Gran Sasso Science Institute (GSSI)

In this lecture we will discuss a research field that brings together problems with strong motivation in quantum physics and mathematical techniques born in the field of harmonic analysis and used for the analysis of dispersing phenomena. In particular, the descriptions that use effective equations of the non-linear Schrödinger type (Gross-Pitaevskii) have a formulation of a fluid-dynamic nature. We will demonstrate a mathematically rigorous theory that justifies these connections and poses new mathematical challenges. In conclusion, we will mention how some ideas of dynamics on fractals, which were one of Andrzej Lasota's topics of interest, can find application in the study of quantum turbulence.

Wydziału Matematyki i Informatyki UMK w Toruniu

The stochastic compactness method finds applications in many recently published papers dealing with existence problems for solutions of stochastic partial differential equations. It consists in approximation of a potential solution with a sequence of stochastic processes that is uniformly tight in topologies proper to the considered problem and then in passing to a subsequence admitting so called almost sure Skorokhod representation with respect to all involved topologies.