**Institution**: Consejo Superior de Investigaciones Científicas

**Position**: Investigador Científico

**Office**: 409 **Phone**:+34 912999 789

**E-mail**: burgos()icmat.es

#### Biographical Review

José Ignacio Burgos is researcher of the Instituto de Ciencias Matemáticas (ICMAT).

His scientific interest lies Arithmetic Geometry, more concretely in Arakelov Geometry and the theory of motives.

Arakelov Theory is at the cross road between Number theory, Algebraic Geometry and Complex Analysis. In Arakelov Theory one considers arithmetic varieties. That is, varieties defined over the ring of integers. An arithmetic variety can be seen as continuous family of varieties, indexed by primes. The main idea behind Arakelov Theory is that we can compactify such varieties by adding, as fiber at infinity, the corresponding complex varieties. The union of the arithmetic variety and the complex variety should behave like a complete algebraic variety. Thus, geometry over the integers (Number Theory) together with analysis on the added fiber (Complex Analysis) should behave like geometry over a compact variety (Algebraic Geometry).

The theory of Motives was postulated by Grothendieck to unify the common properties of all cohomology theories. The search for a theory of motives has proven formidable and we do not have yet a general theory of motives. Nevertheless, many results have already been obtained and the theory of motives is a rich research field with many connections to other areas of Mathematics and Theoretical Physics.

José Ignacio Burgos has many articles in these fields in journals like Inventiones Math., Duke math. Journal and Journal of Algebraic Geometry, with collaborators in France, Germany, Japan, Romania, USA and Argentina. Among other results he has extended Arakelov Geometry to the singular metrics that appear when studying modular varieties, he has proves an arithmetic Riemann Roch theorem for arbitrary morphism and has given the first precise comparison between the regulators of Beilinson and Borel.

Besides his work in Arithmetic Geometry, he has also made contributions in Computer Vision and Quantum Chemistry.