Rigidity in functional spaces, symplectic approximation and geometry on the group of Hamiltonian diffeomorphisms

摘要奖：DMS 1105813，主要研究人员：Lev Buhovski在函数空间中的刚性是辛几何中相对较新的发现，这些项目将继续主要研究人员对辛流形上Poisson括号和其他多线性微分算子的刚性的研究。还将研究一个逼近问题：给定辛流形上的两个光滑函数，是否有可能用另一对泊松括号在一致范数下较小的函数来逼近它们？另一项研究涉及Hofer度量中的哈密尔顿微分同态群。测地线在霍费尔度规及其扰动下的行为是关于这些微分同胚的一类悬而未决的问题的一个例子。辛几何是对哈密顿版本的经典力学的背景结构的研究，上面提到的泊松括号是对函数对的运算，在力学中用于比较机械系统的运动积分，并在量子理论中从经典机械系统传递到对应系统。这个问题从19世纪就开始研究了，但是从20世纪80年代开始，新的工具引入到辛几何中，揭示了许多令人惊讶的现象，包括上面提到的刚性性质，这表明泊松括号运算的定义的微小扰动对它作用于函数对的方式只有很小的影响。

AbstractAward: DMS 1105813, Principal Investigator: Lev BuhovskiRigidity in functional spaces is a relatively recent discovery in symplectic geometry, and these projects will continue the principal investigator's study of rigidity of Poisson brackets and other multilinear differential operators on symplectic manifolds. An approximation problem will also be studied: Given two smooth functions on a symplectic manifold, is it possible to approximate them by another pair of functions whose Poisson bracket is small in the uniform norm? Another line of investigation concerns the group of Hamiltonian diffeomorphisms, in the Hofer metric. The behavior of geodesics under the Hofer metric and under perturbations of it is an example of the kinds of questions that remain open regarding these diffeomorphisms.Symplectic geometry is the study of the background structure for the Hamiltonian version of classical mechanics, and the Poisson bracket referred to above is an operation on pairs of functions that is used in mechanics to compare integrals of motion for a mechanical system and in passing from a classical mechanical system to a counterpart system in quantum theory. The subject has been studied since the nineteenth century, but the introduction of new tools into symplectic geometry from the 1980s onward has opened up many surprising phenomena, including the rigidity property cited above, which shows that small perturbations of the definition of a Poisson bracket operation have only small effects on the way it acts on pairs of functions.ˇ