CAREER: Symplectic and spectral theory of integrable systems

职业：可积系统的辛和谱理论

• 批准号: 1518420
• 负责人: Alvaro Pelayo
• 金额: $ 23.97万
• 依托单位: University of California-San Diego
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-08-01 至 2018-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1518420&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Alvaro Pelayo在动力系统、谱理论和辛几何交叉领域的研究。更具体地说，CAREER奖将资助Pelayo对经典和量子半元系统的研究。经典半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。对数学家来说，半符号系统形成了继环面系统之后的下一类自然系统。半系统保留了环面系统的一些特性，同时表现出更大的灵活性。这种灵活性反映在奇点的存在上，奇点是动态和辛丰富的。半系统通常存在于简单的物理模型中，并且作为分析、偏微分方程、代数几何和辛几何中的例子自然出现。事实上，一个半系统定义了一个奇异的环振，它的基底被赋予了奇异的积分仿射结构。这些奇异仿射结构是辛拓扑和镜像对称中的一个中心概念。用两个交换自伴随半经典算子作用于希尔伯特空间，它们的主符号构成一个经典半经典系统。Pelayo的主要目标之一是研究半半量子系统的谱理论以及它与经典系统的关系。具体地说，Pelayo的计划是努力验证量子半系统的半经典联合谱完全决定了系统；这就是谱猜想，被广泛认为是该领域最引人注目的问题。证明这一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一个独立于猜想的工具。Pelayo研究计划的另一个组成部分是在更广泛的背景下继续研究半系统，特别是关于半系统诱导的奇异拉格朗日纤维的凹凸性和连通性的研究，这对镜像对称有特殊的兴趣。CAREER奖将支持的研究属于动力学和几何的范畴。动力学是研究物体运动的学科。几何学是研究物体和空间形状的学科。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。辛几何植根于物理学，并为研究物理和化学及其量子对应的许多问题提供了一个适当的数学框架。佩拉约在这个主题上的研究展示了数学和物理理论之间的相互作用。事实上，几组从事现代量子光谱学研究的物理学家和化学家对数学方法如何有助于推进他们的研究和预测新的物理现象很感兴趣。他们对了解小分子联合能量动量谱的整体结构特别感兴趣。更重要的是，他们认识到数学不变量在这个问题中所起的关键作用。他们的工作激发了大量的数学问题。物理学家们提出的问题是，是否可以挑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。检测系统光谱中的不变量将使我们能够重建系统，从而预测新的现象。职业生涯奖将支持佩拉约对这个关键问题的调查。这些量子分子光谱学的应用被整合到Pelayo的方法中，该方法采用纯数学方法（辛和光谱理论，微局部分析）来解决应用科学中的问题。更一般地说，Pelayo的研究集中在一种基本类型的物理系统上，即所谓的可积系统。人们可以在许多情况下找到可积系统的直接应用。这些例子包括非线性控制、机器人运动生成、弹性理论、等离子体物理、场论和行星任务设计。为了向公众传达数学研究的相关性，Pelayo多次接受新闻界（拥有数十万听众的电台和杂志）的采访。他评论并倡导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。在该奖项的任期内，佩拉约将继续积极地与公众接触，并将继续在华盛顿大学的外展项目中发挥主导作用。在这方面，他将在华盛顿大学开展一项人才识别计划，以鼓励和培养早期阶段的本科生研究。