CAREER: Symplectic and spectral theory of integrable systems

职业：可积系统的辛和谱理论

• 批准号: 1055897
• 负责人: Alvaro Pelayo
• 金额: $ 45.77万
• 依托单位: Washington University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-07-01 至 2015-01-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1055897&HistoricalAwards=false
• 关键词: CAREER; Symplectic; spectral; theory; integrable

The Faculty Early Career Development Award (CAREER) will support Alvaro Pelayo's research at the intersection of dynamical systems, spectral theory and symplectic geometry. More specifically, the CAREER Award will fund Pelayo's investigations on classical and quantum semitoric systems. A classical semitoric system is an integrable Hamiltonian system with two degrees of freedom for which one component generates a periodic flow. For mathematicians, semitoric systems form the next natural class of systems after toric systems. Semitoric systems retain some properties of toric systems, while at the same time they exhibit a much greater flexibility. This flexibility is reflected in the existence of singularities which are dynamically and symplectically rich.Semitoric systems are commonly found in simple physical models and arise naturally as examples in analysis, partial differential equations, algebraic geometry and symplectic geometry. As a matter of fact, a semitoric system defines a singular toric fibration whose base comes endowed with a singular integral affine structure. These singular affine structures are a central concept in symplectic topology and mirror symmetry. A semitoric quantum system is given by two commuting self-adjoint semiclassical operators acting on a Hilbert space whose principal symbols form a classical semitoric system. One of Pelayo's main goals is to study the spectral theory of semitoric quantum systems and how it relates to classical systems. Concretely, Pelayo's plan is to work towards verifying that the semiclassical joint spectrum of a quantum semitoric system determines completely the system; this is the Spectral Conjecture, widely considered the most spectacular problem in the area. Proving this conjecture requires establishing a number of results in semiclassical analysis, giving a tool for future research in the field, independent of the conjecture. Another component of Pelayo's research plans is to continue studying semitoric systems in a more general context, in particular as it regards to the study of the convexity and connectivity properties of the singular Lagrangian fibrations which semitoric systems induce, and which are of special interest in mirror symmetry.The research that the CAREER Award will support belongs to the context of dynamics and geometry. Dynamics is the study of the motion of bodies. Geometry is the study of shape (broadly understood) of objects and spaces.More specifically, the CAREER Award will support Pelayo's research at the intersection of symplectic geometry, spectral theory and dynamics. Symplectic geometry has its roots in physics, and provides an appropriate mathematical framework to study many problems of physics and chemistry and their quantum counterparts. Pelayo's research on this topic exhibits the interplay between mathematical and physical theories. Indeed, several groups of physicists and chemists working on modern quantum spectroscopy have been interested in seeing how mathematical methods can contribute to advance their research, and predict new physical phenomena. They have been particularly interested in understanding the global structure of joint energy-momentum spectra of small molecules. Even more, they have recognized the pivotal role that mathematical invariants play in this problem. Their works have motivated a large number of mathematical questions. The physicists have asked whether one can single out an optimal set of mathematical invariants that would characterize a physical system and then detect these invariants in the spectrum of the system. The detection of the invariants in the system spectrum will allow us to reconstruct the system and hence predict new phenomena. The CAREER Award will support Pelayo's investigations into this crucial question. These applications to quantum molecular spectroscopy are integrated in Pelayo's approach of employing methods from pure mathematics (symplectic and spectral theory, microlocal analysis) to address problems from the applied sciences. More generally, Pelayo's research is focused on a fundamental type of physical system, the so called integrable systems. One can find direct applications of integrable systems in numerous contexts. Some such examples are nonlinear control, locomotion generation in robotics, elasticity theory, plasma physics, field theory and planetary mission design. In order to communicate this relevance of mathematical research to the general public, Pelayo has been interviewed on several occasions by the press (radio and magazines with several hundred thousand audience members). He has commented and advocated for an understanding of the leading role that mathematicians play in the advance of applied science and technology in society. Pelayo will continue actively reaching out to the general public during the tenure of the award, and will continue to take a leading role at Washington University outreach programs. In this respect he will run a Talent Identification Program at Washington University to encourage and nurture undergraduate research at an early stage.

