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1.Purpose of the COE Programs

Introduction

It is my pleasure to explain to you the plans for our Center of Excellence program.
The Center of Excellence, or CEO, was started in 2002 by the Ministry of Education, Culture, Sports, Science and Technology to promote and establish internationally significant educational and research centers in selected universities for each field of science. The Integrative Mathematical Sciences program has been selected as one of programs in the research field of Mathematics, Physics and Earth Sciences, and started this August.

COE Programs

Our COE program in designed to open up new horizons in mathematics by bridging the gaps between pure mathematics and application-oriented mathematics, and to investigate new concepts and leadership methods for mathematical sciences. Let me explain our organizational strategies for the COE program by itemizing the following:

  1. Integrating pure mathematics, data science and experimental mathematics
  2. Encouraging young researchers internationally
  3. Forming international alliances for education and research in the Mathematical Sciences

COE Educational and Research Programs
Our COE program has three main subjects:

  1. Fundamental Mathematics
  2. Data Science
  3. Experimental Mathematics

The core of the research and educational program is built on pure
mathematics, as supported by data science and experimental mathematics. Data science serves as the interface between various fields with data-intensive phenomena, and experimental mathematics supports the experimental aspect of mathematical sciences. The main objective of our program is to promote the construction of international educational and research centers through integrating these three aspects of mathematical sciences.

Partnerships with the Fundamental Sciences
The Graduate School of Fundamental Sciences covers the various educational and research subjects in natural sciences. Specifically, we have educational and research units for physics, chemistry and life sciences. The Department of Mathematics has partnerships with these educational and research units. We believe these interactions with the other disciplines will be very important for developing the educational and research missions of the mathematical sciences. We also have a partnership with the Graduate School of Economics, which is joining our COE program.

2.Educational and Research Evaluations

History and Evaluations

Let me mention our educational research evaluations briefly.
The Department of Mathematics, which functions as the core institute of the COE program, was established in 1891, and the graduate course of mathematics subsequently began in 1985 with the support of our excellent staff. The Department of Mathematics has been recognized as one of the most successful departments for mathematical sciences in Japan. Our educational and research activities were evaluated by an outside committee in 2002, and we received an excellent evaluation.

Educational and Research Subjects

The Department of Mathematics contains six major educational and research mathematical subjects organized into the following educational and research groups:
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  1. Geometry, Topology and Global Analysis
  2. Combinatorics and Discrete Mathematics
  3. Ordinary and Partial Differential Equations
  4. Probability and Ergodic Theory
  5. Data Science
  6. Experimental Mathematics

The first two groups form the core of the fundamental mathematics curriculum. The second two groups traditionally cover analytical methods in mathematics. The third two groups develop mathematical models to explain observed phenomena. These groups are recognized internationally as significant educational and research groups.

Research Activities

As you can see from the data on research papers, invited talks at workshops and conferences for the past four years, the Department of Mathematics has a successful, internationally recognized high level of activity.

Education

Let us show how successfully we have encouraged students and young researchers. We have produced over 70 students who have found academic positions during the last 25 years. Recently, some of our students have found jobs in private companies and institutions after obtaining their PhD, which is also one of our strategies. In 2002, we educated 70 master course students and 23 PhD students, who are very active in research projects.

Conference Organization

The Department of Mathematics has supported the organization of international workshops every year. Over the past few years, we have had workshops on noncommutative geometry and physics, and an international conference on variational problems. The UK-Japan winter school is a cooperative project of Keio and UK universities to form international networks of students and young researchers, and will be continuing alongside the COE programs.
These achievements in organizing workshops and conferences show that the Department of Mathematics is recognized as a one of the leading educational and research institutions for mathematics.

Visitors

The department has invited domestic and overseas researchers as lectures and colloquium speakers, as well as for joint research projects. The number of visiting overseas researchers is over 50 per year.

3.Plans for the COE Programs

The organization of the COE program will be based on these successful achievements of the Department of Mathematics at Keio University. Our programs will focus on the three fundamental strategies I mentioned in the beginning.
Let me describe some concrete strategies we are planning now.

Center for Integrative Mathematical Sciences
To facilitate our efforts further, the Center for Integrative Mathematical Sciences at Keio will be established under the auspices of the President at the earliest possible opportunity. We have the following seven member advisory board:

  • Prof. J. P. Bourguignon (IHES, director)
  • Prof. K. D. Elworthy (Warwick, MRC, director)
  • Prof. P.Embrechts (Prof., ETH)
  • Prof. R. Mullin (Prof. Univ. of Waterloo)
  • Prof. R. Schoen (Prof., Stanford University)
  • Prof. P. J. Thomson (Director, Statistics Research Associates Ltd.)
  • Prof. A. Weinstein (Prof., UC. Berkeley)

With the recommendations of this advisory board, we will develop the educational and research plans to establish the Center of Integrative Mathematical Sciences at Keio.

Complementary educational and research themes
For the initial activities towards realizing our goal of integrative mathematical sciences, we will select two board research themes:

  1. Noncommutative manifolds and discrete geometric objects in the framework of noncommutative geometry.
  2. Analysis of nonlinear phenomena in the framework of data science.

The first theme represents the relatively pure side of mathematical sciences, including non-commutative geometry and global analysis. The second theme represents the relatively application-oriented side of mathematical sciences.
Since the first theme is strongly motivated by high-energy physics and is expected to have applications to various phenomena in quantum physics, its educational and research activities are not confined to conventional pure mathematics. The second theme attempts to formulate concrete problems that could enrich pure mathematics, through investigations of major problems in the relatively application-oriented side of mathematical sciences, particularly those areas whose main concern is the analysis of large scale nonlinear systems from the data science and system reduction points of view.
These two complementary educational and research themes will be discussed later by Professor Moriyoshi and Shibata.

Educational Plans

As well as our research projects, we have a strong mandate to encourage PhD students and young researchers internationally. For this purpose, we have already started weekly interdisciplinary seminars and colloquia to enhance interdisciplinary exchanges. For the current year, we will also hire 14 research assistants and offer 6 postdoc positions, including two for overseas students, to encourage young researchers. We are planning international internships for PhD students to obtain fruitful experience in frontier areas of research in the mathematical sciences. Let me add that we plan to organize the UK-Japan winter school for 2004 in Durham.

International Cooperation

Another important aspect of our COE program is domestic and international collaborations in mathematical sciences with educational and research institutions overseas. Keio University has many exchange programs with overseas universities, and in particular, our department has been planning actively to have partnerships with overseas mathematical institutes. Furthermore we intend to develop alliances with educational and research institutions overseas. We also invite researchers to give lectures series for graduate students, both to inform the students of progress in recent important problems in mathematics, and to encourage research collaborations with the COE program. We also plan to organize an international workshop each year. For this year, an international workshop on noncommutative geometry and physics will be held at Keio on March, 2004.

4.End

To end my address, let me say that we believe that our programs will propagate several new activities in education and research in the mathematical sciences, including fostering collaborations with institutions and researchers, and creating a stimulating environment for graduate students and young researchers, by providing them with opportunities to work with leading researchers on frontier problems in an international setting. Taken all together, these activities will realize our goal to be a center for integrative mathematical sciences.

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