Over the next ten years the Section for Pure Mathematics in the Department of Mathematics at the University of Oslo will experience changes that might strongly affect its quality as a centre for research and education in the next generation. It is the intent of the present strategic university program (SUP) to take planned measures to ensure that these changes will be for the better, so as to leave the pure mathematics section with a high-quality cluster of academically strong and well-functioning research groups. To achieve this, a number of initiatives will be taken, both in terms of organised research activity, and in effective recruitment of new faculty members. This will involve several of the presently active research groups within the pure mathematics section, which have been selected with a view towards a scientifically meaningful co-ordinated research activity. These initiatives will in turn help to stimulate a strengthening of mathematical education and research on a wider national scale.

The history of mathematical research at the University of Oslo goes back to the early days of the University. Several outstanding mathematicians have during the last 200 years had their base in Oslo. For a long period the University of Oslo was the only one in Norway, and consequently the only place where basic research in mathematics was performed. The expansion in higher education during the last 3-4 decades established a need for more professional mathematicians to educate the huge number of students in mathematics, technology and natural science. In this period the Department of Mathematics at the University of Oslo grew significantly. At the same time the education at the regional university colleges were strengthened and lifted to an academic nivau, in some fields comparable with the universities. This development is the background for the structure that appears in higher education and research in mathematics in Norway today. The reduction in the number of students in the last 5-years period and the consequences for budget figures has made it necessary to consolidate our position and to be more precise in the planning of future research activities in mathematics in Norway in general and in Oslo in particular. The evaluation of the research in mathematics initiated by the Research Council of Norway is a step in this direction, and this Strategic University Program proposal is another.

The pure mathematics section in Oslo presently has functioning research groups in a number of areas of modern pure mathematics, including algebraic geometry, operator algebras, and topology. These all include workers with very strong research records, and have over the last generation constituted most of the core groups of the section. There are also other strong research groups in the section, and its research emphasis has seen gradual shifts over time, but the mentioned three groups form a naturally selected larger cluster within the section because of their possibilities for scientifically meaningful interaction.

Like in most universities, these research groups are now approaching, or even undergoing, drastic long-term changes, mainly because a large part of the academic staff is close to retirement age. The most vigorous researchers remain academically active right up to that age, but within five to ten years many of them will have retired, and the research capacity and quality of the research groups will not automatically be maintained. Of course, the future research groups will be determined by the remaining, mostly younger and middle-aged faculty members, together with the new recruitments to the faculty that we would hope to make. These new groups will then largely define the pure mathematics section in the next generation, i.e., for the next thirty to forty years. Thus this is a time of possible change, update, or redirection of these groups. However, steps should be taken to guide this change in a favourable direction, so as to make sure that the resulting new groups are of high academic quality, function well as research groups, and fit well into the university's combined tasks of research and education.

In the strategic plans of the Department of Mathematics all the involved research groups are given priority. There are no plans to force any reduction in the staff. However the topology group is proposed to be strengthened by one or two permanent scientific positions, due to the high rate of retirements in the next few years in this group.

Engineering colleges and teachers colleges have merged in recent years to become colleges with reasonabley sized groups of mathematicians. To develop the mathematical research potential at the University of Oslo, it will be vital to develop corporations with these groups. The mathematicians at these colleges are recruited from the university so the research relations are often very close. To strengthen these ties an important part of the program are 6 month scholarships for university college mathematicians to work on the project at the university. Thus the guest program will consist of both one year terms for professors from abroad and 6 month scholarships for mathematicians from any of the university colleges named in the project. The university colleges that have appropriate researchers for scientific corporation within the strategic university program are the ones in Agder, Buskerud, Hedmark and Oslo.

The last decades have revealed a need for high-qualified researcher in mathematics in various parts of the Norwegian society, industry, research institutes and educational institutions. This need will continue in the future. It is a necessity that a number of candidates are educated over a period of time. An important outcome of the program is the education of numerous Dr.scient candidates in the subfields of the cluster mentioned in the previous section of this application.

The strategic program will also be compatible with the activity in bringing mathematics out to a general audience and to youngsters. On a long-term basis it is of crucial interest that mathematicians in the program contribute with enthusiasm to show that mathematics can be interesting, amusing and even fun. This is also a topic in the guidelines of the Abel prize, the connection between the recognition of mathematical work at a outstanding level and the task of motivating youngsters for studying mathematics and natural science. The connection between the research and the communication of the results will also be an important task for the program.

