Horario charlas del Jueves 9 de Julio

Abstracts

Giancarlo Lucchini: Kato and Kuzumaki’s conjectures, Serre’s conjectures and the Period-Index problem.

Abstract: Working these last few years with Diego Izquierdo on the arithmetic of fields, we have been led from the study of the Kato-Kuzumaki conjectures to the famous Serre's Conjecture II. The former relate diophantine and algebraic aspects of fields with fixed cohomological dimension via Milnor K-theory, while the latter concerns the arithmetic of linear algebraic groups with small cohomological dimension. The open cases of Serre's conjecture II are moreover related with another very interesting problem on the arithmetic of fields, which is the "period vs. index" problem. These two terms are invariants of central simple algebras, whose relation reflects part of the arithmetic complexity of the base field.

In this talk, I would like to give a panoramic view of how these problems are related and, at the same time, explain all these different ways of measuring the complexity of a field (cohomological dimension, C_i properties, Milnor K-theory, period and index).

José Burgos: The essential minimum of height functions on the projective line.

Abstract: The essential minimum of a height function is the minimal value that the height function can attain at generic points. There are methods to compute upper and lower bounds for the essential minimum. In a joint work with Binggang Qu, Ricardo Menares, and Martin Sombra, we prove using linear programming techniques that the difference between the upper and lower bounds can be made arbitrarily small. Therefore, one can devise a theoretical algorithm to compute the essential minimum with arbitrary precision and thus the essential minimum is a "computable" real number. This result has applications in several classical problems like the integral Chebyshev constant of the unit interval, the spectrums of the Zhang-Zagier and the Faltings heights and the asymptotic behaviour of the length of the shortest vector in the lattice  associated with the Grassmannian Gr(2,4).