Topics in Complex and Real Geometry

Global problems on Nash functions

speaker: Michel Coste (Universite de Rennes 1)

abstract: To have an implicit function theorem in real algebraic geometry, one is led to introduce analytic-algebraic functions. They are named after John Nash, who used them to prove that every compact smooth manifold is diffeomorphic to a component of a real algebraic set. The nice local properties of Nash functions have been known for a long time. Due to the lack of good cohomological properties, their global properties are much harder to obtain. An example of global question is whether a Nash function on a Nash subset of a Nash manifold can be extended to a Nash function on this manifold. The solution to these problems involves a result in commutative algebra (the generalized Néron desingularization) and tools of real algebraic geometry. The course will survey the results obtained in collaboration with Jesus Ruiz and Masahiro Shiota.

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