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

函数空间中的刚性、辛近似和哈密顿微分同胚群的几何

• 批准号: 1105813
• 负责人: Michael Khanevsky
• 金额: $ 12万
• 依托单位: University of Chicago
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-09-01 至 2014-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1105813&HistoricalAwards=false
• 关键词: Rigidity; functional; spaces; symplectic; approximation

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

AbstractAward：DMS 1105813，首席研究员：Lev Buhovski函数空间中的刚性是辛几何中相对较新的发现，这些项目将继续首席研究员对辛流形上泊松括号和其他多线性微分算子的刚性的研究。 一个近似问题也将研究：给定两个光滑的功能辛流形上，是否有可能近似他们的另一对功能，其泊松括号是小的一致范数？ 另一个研究方向是关于霍费尔度规中的哈密尔顿群同态。 测地线在霍费尔度规和它的扰动下的行为是关于这些自同构的各种问题的一个例子。辛几何是对经典力学的哈密顿版本的背景结构的研究，上面提到的泊松括号是对函数对的运算，在力学中用于比较力学系统的运动积分，从经典力学系统到量子理论中的对应系统。 自世纪以来，人们一直在研究这个问题，但从20世纪80年代起，辛几何引入了新的工具，揭示了许多令人惊讶的现象，包括上面提到的刚性性质，它表明泊松括号运算定义的小扰动对其作用于函数对的方式只有很小的影响。ˇ