Département de Mathématiques, Université Paris 13

Laboratoire d'Analyse, Géométrie et Applications, UMR 7539

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Arithmetic, Geometry and Topology

Conference on the occasion of Larry Breen's sixtieth birthday

December 15-17 2004

Institut Galilée, Université Paris 13, France

On the 18th of July 2004, Larry Breen will be 60 years old. As a professor at Université Paris 13 since 1988, he has greatly contributed to the rapid expansion of the Mathematics Department. During his tenure as Director of the LAGA, the Department reached maturity, expanded its scientific activities, and achieved enhanced international standing.

In his research, Breen has devoted his efforts to questions of a cohomological nature, at the interface between algebraic geometry and algebraic topology. His work has had a number of profound applications in arithmetic, algebraic geometry, algebraic topology and category theory. Some of his results have yielded applications in several areas. Thus his early and now classic results on the computation of higher extension groups have proven themselves to be extremely useful both in algebraic geometry (flat duality for surfaces) and in topology (Mac Lane cohomology).

In subsequent work in collaboration with P. Berthelot and W. Messing, he developed the crystalline Dieudonné theory pioneered by A. Grothendieck. The viewpoint of these authors has since been universally adopted and has served for important applications (the Tate full faithfullness theorem in equal characteristic proven by J. de Jong, the classification of finite flat group schemes over a discrete valuation ring developed by C. Breuil, etc).

The notions which we owe to him are noteworthy for their relevance and their simplicity. This is exemplified by the concept of a cubical torsor, which generalizes the cohomological point of view of Grothendieck regarding biextensions and the work of I. Barsotti and D. Mumford on algebraic theta functions. This object has become an essential tool in the study of the degeneracy of abelian varieties and of the compactification of their moduli spaces. It has also been applied to Korteweg-de Vries equations and to loop spaces.

Breen has long been interested in the problem of finding a geometrical interpretation of cohomology with non-abelian coefficients in a very general context. Conversely, he has also been interested in understanding various geometric notions in a cohomological or homotopy theoretic setting. In work extending J. Giraud's thesis, he realized that it is not $G$ itself but rather the crossed module $[G\rightarrow {\rm Aut}(G)]$ that furnishes the correct coefficients for the non-abelian $H^2$. He observed in particular that the stack associated to this crossed module is the monoidal stack of $G$-bitorsors, and that the good non-abelian $H^2$ classifies generalized torsors under this monoidal stack. In order to pursue these ideas and analogies, Breen formalized and studied the concept of a $2$-gerbe and gave applications to $2$-stacks. These new concepts have been used in Langlands theory (J.-P. Labesse's approach for the stabilization of the trace formula) and in theoretical physics.

In recent work in collaboration with Messing, Breen gave a new interpretation of differential calculus in the context of algebraic geometry (differential $n$-forms, the de Rham complex with value in a group). In a subsequent paper with Messing, he developed the global differential calculus of gerbes (connection, curvature and the higher Bianchi identity), extending the work of J.-L. Brylinski for abelian gerbes.

In the spirit of Larry Breen's work, this conference mixes Arithmetic, Geometry and Topology. Its intent is to celebrate a mathematical life blending fine research, strong collaborations and important commitments to the mathematical community at large and to the Université Paris 13 as a whole.