Semiclassical Analysis

PI研究由量子力学、波传播和混沌动力学引发的数学问题，特别是波的振荡和衰减。就像钟声发出褪色的音符一样，波或不稳定的分子以一定的速度振荡和衰变。这两个速率(振荡和衰减率)是系统的特性，而不是测量方法的特性。了解它们的行为在建筑和检测中都是有用的。例如，在工程中，这两个速率的比率称为品质因数，它告诉我们每个循环的能量损失量。了解特定系统的模式的这一比率在例如微电子机械系统(MEMS)的设计中很重要。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)探测可以提供关于黑洞的信息。PI寻找将这些模式的分布和不同系统的几何结构联系起来的统一主题。许多物理系统可以用状态演化来描述。观察到以下相关性：一种是测量一种状态相对于另一种状态的时间演化。时间表示可以用产生功率谱的频率表示(通过进行傅立叶变换)来代替。功率谱的极点出现在不同的环境中，分别称为散射极点(障碍散射)、量子共振(量子散射理论)、准正则模(广义相对论)、Pollicott-Ruelle共振(混沌理论)。这些极点提供了有关长时间行为的信息：实数部分对应于振荡速率，虚数部分对应于衰减率。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一现象，特别是当经典水平上存在混沌行为时。最近，用于研究经典量子对应的方法(微域分析)在纯动力学问题的研究中变得有用，例如Zeta函数的亚纯延拓和X射线层析成像中的问题。

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.