C*-algebra theory is the context of Alain Connes' wide-ranging program of non-commutative geometry. The study of cohomological and K-theoretic invariants of C*-algebras, subtly interpreted as invariants of an underlying, implicit geometry, is a foundation for attacks on many natural, but difficult problems in mathematics and mathematical physics. As one particular case of the many different aspects of this non-commutative geometry program, we find Connes' work on the Riemann hypothesis: that the nontrivial zeros of the Riemann zeta-function all have real part equal to 1/2. This is arguably the most famous unsolved problem in pure mathematics, the solution of which would outshine even Wiles' proof of Fermat's last theorem. Connes' work transforms the perhaps non-intuitive formulation of Riemann's hypothesis into the challenge of proving a trace formula for a certain "adelic flow", or more precisely, the action of an idele class group on an adele class space. In this case the formulation is a clear generalisation of a known result in one context (Abelian Lie groups acting smoothly on compact manifolds) to a more general one (the adelic flow). There is also a parallel between this conjectural trace formula and Atle Selberg's famous trace formula.

It is of course not the explicit aim of the present strategic program to solve the Riemann hypothesis. We know too little about how hard the remaining work is, to be seriously able to set ourselves such an aim. However, the mathematical problems arising in the context of Connes' program are so natural, and so fundamental, that the insight that these issues have such deep implications as the Riemann hypothesis will be likely to attract sustained interest by many of the department's researchers that actively investigate these ideas. The program of first jointly studying and understanding the diverse constituents that naturally assemble into Connes' work on the Riemann hypothesis, and then to pursue the combined insights we gain in this process of learning, is however very well suited to the present expertise of the research groups of the cluster, meets the strategic aims of the section, and is mathematically an ambitious, inspiring, and realistic course to follow.

In more detail, the technical framework of Connes' non-commutative geometry program takes place in the context of C*-algebras, such as the cross-product C*-algebras modelling quotient spaces for group actions, or more general foliated spaces. Likewise the cohomological aspect of the trace formula in the adelic context interacts with an extension of the Brauer theory of central simple algebras to approximately finite algebras, as introduced by Ola Bratteli.

The adelic spaces associated to global fields, like that of the rational numbers underlying in the Riemann zeta-function, are naturally studied in the context of arithmetic algebraic geometry. Likewise most of the "global arithmetic objects" with associated L-functions, which form the natural wider context for the Riemann hypothesis, arise in algebro-geometric contexts. After all, the only form of the Riemann hypothesis that has been proven is the case of varieties over finite fields, as established by André Weil for curves and Pierre Deligne in higher dimensions. This draws heavily on the cohomological theories for algebraic varieties, as inspired by algebraic topology but significantly strengthened in the algebraic context, as well as classical geometric techniques in algebraic geometry, such as the formation of Lefschetz pencils.

The trace formula for the idele group action on the adele class space generalises classical topological trace formulas of Lefschetz and Guillemin, which fall within the domain of the topology group; in particular its members involved with the geometric topology of global structures on manifolds, the differential topology of transversality properties in a wide sense, global analysis as applied to the study of manifolds in terms of flows and related structures. The stratified nature of the adele class space can be compared to the subject matter of singularity theory, and the idele class group action strongly relates to Lie group actions. From another side, the number-theoretic nature of the Riemann hypothesis provides strong ties to classical algebraic K-theory, and the cohomological aspects of Connes' theory are parallel to many of the ideas of the latter theory.

Thus for many aspects, the working seminar will need to draw on the expertise of the algebraic geometry group, the operator algebra group, and the topology group.

