Mathematics was born in Greece 2500 years ago with the study of Geometry. Since then, Geometry has been at the heart of Western civilization and its Philosophy, as Plato prescribed at the Academia. Since then, our society has pursued to unveil the secrets of Geometry.
The development of Geometry, following Gauss, Riemann and Grothendieck, has been instrumental in our modern understanding of our world. From Einstein's General Relativity to the expected formalizations in a unifying theory of the macro and micro study of the world we live in, we rely on geometric tools to describe the phenomena we observe. The
The current research project is framed in the area of Arithmetic Geometry, the part of Mathematics which applies techniques and ideas borrowed from Algebraic Geometry to Number Theory problems, which should be considered as the direct heir to Greek reflections on Geometry. More concretely, during the nineties, arithmetic geometers of the previous century have successfully developed a satisfactory homotopy theory, called motivic homotopy theory. This new framework has been successful enough to provide the motivic analog of Grothendieck's six functors formalism, the basic formalism for satisfactory cohomology theory. These are the modern tools mathematicians use to define "shape" and Many Number Theory problems are related to cohomology theories such as higher K-theory or the Chow ring. A central theorem of the subject is the celebrated Grothendieck's Riemann-Roch theorem and Gillet's extension to higher K-theory. This theorem compares the direct image in cohomology of K-theory and the Chow
ring. Combined with regulators, the Riemann-Roch theorem provides formulas describing both arithmetic and geometric properties. In this research project, I propose to develop Riemann-Roch-type theorems and motivic techniques to attack arithmetic problems.
