Edward Bierstone, Toronto University, Canada, Michel Coste, Rennes 1 University, Michel Merle, Nice University, Tadeusz Mostowski, Warsaw University, Poland
The aim of this conference is to focus on some recent approaches on the real and complex aspects of Singularity Theory, which have led to new developments in the past years. Our idea is to focus on four subjects, by developing both the abstract approach together with the applications already obtained, both in real and complex singularities.
To begin with, let us cite the introduction of the weight structure on the homology of real algebraic varieties (by McCrory and Parusinski), which may be seen as a real analogue of the mixed Hodge structure in complex geometry. The weight structure leads to nice invariants in real geometry like the virtual Poincaré polynomial, and several works are in progress in this direction (in the equivariant setting by Priziac, with respect to the Milnor fibre by Fichou-Parusinski, or for classification purpose by Campesato).
Another main progress concerning the Zariski equisingularity has been made via the arcwiseanalyticity (by Parusinski and Paunescu), leading to new astonishing results such as an
approximation theorem between analytic and algebraic singularities (for complex or real numbers) by Bilski-Parusinski-Kurdyka-Rond or to the definition of an intersection homology in real geometry by McCrory-Parusinski.
A third main theme is around motivic integration, with a particular focus on the recent developments in motivic integration via model theory, leading to a new approach to study Milnor fibres (Hrushovski-Loeser in the complex setting, and more recently Yin in a commun approach in the complex and real case).
Finally, the study of rational continuous functions is now an area of very active research, both in the smooth and singular situation, with recent progress in the direction of hereditarily rational functions of regulous functions (including the semi-algebraic and Nash setting).