The development of geometry has throughout history been closely related to problems in physics. Thus there are natural applications of the new field of noncommutative geometry in physics, in particular in a new space-time geometry and in quantum field theory. This interaction goes, however, in both directions. In recent years ideas from string theory in physics have revolutionised the classical field of enumerative geometry. Up until ten years ago, enumerative geometers were occupied with the task of making Schubert´s calculus rigorous. On the theoretical side intersection theory got a firm footing, while on the practical side most of Schubert´s original calculations were confirmed. One of these calculations was the computation of the number of twisted cubic curves in 3-space which are tangent to 12 given quadric surfaces. This was done by giving a precise description of the parameter space of twisted cubic curves. In particular the intersection ring on this parameter space was determined, from which Schubert's calculations were confirmed. In 1991, a group of physicists presented a power series whose d'th coefficient conjecturally give the number of rational curves of degree d in a smooth quintic threefold. Since then the main focus in enumerative geometry has been threefold. On one hand a lot of effort has gone into understanding the physicist's approach including the so called mirror symmetry conjecture and quantum cohomology. One outcome of the theory of quantum cohomology have been remarkable recursion formulas for the solution of series of enumerative geometry problems. Triggered partly by these results some effort have gone into finding other recursion formulas and generating functions giving other series of solutions to enumerative problems. In particular, several important conjectures say that invariants of algeraic surfaces combined together as generating functions turn out to be modular forms. These include Göttsche's enumerative problems for linear systems, Euler characteristics of moduli spaces and relations with S-duality and other symmetries of string theory. Last but not least there has been a greater effort into finding more effective methods of calculations. Here the interplay of the Hilbert schemes as parameter spaces and Bott's formula on spaces with a torus action has proved instrumental. The transformation of the field of enumerative geometry has also shifted the focus to the broader field of symplectic geometry and to the geometry of algebraic stacks, where the physical problems have a more natural setting or a more precise mathematical formulation.

Algebraic geometry

What separates algebraic geometry from algebra is the geometry. The rings, or arithmetic number systems, studied in algebra are interpreted, loosely or precisely, as rings of functions on some kind of space, and the structure of this space is the associated geometry. From one point of view this is extremely classical, but this perspective is also undergoing a process of reinterpretation, where the kinds of geometry that can be modelled and understood are greatly extended. The research group in algebraic geometry in Oslo (with collaborators elsewhere) is certainly founded in the classical interpretation of algebraic geometry, and contains strong expertise in this area (Ellingsrud, Piene, Ranestad), but does also encompass specialists with insight in the more singular, or deformation local, aspects of geometry (Christophersen, Laudal )

The geometric aspects of algebraic geometry are most immediately apparent when working with commutative algebras (or varieties) over algebraically closed fields, while for varieties over rings like number fields or finite field the geometry takes on an arithmetic flavour, often bordering on number-theory (with some relation to algebraic K-theory). The algebraic geometry group also has expertise in the cycle theory and cohomological theories associated to such geometry, as well as the arithmetic and complexity theoretic aspects of encryption in the context of elliptic curves over finite fields.

The core field of algebraic geometry has been intensively studied since the late 19th century. It provides geometric techniques for describing solutions to equations, and relates to practically every significant branch of pure mathematics and theoretical physics. One explanation for this central role is that most of mathematics is concerned at some level with setting up and solving equations, for example to model problems in applications.

For the moment the algebraic geometry group in Oslo perform active research on Cohomological invariants for Hilbert schemes of surfaces (Ellingsrud), enumeration formulas for singular curves on surfaces (Piene), varieties of power sum presentations of forms (Ranestad) and deformations of rational surface singularities (Christophersen).

Operator algebras

The theory of operator algebras has traditionally two different branches with rather different flavours: the theory of C*-algebras and the theory of von Neumann algebras. These branches can be labelled as "non-commutative topology" and "non-commutative measure theory", respectively, since they reduce directly to the theory of locally compact Hausdorff spaces and the theory of measure spaces in the case that the algebras are commutative. The subject has in the last 35 years developed from a rather special discipline within functional analysis to become a central field in mathematics often described as "non-commutative geometry", see for example the book of Field medallist Alain Connes with this title. There have been exciting recent discoveries in the interaction between operator algebras, geometry, topology, algebraic geometry and quantum groups in pure mathematics and algebraic and conformal field theory, string theory, statistical mechanics in theoretical physics.

For the moment it is active research among operator algebraists in Oslo going on in the fields: Magnetic Laplacian, quantum groups, operator theory in wavelet analysis, C*-dynamical systems, non-commutative ergodic theory, cyclic cohomology and tensor categories. The unifying mathematical concept behind these subjects is the use of noncommutative operator algebras, and noncommutative geometry to understand singular spaces or quantum spaces, replacing classical topological or measure spaces and the associated commutative algebras of continuous or measurable functions.

Topology

The topology group in Oslo has expertise in a range of topics spanning singularity theory, Riemannian geometry, differential topology, algebraic topology, geometric topology and algebraic K-theory.

