IRES Track II: Advanced Studies Institutes in Analysis on Fractal Spaces, Dynamical Systems and Mathematical Physics

Csu Fullerton Auxiliary Services Corporation, Fullerton CA

Investigators

Abstract

Contemporary mathematics has become a truly international endeavor and hence it is important that U.S. students grasp this early on and start cementing their international network of researchers. Under this project, a series of three Advanced Studies Institutes (ASI) will be held in Uzbekistan during 2020-2022. Each ASI will last two weeks and will consist of lectures and tutorials given by distinguished researchers from the United States and Uzbekistan, and short presentations given by graduate student participants. The research focus of the ASIs will be on contemporary topics in Analysis on Fractal Spaces, Dynamical Systems, and Mathematical Physics. The ASIs, through lectures and research related activities, will serve as platforms to incorporate the latest developments in these areas into education and research for U.S. students. The project will expand the U.S. students' horizons, connect them with distinguished mathematicians and graduate students from other parts of the world, and enhance their graduate school experiences and career opportunities. Finally, Uzbekistan has a rich cultural and scientific heritage and presents a unique research environment for the U.S. students. Great scientists of the 8th - 11th centuries, such as Al-Khorezmi and Al-Biruni, made significant contributions to the scientific knowledge that we have today. For example, the word “algebra” comes from the title of Al-Khorezmi's book, while the word “algorithm” comes from his name. The project will have societal benefits by promoting mutual understandings between the United States and Uzbekistan. The ASIs will engage U.S. graduate students in active learning and research at the frontiers of knowledge in broader areas of Mathematical Physics, Analysis on Fractal Spaces, and Dynamical Systems. In recent years, a number of significant results have been established in these areas, including by researchers in the U.S. and Uzbekistan, and the ASIs will serve as venues to disseminate these results and to stimulate new international collaborations. In mathematical physics, the theory of relativistic fluids has witnessed tremendous progress in recent years, with many fundamental problems being addressed, such as constructive proofs of stable multidimensional shock formation without symmetry assumptions and global regularity of the Einstein-Euler system in the presence of a positive cosmological constant. In analysis on fractal spaces, there has been a great progress recently in extending the classical results in analysis and geometry from Euclidean space settings to non-smooth or fractal spaces. Many results in this area often rely on the extensions of classical results in geometry and analysis to fractal settings. In dynamical systems, one-dimensional dynamics has been a flourishing theme over the past several decades. It provides the simplest model for non-linear chaos in its intricate intertwining with regular dynamics. Renormalization is a powerful machinery that was developed in this context to describe the interaction between various scales of dynamical and parameter portraits. The project funds a total of 60 U.S. advanced graduate students and 6 to 9 U.S. researchers. At least half of the participants are recruited from groups underrepresented in sciences including female students and students from academic institutions with limited research opportunities. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.