Quantum structures and rigidity of lagrangian submanifolds

点击翻译按钮获取中文摘要

Symplectic topology is one of the fields of modern mathematics that has seen an extraordinary development in the last 30 years. Its origin come from physics - for example from the classical problem of the stability of the solar system - and various physical concepts continue to motivate its modern evolution. The present proposal is concerned with a number unexpected rigidity properties of some particular spaces - called Lagrangian submanifolds. These spaces are (very extensive !) generalizations of the real numbers viewed as a subset of the complex ones. To have an idea of a typical and very simple rigidity property imagine the reals as a straight line going through the origin in the plane. Suppose that we apply to the plane a transformation which preserves area and which leaves the real line constant outside the interval (-1,1) and which leaves fixed all the points in the plane whose distance from the origin exceeds some fixed large, positive constant K. A moment of thought will show that the image of the reals by this transformation has to intersect the real line inside the interval (-1,1): the real line is thus rigid in the sense that these intersection points can not be avoided when using area preserving transformations of this sort. The types of rigidity considered in this project are considerably subtler and more complicated phenomena. They are also related to various other parts of mathematics: complex analysis, partial differential equations and topology. Altogether this makes for a fascinating subject of investigation !