Joint Research Training in Pure and Applied Mathematics

Transcription

1 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 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 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

3 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 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

4 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 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

5 centers of arithmetic geometry will profit greatly from exchanging researchers particularly in younger generation 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 --minimal models--. The framework of classification theory including algebraic varieties of dimension greater than or equal to 3 was proposed in 1980's. It was just after a big breakthrough in algebraic geometry -- Professor Shigefumi Mori's discovery of a method to find algebraic curves which are topologically copies of sphere --rational curves--. In 1992, Mori received a Fields medal for this discovery and his completion of 3-dimensional classification theory. Three of the leaders of classification theory are Professors Yujiro Kawamata (Tokyo), Yoichi Miyaoka (Tokyo) and Professor János Kollár (Princeton). Kawamata developed many fundamental techniques in this area like the Kawamata-Viewheg vanishing. Miyaoka did important contributions to complete classification theory in dimension 3 based upon his famous Miyaoka-Yau's inequality. Kollár, with Mori and Miyaoka, amplified Mori's method to find rational curves and developed theory of varieties having many rational curves --rationally connected varieties. Besides, they all wrote several 5

6 text books for the classification theory and have enlightened young researchers through ages. Very recently, there was a big breakthrough in this area due to people in younger generations ---the existence of minimal models of algebraic varieties in important classes was established. After that, many young researchers including those in both departments have come in this field. Nevertheless, the existence of minimal models for any algebraic varieties is not established yet. So it is a significant merit for both departments that researchers in both departments aim at solving this problem by collaborating systematically with Princeton/UTokyo Strategic Partnership Multiscale analysis for fluid mechanics and materials sciences In sciences like fluid mechanics and material sciences, one has to handle atomic scale together with macroscopic scale because they are often coupled. Graduate School of Mathematical Sciences of the University of Tokyo is very strong in mathematical analysis on material sciences and fluid mechanics, for example by a group led by Professor Yoshikazu Giga. However, despite necessary multiscale analysis was not popular in the mathematics department of the University of Tokyo. Fortunately, Princeton University has a strong group on multiscale analysis led by Professor Weinan E. It is a very good idea to start collaboration of both groups to develop several new mathematical approaches as well as new theories of mathematical analysis. Of course, through this collaboration young researchers are expected to broaden their scope. For mathematical analysis related to material sciences, Research Center for Integrative Mathematics of Hokkaido University has a very strong tie with Graduate School of Mathematical Sciences of the University of Tokyo through several joint research projects between Professor Yoshihiro Tonegawa (Hokkaido University). The center is also interested in multiscale analysis so it is appropriate to invite as a partner institution. 6

7 3. Plans 3.1. Conformal and CR Geometry Organizers: Alice Chang (Princeton) and Kengo Hirachi (Tokyo) In 2014/2015, we will organize an international workshop in Tokyo. Professor Chang and Yang will visit Tokyo (in the Spring semester of Princeton) together with young researchers. Some of the participant from Princeton will stay in a longer period including the week of workshop. The organizers of Tokyo side include Professor Kengo Hirachi, Akito Futaki and Masahiko Kanai. On the other way, Taiji Marugame (graduate student at Tokyo) will visit Princeton; Professor Yang interested in his master s thesis on CR geometry and this visit should be profitable for both of them. Between 2015/2017, we plan to organize the second workshop in Princeton. In addition to Professors Chang and Yang, Professor Fefferman will be one of the organizers of the meeting. During the whole period of partnership, several young researchers in differential geometry on both sides are invited to stay for some longer period. Faculty members are also invited to the other department to give lectures and/or to collaborate Mathematical Fluid Mechanics Organizers: Peter Constantin (Princeton) and Yoshikazu Giga (Tokyo) Partner Institutions: Research Institute of Nonlinear PDEs (Waseda University, Japan) International Research Training Group (Technical Univ. of Darmstadt, Germany) In 2014/2015 in Tokyo side, we organize a one-week international workshop in Tokyo possibly in November or June on "Recent trends in mathematical fluid mechanics". A couple of key mathematicians will be invited from Princeton. Since University Tokyo and Waseda University have strong ties with TU Darmstadt, by using other source several European mathematicians are invited. Especially young researchers (for example, Dr. Vlad Vicol) are invited to stay in Tokyo a little bit longer, say one or two months to give a series of talks. This is very helpful to start scientific collaborations. 7

