Narrative Project Description Joint Research Training in Pure and Applied Mathematics 1. Project intent. Although individual ideas of researchers are most important in mathematical research, skills for communication, discussion and collaboration with researchers in diverse mathematical, scientific and other fields are increasingly important. To gain such skills, it is essential for mathematical researchers to talk to other mathematicians whose backgrounds are quite different, especially when they are young (e.g. while they are graduate students or postdocs). For this purpose we would like to exchange graduate students and postdoc researchers in various fields of pure and applied mathematics. Moreover, we propose to organize joint workshops to stimulate this process. Strong groups of researchers at U Tokyo and Princeton share significant common interests, including conformal, complex and algebraic geometry, and fluid mechanics. We already have a lot of research communication and collaborations among high-level faculty. This proposed project will further strengthen those collaborations. While the fields of current collaboration will naturally be emphasized, we would like to cover other branches of mathematics as well, since progress often arises from unanticipated interaction between different fields. The proposed joint project will be supplemented by additional support from “Leading Graduate School Programs, Japan Society for the Promotion of Science”, which is run by the Graduate School of Mathematics at the University of Tokyo. 1 2. Proposed topics and background We choose five specific topics and organize projects to achieve our goal. Three topics "Conformal and CR Geometry", "Arithmetic Geometry" and "Algebraic Geometry" are important topics from pure mathematics, while two topics "Mathematical Fluid Mechanics" and "Multiscale Analysis for Fluid Mechanics and Materials Sciences" are two major fields from applied mathematics. For the first four topics Princeton and Tokyo already have strong interactions in research level so we intend to expand and strengthen interactions including young researchers. The last topic is quite new and we plan to start collaboration. The list of projects does not mean that we exclude other projects. As we shall explain in Section 3.6, we will welcome and encourage other individual projects. 2.1. Conformal and CR Geometry In Einstein’s theory of general relativity, the space-time was formulated as a manifold with a metric satisfying a differential equation the Einstein equation, which is called Einstein metric. Since then, the study of the Einstein metric has been a main theme of geometric analysis. Deep theory of nonlinear partial differential equation had been developed in the last quarter of the 20th century, and the existence of Einstein metrics are proved in many cases. More recently the connection of the Einstein metric with conformal geometry has got much attention by the discovery of AdS/CFT correspondence in theoretical physics due to Juan Maldacena (Institute for Advanced Study). In mathematical formulation, this is a correspondence between an Einstein manifold (a manifold with a Einstein metric) and the conformal geometry given on the boundary at infinity of the manifold. This point of view provides many new geometric invariants and interesting questions in analysis; in particular, the existence of Einstein metric in this setting is the main open question. Such a correspondence is classically known in the complex analysis, where the geometry of open complex manifold is determined by CR geometry of the boundary; the complex manifold support Einstein metric and CR geometry can be seen as the boundary at infinity. The AdS/CFT correspondence also gives a strong stimulation in this classically studied field. We 2 remark that the basic formulation of these correspondences were obtained, prior to Maldacena's discovery, by Professors Charles Fefferman (Princeton) and Robin Graham (University of Washington), which is now called the theory of the ambient metric. Princeton University and University of Tokyo have strong groups of researchers with common interest in this field. On Princeton side, Professors Charles Fefferman, Alice Chang, Paul Yang are leading researchers in the ambient metric, Conformal and CR geometry; especially on the nonlinear analysis of the geometric invariants arises from the Einstein metric. Professor Gang Tian is also working on Einstein metrics on complex manifolds. On Tokyo side, Professors Kengo Hirachi, Akito Futaki and young researchers working on conformal and CR manifold in connection with Einstein metrics, application of representation theory and geometric invariant theory to the ambient metric and the Einstein metric. There are several approaches to the Einstein equation and interaction between different viewpoints offers a chance for a breakthrough. One example is the recent discovery of Q-prime curvature in CR geometry, which was first introduced by Yang (Princeton) in a problem of geometric analysis in 3-dimensions and has been generalized to higher dimension by Hirachi (Tokyo) via the ambient metric and makes its geometric meaning clearer. It is also very profitable to graduate students and postdocs working in this field to visit other department to learn different aspect of the geometry; a system of exchanging young researchers is ideal for this purpose. 2.2. Mathematical Fluid Mechanics Fluid Mechanics has always been a challenging area for mathematical analysis. For example, it is by now a famous open problem (known as Clay's Millennium problem) whether or not there exists a smooth 3-dimensional Navier-Stokes flow for large initial data. Besides such a classical problem, there are several new areas as complex fluid. Fortunately, both Princeton University and University of Tokyo are strong in this area. For example, Professor Peter Constantin (Princeton) is one of the top leaders on mathematical fluid mechanics and Professor Charles Fefferman (Princeton) contributed a lot by developing his real analysis techniques. In the meanwhile, Professor Yoshikazu 3 Giga (Tokyo) also gave several fundamental contributions on the Navier-Stokes equation, especially the region of fluid is a domain not a whole spaces. Since the field is very wide, it would be very important to learn several approaches developed by different schools to achieve real breakthrough. Although mathematicians related to this topic in Princeton and Tokyo knew each other well, so far scientific interaction is not very large. Since both places are world center of mathematical fluid mechanics, exchanging researchers especially young researchers give a significant merit for both sides. 2.3. Arithmetic geometry Although the number theory has a long history starting with the mathematics itself, algebro-geometric method introduced to number theory in the beginning of the 20th century opened a new world, now called arithmetic geometry. It eventually enabled a proof of Fermat's last theorem at the end of the century. The efforts by many great mathematicians aimed at solving the Weil conjecture resulted in settling the foundation tools of arithmetic geometry including schemes, etale cohomology etc. This branch of mathematics has been intensively studied for nearly a century with great successes but we still have many more important open problems, including the Langlands programme, the Tate conjectures, the Birth-Swinnerton conjecture. Princeton University and University of Tokyo are two of the world centers of research in this area. On Princeton side, Professors Nick Katz, Shou-wu Zhang, Sophie Morel are world-leading researchers in arithmetic geometry. Professor Peter Sarnak (Princeton and IAS) is also working in a close field. On the other side, we also have a strong research group on arithmetic geometry in Tokyo consisting of Professors Takeshi Saito, Takeshi Tsuji, Atsushi Shiho, Naoki Imai and young researchers working on Galois representation, etale cohomology, crystals, Langlands program etc. The research areas on both sides have large intersection as well as independents directions. Such combination is ideal to realize significant and unexpected breakthrough. Although mathematicians working in this field in the two universities knew each other reasonably well, so far scientific interaction has not been made on regular basis. The two world 4 centers of arithmetic geometry will profit greatly from exchanging researchers particularly in younger generation. 2.4. Algebraic Geometry Princeton University and University of Tokyo are two of the world centers of research in classification theory of algebraic varieties, which is one of the main area of algebraic geometry. Algebraic geometry is a classic area of mathematics studying zero sets of polynomial equations called algebraic varieties. Its origin is the introduction of analytic geometry by Decarte but it was more amplified by Riemann's study of 1-dimensional algebraic varieties --algebraic curves-- based upon his theory of algebraic functions and abelian integrals after Gauss, Abel, and Jacobi. After great successes in studies of algebraic curves, the Italian school studied 2-dimensional algebraic varieties --algebraic surfaces-- in the beginning of the 20th century. It is the origin of classification theory of algebraic varieties. Classification theory aims at classifying algebraic varieties X into classes by the sets of ratio of polynomial functions on them --rational function fields k(X)--, and constructing good representatives in each class