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

点击翻译按钮获取中文摘要

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.