The group in topology and algebraic K-theory (Jahren, Rognes) is largely using algebraic topological methods to study the relation between geometric topology and algebraic K-theory. This translates the global geometry of manifolds to the algebraic K-theory of their group rings over fundamental groups, or more of precisely S-algebras over loop groups, which in turn leads to the cohomological study of the arithmetic algebraic geometry underlying such S-algebras. These can be very non-commutative, but there are core examples which are only non-commutative in a controlled, deformed sense (commutative S-algebras). Here the arithmetic geometry is largely explained by the chromatic ideas of modern stable homotopy theory, with each point in the algebraic sense resolving into an infinite nested tower of finer geometric structures. This again provides examples of the more delicate and powerful notions of geometry that are common to the interests of the three research groups in this cluster.

Rognes has also done work on the number-theoretic aspects of algebraic K-theory, which bear on the arithmetic aspects of Connes' program.

The group in singularity theory (Holm Brodersen) works on stability of singularities of smooth mappings, which can be treated as an aspect of transversality in jet bundles. These methods, and related topics in global analysis (geometric asymptotics), provide many of the classical examples and motivating results that the present program will try to generalise to an adelic setting.

Non-commutative geometry

Non-commutative geometry is a discipline in formation. Recent interactions between areas of physics and mathematics put new objects in the centre of scientific attention, knots and braidings, Hopf algebras and quantum groups, quantized algebras and spaces. In recent years it became clear that a new geometry maybe underlying many of these new developments, and several people have been trying to establish the foundations of non-commutative geometry. The goals are to elaborate its links to categorical methods, algebraic and topological methods, C*-algebra theories and noncommutative differential geometry. Then apply the synthesis obtained to the representation theory of algebras and quantum groups, to holonomicity and dimension theory over rings of differential operators, to algebraic Lie representations and to physics again, e.g. the renormalization problem. Recent developments in physics related to quantum groups have had an immense impact on algebra, bringing new deep problems and meaning to non-commutative algebra. The geometric background present in most phenomena transpires in the mathematical formalism. However, this geometry remained "virtual" because a suitable geometry connected to the non-commutative algebra was lacking. Developing a geometric counterpart to the non-commutative algebra, often worded in terms of deformations or quantizations of algebras, requires an abstract foundation allowing the desired generality of its applicability, but should also allow concrete calculations, either in the algebras or in (co)homology groups, reflecting real geometric aspects.

The research group in Oslo are mainly concentrated on studying the following topics: Applications of non-commutative geometry to the representation theory of algebras, quantized algebras and quantum groups, study of particular cases, e.g. application of non-commutative geometry to algebras satisfying polynomial identities and in particular to quantum groups at roots of unity (Sletsjøe), the local aspect of representation theory of algebras, non-commutative schemes and deformation theory (Laudal) and non-commutative geometry in the group theory setting (Borge).

Concerning the academic quality of the future researchers, this is of course a matter of whom the department will be able to hire. Given the sporadic nature our opportunities to do so, this is not a continuous process, and there is a great danger that the best candidates that develop at the masters level, the doctoral level, or at the post doctoral level, are not necessarily available for hiring at the time when the university wishes to replace a retired faculty member. More concretely, the last faculty member to join the mentioned research groups did so as a post doc. in 1992, and we have not been able to hire a single one of the other candidates developing in the intervening ten years. Hiring the best candidates is necessarily a competitive matter, where alternatives like better-paid positions in private companies or more continuously available positions among the many universities abroad mean that we cannot expect the candidates we wish to hire to simply wait around for a position to appear in Oslo.

Thus while we would like over the next ten years to be able to hire the best candidates developing over that period (or already developed), as long as we are without any system of continuing support for such potential future faculty members we are in practice only able to hire from a rather random sub-selection of that group, namely, those who have not yet settled elsewhere when a position appears. A similar problem occurs at each earlier stage in the recruitment process, when finished masters students have to wait for doctoral fellowships, or finished doctorates must wait for post doctoral support. The well-established solution to this problem is a predictable sequence of temporary fellowships or positions, i.e., doctoral fellowships, post doctoral fellowships, and temporary visiting researcher positions. Knowledge that such a chain of preliminary positions are available will make it significantly easier and more sensible for the best candidates to remain within the research community, and the hiring process for such positions also allows for the selection of these better candidates from a larger body of applicants.

Again, these considerations also apply at the department level, but the present cluster of research groups is important enough to mathematical research in Norway that the combined number of candidates developing in these areas is so large that there is significant demand for such temporary positions associated with the cluster. Likewise, a majority of the active mathematical researchers in university colleges outside of the universities, which might be suitable candidates for future positions, have interests overlapping with those of the cluster. Thus the further development of these groups is enough of a strategic long-term matter that it makes sense to do so in the framework of a larger program. This is especially so since the university has offered only one post doctoral fellowship in pure mathematics from 1993 to 2001, which furthermore was restricted to female applicants.

