation expands on this notion. Here, some decisions can be made after the full data is known. Using an adjustable robust model, we develop a heuristic approach to solving a transportation problem where the demands for materials transported from a source to a sink are uncertain.

Abstract: Flow polytopes are a kind of polytopes that was born in the study of optimization, due to this an interesting question about them is related to their triangulation, beacuse there are a lot of famous lattices that can be studied as the dual of the triangulation of these flow polytopes, but the construction is only related to the idea behind them. In this presentation, we will talk about the possibility of doing a general construction of this dual for any flow polytope and its triangulations.

Abstract: We study subsets of F_pⁿ that do not contain progressions of length p or in other words full lines. We denote by r_p(F_pⁿ) the cardinality of such subsets containing a maximal number of elements. A starting point is the lower bound r_p(F_pⁿ) ≥ (p-1)ⁿ, which is achieved by a hypercube of side length p-1. We will show that this is not optimal for n ≥ 3. The main focus of this talk will be on the asymtotic behaviour of r_p(F_pⁿ) as n tends to infinity. We will present a construction that gives an improvement for every p ≥ 5, and also gives the first lower bounds cⁿ for r_p(F_pⁿ) with c=p-o(1), as also p tends to infinity.