Quantum structures and rigidity of lagrangian submanifolds

辛拓扑学是近30年来现代数学中取得了非凡发展的领域之一。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着它的现代发展。本文讨论了一些特殊空间——拉格朗日子流形的一些意想不到的刚性性质。这些空间是（非常广泛的！）实数的概括，被视为复数的子集。为了理解一个典型的，简单的刚性性质把实数想象成一条穿过平面原点的直线。假设我们对平面进行一个变换，这个变换保留了面积，使实数线常数在（-1,1）区间之外，并且使平面上所有到原点的距离超过某个固定的大的正常数k的点保持不变。稍作思考就会发现，通过这个变换得到的实数的像必须在（-1,1）区间内与实数线相交：实线因此是刚性的，因为当使用这类保面积变换时，这些交点是无法避免的。在这个项目中考虑的刚性类型是相当微妙和复杂的现象。它们还与数学的其他各个部分有关：复分析、偏微分方程和拓扑学。总之，这是一个引人入胜的研究课题！

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 !