A number of such temporary positions offered with the intent of ensuring that the university is eventually able to hire as good candidates as possible, over the long run, is thus a core part of this strategic program. Since mathematics research is relatively inexpensive in terms of equipment costs, it is also the main part of the financial side of the program. Likewise, the ability to hire visiting researchers, forreign as well as Norwegian, will make it significantly easier for candidates presently in the university college sector, but suitable for university research, to maintain and uphold their good contacts with such research. Finally, there is also a shortage of departmental doctoral fellowships, so as to avoid losing promising masters students that wish to pursue an academic career, we also wish to offer doctoral fellowships over 3 years each.

The second main aspect of the strategic program concerns the further development of functioning interaction and co-operation within the research groups, or within the larger cluster. Achieving such functioning groups also in the future, as opposed to having a collection of isolated individuals, is also an important aim for the pure mathematics section in the coming generation. Such communication certainly strengthens a research group in the long run, as most researchers benefit from being aware of and knowledgeable about their colleagues' expertise, or even learning from it. At this time when members of the department retire at a steady rate, it is perhaps particularly important that these are able to pass on their accumulated insights and knowledge to their younger colleagues, who are after all among the few with the real capacity to receive such understanding.

The pure mathematics section presently has two kinds of arenas for research interaction: (1) the (very) specialised seminars of the smaller research groups, such as the algebra seminar, the C*-seminar, and the topology seminar, and (2) the (very) general departmental colloquium, which is a forum for somewhat popularised talks aimed at a wide mathematical audience. We find that professional research development at the research group level, and between adjacent research groups, will be better served by a "working seminar" operating as an intermediate arena, between the two mentioned above. For the specialised seminars most often concentrate on recent individual research achievements, and do often not provide enough context and background for the non-specialist working in an adjacent area. And the colloquium talks may serve well to spark an interest in an unfamiliar area, but does not delve deeply enough into the challenges of serious research to really form a foundation for further new work in an area.

Instead, for the faculty members of the present cluster of research groups, longer term interaction and co-operation can be stimulated by a high-level joint working seminar, where we take on a larger-scale research program of current interest. Such current research programs will most often involve deeper ideas and techniques from a number of different parts of modern mathematics. Thus by working together over time, the specialists of each research group can start by digesting and presenting the aspects of the larger-scale program that are the most familiar to them to the remainder of the working group. This will form a natural focussed forum for each researcher to present some of his or her wider knowledge, without the possibly too ambitious format of the specialised seminars, where one mostly only presents one's own work assuming that the wider context is well known. By grouping these lectures around a larger-scale topic of common interest, these presentations can also be both sufficiently accessible and sufficiently interesting to a wider group, and foster more communication among the (presently temporary or permanent) future faculty members. And then, once the context for the larger-scale research program has been sufficiently developed and understood by several faculty members, one has established a solid foundation and ripe potential for doing new and interesting work on these problems. Such new work, done by an interacting group of experts on a problem spanning over several speciality areas, will then certainly go well beyond what each individual researcher can expect to do alone. Also beyond the possible solution of a specific mathematical problem, the experience of such a high-level working seminar will strongly help to maintain the well-functioning of the research groups involved, as co-operative groups, also in the future.

We thus intend to organise a "working seminar" run by and mostly for the cluster members, with two hours of lectures per week, each semester.

In addition to the local working seminar it will also be crucial for the scientific development of the cluster members and the research students to get motivation and impulses from a wider range of mathematicians. The members of the cluster attend on a regular basis international conferences and workshops, but it will also be important to arrange such meetings in Norway. The Department of Mathematics has experience in arranging well-organised conferences (most recently the Abel Bicentennial Conference 2002) of various size and research summer schools (on a yearly basis in Nordfjordeid).

The results of the program will mainly be measured in a collection of deliverables:

• articles in international refereed journals
• talks at international conferences and workshops
• education of masters in pure mathematics
• education of doctorands in pure mathematics
• research done by post docs
• hosting of internationally leading guest scientist
• hosting of mathematicians from the collaborating university colleges
• arrangements of workshops, conferences and summer schools
• a working seminar

Communication of the results of the program will be done through reports available on the website of the program.