教师早期职业发展奖（CAREER）将支持阿尔瓦罗·佩拉约在动力系统，谱理论和辛几何的交叉点的研究。更具体地说，职业奖将资助佩拉约对经典和量子混沌系统的研究。经典的哈密顿系统是一个具有两个自由度的可积哈密顿系统，其中一个分量产生周期流。对于数学家来说，复曲面系统是继复曲面系统之后的下一类自然系统。半导系统保留了复曲面系统的一些特性，同时它们表现出更大的灵活性。这种灵活性反映在奇异性的存在是动态和辛丰富。半系统通常在简单的物理模型中发现，并自然出现在分析，偏微分方程，代数几何和辛几何的例子。事实上，一个复曲面系统定义了一个奇异复曲面纤维化，它的基被赋予一个奇异积分仿射结构。这些奇异仿射结构是辛拓扑和镜像对称中的中心概念。一个量子系统是由两个可交换的自伴半经典算符作用在一个Hilbert空间上，它们的主符号构成一个经典的量子系统。佩拉约的主要目标之一是研究量子系统的光谱理论，以及它与经典系统的关系。具体地说，佩拉约的计划是致力于验证量子动力学系统的半经典联合光谱完全决定了系统;这就是光谱猜想，被广泛认为是该领域最引人注目的问题。 证明这个猜想需要在半经典分析中建立一些结果，为该领域的未来研究提供独立于猜想的工具。Pelayo研究计划的另一个组成部分是继续在更一般的背景下研究混沌系统，特别是关于混沌系统诱导的奇异拉格朗日纤维化的凸性和连通性的研究，以及镜像对称的特殊兴趣。CAREER奖将支持的研究属于动力学和几何学的背景。 动力学是研究物体运动的学科。几何学是对物体和空间的形状（广泛理解）的研究。更具体地说，CAREER奖将支持Pelayo在辛几何，谱理论和动力学交叉领域的研究。辛几何起源于物理学，它为研究物理学、化学和量子力学的许多问题提供了一个合适的数学框架。佩拉约的研究对这一问题的展品之间的相互作用数学和物理理论。事实上，几组研究现代量子光谱学的物理学家和化学家一直对数学方法如何促进他们的研究并预测新的物理现象感兴趣。他们对理解小分子的联合能量-动量谱的全局结构特别感兴趣。更重要的是，他们已经认识到数学不变量在这个问题中发挥的关键作用。他们的工作激发了大量的数学问题。物理学家们提出了这样一个问题：是否可以挑选出一组最优的数学不变量来表征一个物理系统，然后在系统的光谱中检测这些不变量。 检测系统谱中的不变量将使我们能够重建系统，从而预测新的现象。职业奖将支持佩拉约对这一关键问题的调查。 量子分子光谱学的这些应用被集成到佩拉约采用纯数学方法（辛和光谱理论、微局部分析）来解决应用科学问题的方法中。更一般地说，佩拉约的研究集中在一个基本类型的物理系统，所谓的可积系统。人们可以在许多情况下找到可积系统的直接应用。一些这样的例子是非线性控制，机器人运动生成，弹性理论，等离子体物理学，场论和行星使命设计。为了沟通这一相关的数学研究，以广大公众，佩拉约已多次接受采访的新闻界（电台和杂志与几十万观众）。他评论并主张理解数学家在社会应用科学和技术进步中发挥的主导作用。佩拉约将继续积极接触广大公众在任期内的奖项，并将继续采取在华盛顿大学外展计划的主导作用。在这方面，他将在华盛顿大学开展一项人才鉴定计划，以鼓励和培养早期阶段的本科生研究。