8 Of course, graduate students are also invited to this workshop and to stay a longer period to participate activity in University of Tokyo. In 2015/2016 in Princeton, several young researchers related to Professor Yoshikazu Giga and others are invited to stay in Princeton and participate activities of the group of Professor Peter Constantin. Also several faculties like Professor Norikazu Saito (Tokyo) are invited to visit Princeton to deliver talks there and expect to stay there a few weeks or one month. In 2016/2017, similar style will be expected but depending on the project. We plan to use several grants for these projects, especially hiring researchers. We use support from this Princeton--Tokyo partnership only for traveling expenses to exchange students, post-doc researchers and faculties Arithmetic geometry Organizers: Nicolas Katz (Princeton) and Takeshi Saito (Tokyo) During the whole period of partnership, several young researchers in arithmetic geometry working on both sides are invited to stay for some longer period and to participate in joint research activity on the other side. Also some faculty members are invited to visit the other for short stays and give lectures. We will also have a one-week international workshop in Tokyo on arithmetic geometry, the exact period to be discussed. It will cover several branches of arithmetic geometry where the faculty members of the two departments are playing leading roles. We will also invite some of strong mathematicians working actively in the relevant fields, in order to have more stimulating activities. We will give opportunities for young researchers including students in the two departments to present talks on their own work Algebraic Geometry Organizers: János Kollár (Princeton) and Yujiro Kawamata (Tokyo) In , we will organize an international workshop in Princeton. Both universities have many graduate students of algebraic geometry, and they are also 8

9 invited to this workshop and to stay a longer period to participate activity in Princeton University. There are already scientific collaborations between young researchers of both universities. For example, Yoshinori Gongyo (Reseach associate) and Shunsuke Takagi (Associate Professor) in University of Tokyo and Zsolt Patakfalvi (instructor) in Princeton University did joint works. So Princeton/UTokyo Strategic Partnership is helpful for making these collaborations systematically Multiscale analysis for fluid mechanics and materials sciences Organizers: Weinan E (Princeton), Yoshikazu Giga (Tokyo) Partner Institution: Research Center for Integrative Mathematics (Hokkaido University, Japan) In 2014/2015, U Tokyo will invite Professor Weinan E for a series of lectures and stay there for several weeks possibly as a summer school of June or July. We shall discuss concrete individual research plan at the occasion of his visit. In 2015/2016, we start to exchange young researchers for both sides following the plan discussed. The budget of U Tokyo-Princeton partnership only covers a part of traveling and local expenses. In 2016/2017, we continue to this project. Through this process we expect to expand our area of researches significantly and improve its quality Other fields We also plan to encourage people in other fields to join the program. For example, Professor Yoshiko Ogata (Tokyo) plan to visit Professor Elliott Lieb (Princeton) for a month in Ogata is a well-known expert on C* algebras and has recently proved an important conjecture of von Neumann about operators in quantum mechanics. There are strong groups of researchers in the operator algebra and theoretical physics on both sides. This visit will be a beginning of the partnership in this field. 9

10 4. Budget policy and support form the departments We shall spend the grant from this program for inviting people from Princeton/Tokyo for a long-term stay or a short visit for participating the joint workshops. It will cover traveling expenses as well as local expenses. Both departments agree to offer library privileges and (shared) offices for the visitors in the project; they also cover the const for administrative staffs who will help the project. In addition, Department of U Tokyo agreed to support up to $5,000 to help the organization of the workshops in this project. This project is consistent with the program FMSP* ran by the Graduate School of Mathematical Sciences at U Tokyo. FMSP will partially support the travel of graduate students at Tokyo to visit Princeton and the workshops held at Tokyo. Although it is not explicitly claimed, research grants of individual faculties will be used to support the travel for collaborations and the workshops. This will allow us to use the main portion of the grant form the project for the benefit of young researchers and students. * FMSP (Frontiers of Mathematical Sciences and Physics) is an educational program for graduate students ran by the Graduate School of Mathematical Sciences in collaboration with two other institutions of University of Tokyo. This program is supported by Japan Society for the Promotion of Science, and Ministry of Education, Culture, Sports, Science and Technology, Japan. 10

11 5. Project Committee We form a Project Committee to organize everything in a smooth way. Committee Chairpersons: Charles Fefferman (Princeton), Kengo Hirachi (Tokyo) Committee Co-chairpersons: Peter Constantin (Princeton), Yoshikazu Giga (Tokyo) Committee members: Sun-Yung Alice Chang (Princeton), Nicholas Katz (Princeton) János Kollár (Princeton) Yujiro Kawamata (Tokyo), Takeshi Saito (Tokyo) Attachments CV s of the project committee members. Letters of participation from the partner institutions: 1. Technicshe Universitat Datmstad 2. Hokkaido University 3. Waseda University 11