Variational Questions in Mathematics and Physics

在数学、物理、工程、经济学和其他学科的大量问题中，人们寻求满足某些要求或约束的解决方案，更重要的是，人们寻找最优的解决方案。解决这些问题的数学理论是变分法。变分问题深深植根于自然界，因为系统倾向于处于能量尽可能低的状态。一个例子是一个激发态原子在发光的同时降到它的最低能态。同样，热液体通过冷却趋于与环境平衡。这个领域的研究目的有两个：描述一个系统趋向的状态，并解释系统是如何达到那个状态的。以原子为例，这意味着找到最低能态并描述它是如何衰变到那个态的；在液体的例子中，这意味着描述热量如何从热流向冷。这项研究将在新的和相关的例子中探讨这些问题。在理解诸如粒子碰撞气体之类的系统如何趋于平衡方面已经取得了相当大的进展。在研究中，研究者将把这些想法扩展到量子力学系统，并分析它们如何趋于平衡。在量子力学中，这些问题在很大程度上是开放的，它们的答案将为量子信息论中的各种问题提供一些启示。另一个需要研究的问题是磁场，它将电子限制在一个区域内。挑战在于找到某种精确意义上的最佳场，并描述电子如何分布。预计磁场的线条会有有趣的图案，类似于核聚变研究中产生的图案，在核聚变研究中，人们使用磁场来限制热等离子体。研究生将接触并参与研究水平数学的几个问题。作为国际数学物理协会的秘书，首席研究员被期望对社区产生重大影响。该项目解决了变分法中的各种问题。一个问题是研究三维狄拉克方程的零模态。零模在量子场论和证明物质稳定性的问题中都是重要的。众所周知，如果磁场的“3/2范数”很小，那么磁场就不能支持零模式。类似的结果也适用于向量势的“3范数”。目的是找出这些量存在零模态的尖锐必要条件。据推测，最优场的场线与霍普夫振动相似。在这方面，我们将继续研究当磁矩大于2时，耦合麦克斯韦-泡利方程解的可能爆破。一个不同的研究方向将是理解某些量子力学系统中达到平衡的方法。这样的系统可以用林德布莱德方程来描述。总的来说，关于接近平衡的速率我们说的不多，但是有一些有趣的量子马尔可夫算子，它们类似于经典的卡茨主方程。在这些情况下，期望定量确定接近平衡的速率。一种可能性是通过计算发电机的间隙来做到这一点。另一个尝试是证明熵平衡的方法。这将通过建立经典熵不等式的类似物来实现。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

In an overwhelming number of questions in mathematics, physics, engineering and economics and other disciplines one seeks solutions that satisfy certain demands or constraints and importantly, one looks for an optimal one. The mathematical theory that addresses these questions is the calculus of variations. Variational questions are deeply rooted in nature in the sense that systems tend to settle in a state that has energy as low as possible. An example is an excited atom that falls to its lowest energy state while emitting light. Likewise, a hot liquid tends to an equilibrium with the environment by cooling down. The aim of research in this area is always twofold: describe the state to which a system tends and explain how the system reaches that state. In the case of the atom this means finding the lowest energy state and describing how it decays to that state; in the case for the liquid this means describing how heat flows from hot to cold. The research will pursue such questions in relation to new and relevant examples. There has been considerable progress in understanding how systems such as gases of colliding particles tend to their equilibrium. In research the investigator will extend these ideas to quantum mechanical systems and to analyze how they tend to their equilibrium. In quantum mechanics these questions are to a large extent open and their answers will shed some light on various issues in quantum information theory. Another question to be pursued concerns magnetic fields that keep electrons confined to a region. The challenge will be to find fields that are optimal in some precise sense and to describe how the electrons are distributed. It is expected that the fields’ lines have interesting patterns that resemble the ones created in fusion research where one uses magnetic fields to confine hot plasma. Graduate students will be exposed to and contribute to several questions in research level mathematics. The principal investigator is expected to significantly impact the community, as evidenced by their role as secretary of the International Association for Mathematical Physics.The project addresses various questions in the calculus of variations. One question is to study zero modes of the three-dimensional Dirac equation. Zero modes are important in quantum field theory and in the question of proving the stability of matter. It is well known that if the ‘3/2 norm’ of the magnetic field is small then the magnetic field cannot support a zero mode. A similar result holds for the ‘3 norm’ of the vector potential. The aim is to find sharp necessary conditions on these quantities zero modes exists. It is conjectured that the optimal fields have field lines that look like the Hopf fibration. In this connection investigations will be pursued concerning possible blow up of solutions of the coupled Maxwell-Pauli equations where the magnetic moment is larger than 2. A different research direction will be to understand the approach to equilibrium in certain quantum mechanical systems. Such systems can be described by Lindblad equations. In general, not much can be said about the rate of approach to equilibrium but there are interesting Quantum Markov Operators that are analogs of the classical Kac master equation. In these cases, quantitative determination is expected of the rate of approach to equilibrium. One possibility is to do this by computing the gap of the generator. Another attempt is to prove approach to equilibrium in entropy. This will be achieved by establishing analogs of classical entropy inequalities.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.