The present project is the sequel of MTM2015-69135-P which has had a great impact inside and outside the mathematical community. Just to mention some indicators: In the last 10 years our research team of 12 permanent members at UPC has a total of 192 publications and 1438 citations according to Scopus and several members have h-index between 9 and 12. We now present an extended group with 8 Phd students, 5 postdocs and 5 senior researchers abroad in the working group, so a total of 30 members. Our research is addressing fundamental questions in several areas of Mathematics focusing on Geometry, Algebra, Topology and it is multifaceted with a view towards interactions among different research branches and applications to Biology, Celestial mechanics, Robotics and Physics. Our publications in these areas have a strong mathematical component and the impact of the applications to those disciplines is reflected by the high-impact journals where these results are published like Systematic Biology or Physical Review Letters. In the last decade we have become active contributors in interdisciplinary science and we are now focused on both a theoretical point of view and the transversal applications: Yanos conjecture, Shimodas conjecture, singular Weinstein conjecture, the Riemann-Hilbert correspondence for D-modules over singular varieties, Baez conjecture, minimal models and formality of operads, phylogenetic reconstruction and machine learning, reconstruction in robotics, quantization and rigidity and flexibility for singular objects are just some of the challenging goals on our to-do-list for the next 4 years. One of the strengths of our group is the interaction among different lines either in the theoretical or the multidisciplinary sides. This interaction is bidirectional. More theoretical mathematics are applied to multidisciplinary sciences for instance algebraic statistics is an interdisciplinary field that uses tools from algebraic geometry, commutative algebra, combinatorics, and their computational sides to address problems in statistics and its applications. Sometimes multidisciplinary sciences serve as an inspirational white canvas where formulas are written in pure maths enhancing theoretical advances. This is the case of the Weinstein conjecture on the existence of periodic orbits considered in this project where inspiration comes from the restricted 3-body problem in Celestial Mechanics where periodic orbits accumulate close to the line at infinity (observed by Poincaré) giving rise to singularities which play a central role in the project. In this project this multidisciplinary to theoretical inspirational trend appears in the following binomials: Robotics/persistent homology, Phylogenetics/embedding of Markov matrices and quantum states/geometric quantization. Interplay among other lines as Algebraic Geometry and Commutative algebra and also Geometry and Dynamical Systems will also be exploited: Algebraic geometry and commutative algebra have always been closely intertwined but recently there has been a big interest from the birational geometry community in the use of algebraic positive characteristic methods as a complementary of resolution of singularities. Geometry, Topology and Dynamics will interplay at different levels including applications of the h-principle to Fluid Dynamics, Floer homology to periodic orbits, group actions to perturbation and KAM theory and to planar polynomial vector fields.
PLAN ESTATAL DE INVESTIGACIÓN CIENTÍFICA Y TÉCNICA Y DE INNOVACIÓN 2017-2020
PROGRAMA ESTATAL DE GENERACIÓN DE CONOCIMIENTO Y FORTALECIMIENTO CIENTÍFICO Y TECNOLÓGICO DEL SISTEMA DE I+D+I
SUBPROGRAMA ESTATAL DE GENERACIÓN DE CONOCIMIENTO
PROYECTOS DE I+D DE GENERACIÓN DE CONOCIMIENTO (ANTIGUES EXC)