Semiclassical Analysis

PI研究由量子力学、波传播和混沌动力学引起的数学问题，特别是波的振荡和衰减。就像钟发出的声音会逐渐消失一样，波或不稳定的分子也会以一定的速率振荡和衰减。这两种速率（振荡速率和衰减速率）是系统的特性，而不是测量系统的方式。了解它们的行为在构造和检测中都很有用。例如，在工程中，这两种速率的比值被称为质量因子，它告诉我们每个循环的能量损失量。在微机电系统（MEMS）的设计中，了解特定系统模式的这个比率是很重要的。在不同的尺度上，类似的模式出现在黑洞碰撞产生的引力波中，它们的（假设的）探测可以提供有关黑洞的信息。PI寻找统一的主题，将这些模式的分布和不同系统的几何形状联系起来。许多物理系统可以用状态演化来描述。观察到以下相关性：一个测量一个状态相对于另一个状态的时间演化。时间表示法可以用频率表示法（通过傅里叶变换）代替，从而产生功率谱。功率谱的极点出现在不同的环境中，被称为散射极点（障碍物散射）、量子共振（量子散射理论）、准正态模式（广义相对论）、波利科特-鲁埃尔共振（混沌理论）。这些极点提供了关于长时间行为的信息：实部对应于振荡速率，虚部对应于衰减速率。PI在上面提到的不同设置中研究这些极点。一个反复出现的主题是经典/量子（波）对应的使用，它表明系统的“经典”属性和波的属性之间存在微妙的相互作用。PI在许多情况下研究这种现象，特别是当混沌行为存在于经典水平时。最近，为研究经典量子对应（微局部分析）而开发的方法在研究纯动力学问题（如zeta函数的亚纯延拓和x射线断层扫描问题）中变得有用。

The PI studies mathematical problems motivated by quantum mechanics, wave propagation, and chaotic dynamics, in particular, oscillations and decay of waves. Just as a bell sounds a fading note, a wave or an unstable molecule oscillates and decays at certain rates. These two rates (of oscillation and of decay) are properties of the system and not of the way in which it is measured. Understanding their behaviour can be useful both in construction and in detection. For instance, in engineering, the ratio of the two rates is called the quality factor and tells us the amount of energy loss per cycle. Knowing this ratio for modes of specific systems is important in design of, for instance, microelectromechanical systems (MEMS). On a different scale, similar modes appear in gravitational waves generated by colliding black holes and their (hypothetical) detection could provide information about black holes. The PI searches for unifying themes connecting the distribution of these modes and geometries of various systems.Many physical systems can be described using evolution of states. The following correlations are observed: one measures the time evolution of one state against another state. The time representation can be replaced by the frequency representation (by taking a Fourier transform) which produces the power spectrum. The poles of power spectrum appear in different settings and are called scattering poles (obstacle scattering), quantum resonances (quantum scattering theory), quasinormal modes (general relativity), Pollicott--Ruelle resonances (chaos theory). These poles provide information about long time behaviour: the real part corresponds to the rate of oscillations, and the imaginary part to the rate of decay. The PI studies these poles in the different settings mentioned above. One recurrent theme is the use of the classical/quantum (wave) correspondence which suggests subtle interplay between "classical" properties of the system and properties of waves. The PI investigates this phenomenon in many settings, in particular when chaotic behaviour is present on the classical level. Most recently methods that were developed for the study of classical quantum correspondence (microlocal analysis) became useful in the study of purely dynamical problems such as meromorphic continuations of zeta functions and problems in X-ray tomography.