Quantum structures and rigidity of lagrangian submanifolds

辛拓扑学是现代数学的一个领域，在过去的30年里得到了非凡的发展。它的起源来自物理学-例如来自太阳系稳定性的经典问题-各种物理概念继续推动其现代演变。本建议是关于一些意想不到的刚性性质的一些特殊的空间-所谓的拉格朗日子流形。这些空间是（非常广泛！）把真实的数看作复数的子集。为了理解一个典型的、非常简单的刚性性质，可以把实数想象成一条通过平面原点的直线。假设我们对平面应用一个变换，该变换保持面积不变，使真实的直线在区间（-1，1）之外保持常数，并且使平面上所有距原点的距离超过某个固定的大的正常数K的点保持固定。稍加思考就会发现，通过这种变换，实数的像必须与区间（-1，1）内的真实的直线相交：因此，真实的直线是刚性的，因为当使用这种面积保持变换时，这些交点无法避免。在这个项目中考虑的刚性类型是相当微妙和复杂的现象。它们也与数学的其他部分有关：复分析，偏微分方程和拓扑学。总之，这是一个迷人的调查主题！

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 !