Quantum Friction, Resonance Problems and New Methods in Geometric Flows

量子摩擦问题涉及在玻色-爱因斯坦凝聚体中入侵的经典粒子的运动。PI在偏微分方程（PDE）模型上考虑问题。具体来说，PI使用了一个非线性薛定谔方程与经典粒子的轨迹耦合，该方程是在取玻色爱因斯坦凝聚体的平均场极限后得到的。PI将证明以下三个结果：(1)如果经典粒子的初始速度高于玻色爱因斯坦凝聚中的声速，即超音速，那么粒子将由于摩擦而减速，直到速度达到声速；(2)如果粒子的初始速度是亚音速，那么随着时间趋于无穷，粒子将以弹道方式运动；(3)整个系统将收敛于某个惯性模式。从技术上讲，PI必须更好地理解积分微分方程，以及非线性偏微分方程中的其他技术，例如费米黄金法则。第二个问题是共振问题。PI想在线性和非线性薛定谔方程的背景下考虑这个问题。在微扰展开中，简并或共振的存在常常使问题变得困难。PI希望能够更好地理解范式变换，然后解决薛定谔方程中的一些问题。第三个问题是发展几何流的新方法，特别是平均曲率流和里奇流。与Huisken等前人的研究不同的是，PI主要利用调制方程和谱分析来进行近乎精确的估计，而不是最大原理和熵估计。希望PI可以解决一些开放的问题。例如，在平均曲率流的情况下，演化的曲面将在有限时间内坍缩，并在坍缩点周围形成一个圆柱体。然而，圆柱体是否独一无二是一个悬而未决的问题。PI希望解决这个问题，在利玛窦流中也有类似的问题。量子摩擦现在有许多应用。一个例子是测试粒子的速度，例如中微子，通过将粒子射向某种介质，例如氩气。这种现象被称为切伦科夫辐射（1958年诺贝尔奖）。尽管切伦科夫辐射很重要，但对它的数学理解并不令人满意。在更广泛的背景下，问题是在非平衡统计力学和量子流体类，这是目前流行的。第二个问题，共振问题，将加深对范式变换的理解，并有助于解决其他问题，例如动力系统（特别是KAM理论）和量子力学中的自旋模型。对于第三个问题，近年来人们利用平均曲率流对不同表面的拓扑结构进行分类，并在广义相对论中对质量量进行了估计。PI的技术提供了关于表面演变的更精确的信息。希望PI的方法能在那里得到应用。

The problem of quantum friction concerns the motion of an invading classical particle in the Bose Einstein condensate. The PI considers the problem on a partial differential equation (PDE) model. Specifically the PI uses a nonlinear Schrodinger equation coupled to the trajectory of a classical particle, obtained after taking the mean field limit on the Bose Einstein condensate. The PI will prove the following three results: (1) if the initial speed of the classical particle is higher than the speed of sound in the Bose Einstein condensate, i.e. supersonic, then the particle will decelerate due to the friction until the speed reaches the speed of sound, (2) if the initial speed of the particle is subsonic, then the particle will travel ballistically as the time goes to infinity, (3) the whole system will converge to some inertial mode. Technically, the PI has to develop a better understanding of integro-differential equations, together with other techniques in nonlinear PDEs, for example Fermi Golden rules. The second problem is the resonance problem. The PI wants to consider the problem in the context of linear and nonlinear Schrodinger equations. Very often the presence of degeneracy, or resonance, in perturbation expansions makes the problem hard. The PI hopes to develop a better understanding of normal form transformations, and then to tackle some of the problems in Schrodinger equations. The third problem is to develop a new method for geometric flows, specifically mean curvature flow and Ricci flow. Different from the previous works, by Huisken for example, the PI mainly uses modulation equations and spectral analysis to perform almost precise estimates, instead of the maximum principle and entropy estimates. Hopefully the PI can solve some open problems here. For example, in the context of mean curvature flow, the evolving surface will collapse in finite time and form a cylinder around the collapsing point. However, whether the cylinder is unique is an open problem. The PI hopes to solve it, also a similar problem in Ricci flow.Quantum friction has many applications nowadays. One example is to test the speed of particles, for example neutrinos, by shooting the particle to some medium, for example argon. This phenomenon is known as Cerenkov radiation (Noble prize 1958). Despite its importance, the mathematical understanding of Cerenkov radiation is not satisfactory. In a broader context, the problem is in the class of non-equilibrium statistical mechanics and quantum fluid, which are popular at the moment. The second problem, the resonance problem, will deepen the understanding of normal form transformations, and help to tackle other problems, for example, in dynamical system (specifically KAM theory), and spin model in quantum mechanics. For the third problem, in the recent years, people have applied mean curvature flow to classify topological structures of different surfaces, and have estimated the amount of mass in general relativity. The PI's techniques provide more precise information on the evolution of the surfaces. Hopefully the PI's method will find applications there.