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努力，因此，美国学生及早掌握这一点并开始巩固他们的国际研究网络是很重要的。根据该项目，将于2020-2022年在乌兹别克斯坦举办一系列三所高级研究院(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的讲座和教程，以及研究生参与者的简短演讲组成。ASIS的研究重点将集中在当代的分维空间分析、动力系统和数学物理主题上。ASIS将通过讲座和研究相关活动，作为将这些领域的最新发展纳入美国学生教育和研究的平台。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，并增强他们在研究生院的经历和职业机会。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，对我们今天所拥有的科学知识做出了重大贡献。例如，Al-Khorezmi的书名中有一个词叫“代数”，而“算法”这个词则来自他的名字。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将让美国研究生在数学物理、分数维空间分析和动力系统等更广泛领域的知识前沿积极学习和研究。近年来，包括美国和乌兹别克斯坦的研究人员在内，在这些领域已经取得了一些重大成果，ASIS将作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展，解决了许多基本问题，如在没有对称性假设的情况下稳定的多维激波形成的构造性证明，以及存在正宇宙常数时爱因斯坦-欧拉系统的整体正则性。在关于分形空间的分析中，最近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的进展。这一领域的许多结果往往依赖于几何和分析中的经典结果到分形环境的扩展。在动力系统中，一维动力学在过去的几十年里一直是一个蓬勃发展的主题。它为复杂交织的规则动力学中的非线性混沌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，它描述了不同尺度的动力学和参数画像之间的相互作用。该项目总共资助了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界代表性较低的群体，包括女学生和研究机会有限的学术机构的学生。这一奖项反映了NSF的法定使命，并通过使用基金会的学术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.