Dispersive partial differential equations: between a deterministic and a probabilistic approach

大自然有多种表达方式，但波浪现象无疑是其力量最普遍的展示之一。从平静的海洋中央形成的反常波，到黑洞的奇点，再到通过光纤电缆的信号，再到玻色-爱因斯坦凝聚体，波动现象出现在许多不同的物理环境中。显然，通过只知道几个测量数据就能够准确预测波动现象，将在许多方面影响我们的世界：从疏散计划，到对我们深部宇宙的理解，从更好和更清晰的通信工具，到对微观宇宙的理解。获得对波动现象的准确预测的根本困难在于，它们是通过数学恒等式来表示的，而这些恒等式太复杂了，我们无法使用经典工具进行研究。研究这些复杂的解析结构，以便分离出可用于计算机建模的基本部分，一直是PI的长期研究目标。近年来在这方面取得重大进展的一个例子是，我们在过去十年中享受到了越来越准确的天气预报。PI的研究集中在发现和发明新的抽象数学工具，这些工具可以有效地用于波浪现象研究的几个方面。最值得注意的是，研究某些波动系统的长期行为，控制某些数值近似的误差，并确定一个特定的结果是波浪现象的唯一可观测表达式。近年来，为了成功地对涉及波浪的复杂问题进行数学分析，人们必须准备好使用数学的几个领域的工具，这一点变得很清楚。基于这一原则，国际和平研究所以“跨学科”的方式进行研究。经典的谐波和傅立叶分析技术是理解复杂波相互作用的基本需要，PI曾在这些技术中接受过训练，随后经过多年的改进和完善。然而，最近，PI用经典概率、几何、数论和动力系统的思想来补充这种分析，以捕捉手头问题的更微妙的方面。公平地说，在过去的二十年里，由于许多研究人员采用了这种跨学科的方法，色散和波动方程这一学科经历了前所未有的研究活动。经验观察与抽象的数学描述之间的差距正在日益缩小。

Nature has many ways of expressing itself but certainly wave phenomena are some of the most ubiquitous demonstrations of its power. From the freak waves forming in the middle of a calm ocean, to the singularities of a black hole, to a signal which passes through fiber optics cables, to Bose-Einstein Condensates, wave phenomena arise in many distinct physical settings. Clearly being able to accurately predict wave phenomena by knowing only few measurements would impact our world in many ways: from evacuation plans, to the understanding of our deep universe, from better and clear communication tools to the understanding of the micro-cosmo. The fundamental difficulty in obtaining accurate predictions for wave phenomena lies in the fact that they are expressed via mathematical identities that are too complex for us to study using classical tools. It has been the PI's long time research goal to investigate these complicated analytic structure in order to isolate the fundamental parts than can then be used for computer modeling. An example of the great advances that have been made in recent years in this direction is the ever more accurate weather predictions that we have been enjoying in the last decade.The PI's research is focused on the discovery and invention of new abstract mathematical tools that can be effectively used in several aspects of the study of wave phenomena. Most notably, to study the longtime behavior of certain wave system, to control the error of certain numerical approximations and to determine that probabilistically, a particular outcome is the only observable expression of a wave phenomenon. In recent years, it has become clear that in order to conduct a successful mathematical analysis for complex problems involving waves, one has to be ready to use tools from several areas of mathematics. Based on this principle, the PI has conducted research with an "interdisciplinary" approach. The classical harmonic and Fourier analysis techniques, in which the PI was trained and that subsequently refined and sharpened over the years, are of fundamental necessity for understanding complex wave interactions. Recently, however, the PI has complemented this analysis with ideas from classical probability, geometry, number theory and dynamical systems in order to capture more subtle aspects of the problem at hand. It is fair to say that in the last twenty years, thanks to this interdisciplinary approach, which many researchers have adopted, the subject of dispersive and wave equations has experienced an unprecedented level of research activity. Increasingly, the gap between empirical observations and abstract mathematical descriptions is being narrowed.