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Résumé |
I will give a gentle introduction to some recent applications of o-minimal
techniques to questions in Hodge theory and functional transcendence, due to
Bakker, Brunebarbe, Klingler, and Tsimerman. Roughly speaking, a structure is a
collection of subsets of the power set of R^n, one for each n, that are definable
by first order formulas involving the operations and the order coming from the
real numbers, as well as function of a certain class chosen beforehand (e.g. the
real exponential). Such a structure is called o-minimal if the subsets of R are
finite unions of points and open intervals. This tameness condition allows one to
develop topology or geometry without the usual pathologies that one encounters
while working with arbitrary spaces. For example, if a closed subset of an
analytic variety is definable in some o-minimal structure, then it is
automatically algebraic, whether the ambient space is proper or not. I will
explain why period maps coming from Hodge theory are definable in a suitable o-
minimal structure, and a few spectacular consequences of this result. |
