Jean-Benoît Bost - Biography#

Jean-Benoît Bost was born in 1961. He is currently a full professor at University Paris-Sud XI (Orsay).

His area of expertise is very broad (mathematical physics, non-commutative geometry, algebraic geometry, and number theory) and we'll just sketch some aspects of his contributions.

His first works are in mathematical physics (string theory) where he has several contributions with different co-autors (Jolicoeur, Nelson, Alvarez-Gaumé, Moore and Vafa).

He then made some important contributions in K-theory and non-commutative geometry (Principe d'Oka, K-théorie et systèmes dynamiques non commtuatives, Invent. Math. 1990). He wrote a seminal paper with Alain Connes (Hecke algebras, type III factors, and phase transitions with spontaneous symmetry breaking in number theory, Selecta Math. 1995). In this paper a C*- dynamical system whose partition function is the Riemann zeta function is constructed and appears as a particular case of a very general construction which has some interesting implications in mathematical physics, number theory, and non-commutative geometry.

Bost then wrote with Bismut a long and deep paper on the analytical aspects of determinant line bundles on degenerating curves (Fibrés déterminants, métriques de Quillen et dégénéréscence des courbes, Acta Math. 1990). It was the starting point of his interest for Arakelov theory where he has several important contributions. His work with Gillet and Soulé on heights of projective varieties (JAMS 1994) is of constant use in diophantine geometry. He was the first to use systematically Geometric Invariant Theory (GIT), in the theory of heights of cycles (Semi-stability and heights of cycles, Invent. Math. 1994).

Bost also explained how classical diophantine geometry should be interpreted in geometric terms via Arakelov theory. His theory of evaluation maps and the associated slope estimates is now a basic tool in diophantine geometry with a wide range of applications. It replaces the classical technique of auxiliary polynomials adding the flexibility allowed by modern algebraic geometry, notably Néron models, GIT and work of Moret-Bailly. It was introduced by Bost in his Bourbaki report on the work of Masser and Wüstholz, where the proof of the "isogeny estimates" (a crucial step in this approach to the Mordell conjecture) is written in this more conceptual way and using the flexibility of the method, several statements are made effective. This theory was used by several authors including Viada, Gaudron, Gasbarri, Graftieaux to prove deep and interesting diophantine results.

The main application of this method was given by Bost in a series of papers starting with (Algebraic leaves of algebraic foliations, IHES 2001), the last one with Chambert-Loir), where the slope estimates are applied systematically to prove some algebraicity criteria concerning germs of formal subvarieties of algebraic varieties defined over number fields related to the Grothendieck-Katz conjecture. Some applications to the algebraicity of leaves of algebraic foliations are obtained and the link with positivity in Arakelov theory is explained.
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