This research centers on developing Gaussian methods for the study of rare events, mainly depending on small values of a positive random quantity. The major objectives are to extend the understanding of related topics and build a general theory based on systematic study of various techniques and applications. The recent completion of the connection between small ball probabilities and metric entropy problems allows applications of tools and results from functional analysis to provide the most powerful method available at this time for estimating the lower bound for Gaussian processes. In turn, it suggests many further questions connected to applications in probability theory and geometric functional analysis. Similar relationships will be studied for other random processes such as stable processes and Gaussian chaos. The overview on rare events depending on small values unifies many problems from diverse areas in mathematics. The very successful applications of Gaussian methods to lower tail probabilities, the Brownian pursuit models and the first exit times will be expanded to a detailed study of zeros of random polynomials, balancing vectors, and Hadamard matrices. The primary focus of this research is a better understanding of rare random phenomena related to Gaussian processes and others which serve as models in many applications. These types of problems often arise in estimating the chances for rare events to occur in areas where such events are of fundamental importance, such as weather, economic indices, epidemics, etc. This research should improve our understanding of rare random events and provide basic tools for the study of our random environment.
