Abstract:

L'objectiu general del grup és aprofundir en l'estudi d'estructures geomètriques i les seves aplicacions. Les estructures geomètriques considerades són varietats algebraiques, simplèctiques o diferenciables i les seves aplicacions es centren principalment als camps de la biologia, la robòtica, la física, els sistemes dinàmics i la mecànica celeste. Per fer-ho utilitzarem diverses eines (geomètriques, algebraiques, topològiques, aritmètiques, diferencials i computacionals) i en moltes ocasions fusionarem tècniques provinent de diversos àmbits. Membres del grup treballen dintre d'equips pluridisciplinars i en línies de recerca transversals.]]>

Abstract:

This project focuses on various aspects of the areas of algebraic geometry, symplectic geometry, commutative algebra, algebraic topology and their interactions, and their applications to biology, physics and robotics. We propose to approach the problems considered both from the traditional view of research in mathematics and from a multidisciplinary perspective on the part of applications. This project is the continuation of MTM2012-38122-C03-01, which has been very successful and has had a major impact both within the mathematical community and other disciplines and even outside the scientific community (to name a few indicators, we have 160 publications since 2005, the group's work has received more than 700 citations according to Scopus and there are members of the group with h-index between 6 and 8). The current group has 23 researchers; 15 of them doctors (of which 11 are senior and 4 are postdoc) and 8 tPhD students. In the last decades, algebraic geometry has experimented a spectacular development in interaction with affine areas of research. For instance, some ideas of algebraic geometry find its generalization in symplectic geometry; in Poisson Geometry, the algebraic techniques are omnipresent in the study of generalized complex geometry also the algebraic topology techniques are present in the modern homotopy theory of schemes. Other examples of interaction of disciplines is observed in the recent applications of geometric analysis techniques to the realm of Poisson Geometry and Complex geometry. In this project we will keep on exploring the connection among these areas and we will endeavour to solve important problems relative to known conjectures such as Xiaos conjectures for irregular varieties in Algebraic Geometry, Voevodskys nilpotence conjecture in K-theory or the conjecture of Guillemin-Sternberg concerning quantization and reduction for symplectic manifolds and their generalizations, or the Sturmfels-Sullivant conjecture on phylogenetic varieties. Given the success of the results obtained in the preceding project, in this project we plan to consolidate the interdisciplinary aspects of the project. A strong mathematical component is observed in the publications of the group in the areas of biomathematics, robotics and physics. Notwithstanding, those results can really be applied to those disciplines as observed in the list of high-impact journals where these results are published (Nature Methods, Systematic Biology, Molecular Biology and Evolution, BMC Evolutionary Biology, International Journal of Computer vision, Physical Review Letters, Journal of Cosmology and Astroparticle Physics, Physical Review D, Physical Reviews E). Our team collaborates with various national and international groups. We enclose details of the collaborators in the scientific proposal and highlighting now the following national collaborative centers: ICMAT, IRI, Centre for Genomic Regulation, UAB and UB and international: MIT-Northeastern, U. Pavia, U. Bayreuth, U.Kansas, U .Leicester, KU Leuven, UC Berkeley, Mathematical Institute of the Polish Academy of Sciences, U. Tasmania, New Zealand Biomathematics Research Centre.]]>