Contemporary problems in science, engineering and technology increasingly demand the analysis of highly complex systems that feature both high-dimensional random dynamics or interactions and a large amount of observed data. In order to obtain reliable predictions in such systems, it is essential to exploit large-scale stochastic models and observed data in an integrated fashion. The goal of this proposal is to initiate a systematic study of how conditioning on observed data affects the properties of large-scale stochastic models such as interacting particle systems, stochastic partial differential equations, and Markov random fields. Research will focus on developing the foundations of a conditional ergodic theory for infinite-dimensional Markov processes and of conditional infinite Gibbs measures; on the investigation of probabilistic phenomena such as conditional phase transitions; on developing connections with problems in measure theory, statistical mechanics, and high-dimensional probability; and on potential applications to the design and analysis of Monte Carlo algorithms for filtering and prediction in high-dimensional systems, where classical methods are known to fail.
Large-scale forecasting problems arise in a myriad of important applications such as weather forecasting, geophysical and oceanographic data assimilation, image analysis, traffic forecasting, and prediction in networks. Such problems have a direct impact on our daily lives, and arise in crucial areas of our society such as national security, energy resource management, climate prediction, and medical imaging. The broad goal of this project is to develop a systematic understanding of the interplay between complex models, randomness, and observed data that lies at the heart of any forecasting problem. By focusing on the fundamental structures that are common to a diverse range of applications, mathematicians can provide unique insights and new directions to complex problems and provide an impetus for developing interdisciplinary connections. At the same time, a strong workforce in the mathematical sciences is of crucial importance to the future of technological innovation and education. An integral part of this project is formed by a range of educational, mentoring and outreach activities aimed at increasing student interest and diversity in the mathematical sciences across pre-college, undergraduate and graduate student levels, and at training the next generation of researchers and educators.
