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Introduction

Our current project continues the geometric study of algebraic, symplectic and arithmetic varieties (DGICYT PS90-0069, PB93-0790, PB96-0234, BFM2000-0799, BFM2003-06001, BFM2003-02914,  DGI MTM2006-14234, MTM2012-38122-C03/FEDER) as well as its applications to Robotics, Computational Biology and Materials Science started in the last project. 

The topics and objectives are summarized in 4 research areas:

Algebraic geometry. This involves the study of irregular varieties of Euler characteristic 1 or of Albanese general type, bounds for the slope of fibred varieties, vector bundles without intermediate cohomology on cubic surfaces and 3folds, properties of the germs of singular analytic morphisms between surfaces (from the local and global point of view), analytical classification of sandwiched surface singularities, representation theory for 2-groups, the problem of absolute factorization.


Arithmetic geometry. This involves the study of computation of the height of Toric varieties, application of Langlands correspondence to Arakelov theory, Arakelov motivic cohomology, arithmetic Riemann-Roch  theorems for arbitrary proper morphisms.

Applications.

  • Computational biology: This involves the development of tools and algorithms for phylogenetic reconstruction using the information from the algebraic varieties associated to evolutionary models.
  • Robotics: We use tools based in topology and algebraic geometry to develop Hybrid Vision Camera Systems and to study singularities of parallel manipulators.
  • Materials science: This involves the use of algebraic and geometric tools in materials science.