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

当代数学已经成为一项真正的国际事业，因此美国学生尽早掌握这一点并开始巩固他们的国际研究人员网络是很重要的。根据该项目，2020-2022年期间将在乌兹别克斯坦举办一系列三个高级研究所（ASI）。每个ASI将持续两周，由来自美国和乌兹别克斯坦的杰出研究人员提供讲座和辅导，以及研究生参与者的简短报告。ASIs的研究重点将是分形空间分析、动力系统和数学物理等当代主题。通过讲座和相关研究活动，美国科学研究所将成为一个平台，将这些领域的最新发展融入美国学生的教育和研究中。该项目将扩大美国学生的视野，将他们与来自世界其他地区的杰出数学家和研究生联系起来，并增加他们在研究生院的经历和就业机会。最后，乌兹别克斯坦拥有丰富的文化和科学遗产，为美国学生提供了独特的研究环境。8 - 11世纪的伟大科学家，如Al-Khorezmi和Al-Biruni，为我们今天的科学知识做出了重大贡献。例如，“代数”这个词来自于Al-Khorezmi的书名，而“算法”这个词来自于他的名字。该项目将通过促进美国和乌兹别克斯坦之间的相互理解而产生社会效益。ASIs将吸引美国研究生积极学习和研究数学物理、分形空间分析和动力系统等更广泛领域的前沿知识。近年来，包括美国和乌兹别克斯坦的研究人员在内，在这些领域取得了一些重要成果，亚洲发展研究所将成为传播这些成果和促进新的国际合作的场所。在数学物理中，相对论流体理论近年来取得了巨大的进步，许多基本问题得到了解决，例如，在没有对称假设的情况下稳定多维激波形成的建设性证明，以及存在正宇宙常数时爱因斯坦-欧拉系统的全局规律性。在分形空间的分析中，近年来在将分析和几何中的经典结果从欧几里得空间扩展到非光滑或分形空间方面取得了很大进展。这一领域的许多结果往往依赖于几何学和分析中的经典结果对分形设置的扩展。在动力系统中，一维动力学在过去几十年中一直是一个蓬勃发展的主题。它为非线性混沌与规则动力学的复杂交织提供了最简单的模型。重整化是在这种背景下发展起来的一种强大的机制，用于描述各种尺度的动态和参数肖像之间的相互作用。该项目共资助60名美国高级研究生和6至9名美国研究人员。至少一半的参与者来自科学领域代表性不足的群体，包括女学生和来自研究机会有限的学术机构的学生。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.