The Research Plan

The main vision
We want to create a national research centre in mathematics where top class scientists from different areas can work together with a joint prospectus and common goals. The main focus will be on mathematics motivated from applications, with emphasis on problems arising from modern scientific computing. Our belief is that the structure of a centre like this will provide new fruitful collaborations and lead to progress in research areas of importance to science, industry, finance, and society in general. The centre will provide a new, original, and innovative scientific environment for mathematical research in Norway.

We firmly believe that ground-breaking research will depend on strong links between theory and applications. Classical theorists and numerical analysts with a strong commitment to theory will be merged in the centre, the main focus being on topics which can lead to the future development of computational mathematics. We also believe that it is of fundamental importance that part of the research group has a background in scientific disciplines other than mathematics.

The activity will be built upon four pillars represented by different research areas, geometry, stochastic analysis, differential equations, and applications in the physical sciences. These areas all represent strong activities at the University of Oslo today. However, our vision is that strong interaction between these fields will create a leading group in mathematics for applications. We believe that significant progress will arise as a result of a balance between theory, computation, and applications.

The centre plans to organize workshops and symposia together with industry, where new developments in mathematics and its applications are presented. This will mainly be done through our close cooperation with SINTEF Applied mathematics. The aim is to bridge the gap between novel results in basic research and their use in potential technological innovations. We also hope to expose essential new research directions in close collaboration with industry.

In addition to the research itself, a major purpose of the centre will be the training of new researchers. This will not be a new situation for the members of the centre, since they have successfully supervised a large number of candidates during the past two decades. Based on the idea of combining theoretical mathematics with a wide range of applications, the centre will create an ideal atmosphere for research education in areas of practical importance for the future. Hence, a main “product” of the centre will be a large number of new researchers, trained and inspired by international experts, in an exciting and stimulating environment. Their future contributions to society, both in industry and in the research sector, will represent important benefits of the centre.

The Research Group
The research activity will be built around ten principle investigators. Eight of them hold full time positions at the University of Oslo, one has his ordinary position at the Norwegian University of Science and Technology (NTNU) in Trondheim, and one is a leading investigator at SINTEF Applied mathematics. In addition to these ten, a number (10-15) of other investigators, either from the University of Oslo or other institutions, will be associated to the centre.

The principal investigators are:

• Tom Lyche, Dept. of informatics, UiO
• Ragni Piene, Dept. of mathematics, UiO
• Tor Dokken, SINTEF Applied mathematics
• Bernt Øksendal, Dept. of mathematics, UiO
• Fred Espen Benth, Dept. of mathematics, UiO
• Helge Holden, Dept. of mathematical sciences, NTNU
• Ragnar Winther, Dept.s of informatics and mathematics, UiO
• Geir Ellingsrud, Dept. of mathematics, UiO
• Morten Hjorth-Jensen, Dept. of Physics, UiO
• Mats Carlsson, Institute of theoretical astropysics, UiO

Lyche, Piene, and Dokken will be heading the activity in geometry. Øksendal and Benth are the the pricipal investigators in stochastic analysis. Holden and Winther both work with differential equations, while Ellingsrud will focus on the interplay between geometry, differential equations, and physics. Finally, Hjorth-Jensen and Carlsson will represent the physical sciences.

Planned centre activities
The centre participants, as well as the centre itself, will contribute to, participate in, or arrange a number of conferences, workshops, educational programs, international networks etc. At present time we only have partial overview of this activity, and we will have to return to a more specific list later in annual reports and plans.

However, we will specifically mention that during 2003 we will arrange a conference which will focus on the interdisciplinary aspects of our research activity. The purpose of the conference is to strengthen the ties between the different pillars, to further identify the main research problems, and to improve the relations with key international researches. We will regard this as a “kick-off”- event, and we intend to invite some major international speakers. A “follow-up” conference will be arranged two or three years later with the purpose of returning to most of the same topics, evaluating the progress in these areas.

Our research plan are divided into four main projects, roughly representing the four pillars of the centre. However, each project is composed of a variety of challenging research problems (subprojects), and most of them require interactions from at least one of the other pillars. We can only expect full activity in some of the subprojects. At the present time it is not desirable to choose which subprojects that will be given full priority. This will depend on, at least partially, the interests and qualifications of the candidates recruited for the Ph.D. stipends and post.doc. positions.

Anyway, successful attempts to attack the projects should lead to the development of a research group with highest international standards in modern applied mathematics, in particular in the direction of computational mathematics. The creation of such a research group in Norway can be seen as the ultimate goal of the Centre.

The Departments of Mathematics and of Informatics at the University of Oslo are currently making sweeping changes in the way computional mathematics is being taught. In the last years, several new basic mathematics courses, which combine traditional calculus and programming, have been taught, and more courses are being added. An entirely new educational programme leading to Bachelor, Master, and Ph.D. degrees in “Applied Mathematics and Scientific Computing” started in the autumn of 2002.

This newly established educational programme entails the development of a series of new courses which combine mathematics, computer science, and applications to the physical sciences. One of our aims is the development of a national graduate curriculum which encompasses the newly established programme. We already offer permanent advanced courses in partial differential equations, functional analysis, numerical analysis, Monte Carlo methods, stochastic differential equations, spline methods, algebraic geometry, and more. In addition, we plan to establish courses in computational physics and fluid mechanics, specific courses on modern tools within high-performance computing, such as the use of parallel algorithms and machines, advanced use of object-oriented languages in numerical projects, and so on.

The combination of such an extensive graduate programme and the research activity is expected to provide excellent educational and training possibilities for graduate students (as well as for post-doctoral fellows, visitors, and researchers in general). We therefore consider the Centre to be well positioned and prepared to meet the requirements of a PhD-school (or Researcher school) as newly proposed by the Research Council. Anyway, we plan to educate a significant number of new researchers.

The main goals of the research activity for the next ten years can be summarized as
follows:

• To create a national research centre with an atmosphere where the participants continue to do first rate research in geometry, stochastic analysis, differential equations, and applications in physical sciences.
  
  
• To arrange a "kick-off" conference in 2003 and a "follow-up" conference two to three years later which will focus on the interdisciplinary aspects of our research activity.
  
  
• To create interdisciplinary research activities where we focus on the interplay between
  i) computational geometry and algebraic/differential geometry
  ii) computational geometry and differential equations
  iii) stochastic analysis and differential equations
  iv) stochastic analysis and physical sciences
  v) differential equations and physical sciences.
  
  
• As a national centre in applied mathematics our intention is to cooperate with a variety of research groups in Norway which are in need of such expertise. The cooperation with SINTEF Applied mathematics will provide the centre with an open channel to industrial problems. If our capacity permits, the centre may also be involved in other joint projects, for example in basic petroleum research or neuroscience.
  
  
• A major purpose of the centre will be the training of new researchers. Our expectations is that approximately 25 Ph.D. candidates associated the centre will graduate during the ten year period, with approximately ten completing their degree during the first five years. Similar figures exist for the estimated number of Post.docs. (These estimates also includes a few candidates who work with some of the participants of the centre as advisors, and therefore will be associated the centre when it starts to operate, but have begun their Ph.D. studies before the centre existed.)

In the following sections, we present the four main projects with its main goals.

Project 1: Geometric Modelling
Project 2: Stochastic analysis and finance
Project 3: Nonlinear partial differential equations - theory and numerics
Project 4: Physical Applications