New contexts
for geometry and arithmetic
in Pure Mathematics
John Rognes (project leader)
Ola Bratteli, Kristian
Ranestad, Arne B. Sletsjøe
University of Oslo
Introduction...................................................................................... 3
Background........................................................................................ 3
Mathematics at the
University of Oslo................................................................................ 3
Human resources.................................................................................................................. 3
Collaborating
institutions................................................................................................. 4
Education of
researchers..................................................................................................... 4
Outreach............................................................................................................................. 4
New contexts for
geometry and arithmetic.................................... 5
Connes' work on the
Riemann hypothesis............................................................................. 5
Inspirations from
mathematical physics................................................................................ 6
Subfields.............................................................................................................................. 6
Related activities.............................................................................. 8
Evaluation of
mathematics, RCN......................................................................................... 8
The Mittag-Leffler
institute in Stockholm.......................................................................... 8
Strategic plans for the
Department of Mathematics, University of Oslo.............................. 8
OMATS................................................................................................................................. 8
Networks............................................................................................................................. 8
Organisation of the
program............................................................ 9
Temporary positions............................................................................................................. 9
A research forum................................................................................................................ 10
A working seminar............................................................................................................. 10
Meetings............................................................................................................................ 11
Deliverables..................................................................................... 11
Budget............................................................................................... 11
Personnel.......................................................................................... 11
This proposal for a Strategic University
Program in pure mathematics was submitted to the Research Council of Norway,
June 14, 2002.
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.
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).
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.
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 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).
In spring 2002 RCN will
evaluate research in mathematics in Norway. The report of the international
evaluation committee will not be presented before fall 2002. In the discussion
with the committee the possibility of applying for a strategic university
program in pure mathematics was mentioned, also by members of the committee. As
background for the international evaluation committee the department has
published a self-evaluation report. In this report it is stated: "The main
goals for the Department of Mathematics are to consolidate its position as the
largest mathematical department in Norway, and to strengthen its position in
selected areas. To achieve these goals it is of vital importance to improve the
recruitment. This is currently a serious problem on all levels."
At the Mittag-Leffler
institute in Stockholm the forthcoming year there will be a program with close
relations to the proposed SUP. For the academic year 2003-04 the subject is
Non-Commutative Geometry, and two of the organisers, Arnfinn Laudal and Arne B.
Sletsjøe, are members of the research groups included in the SUP.
The Department of Mathematics
addresses in its strategic plan the following statement: "The field of
mathematics should be given priority at the University of Oslo in the
forthcoming 10 years. To achieve this aim it is necessary that all subfields of
research is maintained at a reasonable level. Regarding the section´s
task of been a national centre of mass education and research in mathematics it
is of crucial importance that the section maintain its wide scientific range
and at the same time contains internationally high-ranked researchers in some
particular fields."
OMATS (Oslo Mathematics
Doctoral Training Site) at the University of Oslo has status as an EU Marie
Curie Training Site for pre doc students in algebraic geometry, operator
algebras and algebraic K-theory/geometric topology. Fellowships are awarded for
a period between 3 months and 1 year. This training site was among the first of
this kind to be established in Norway.
The researchers connected to the proposed SUP are node contacts or
members of several European networks:
QSNG is the
acronym for an EU research training network in operator algebras with a node in
Oslo. The program started in 2002 and will last until 2005. The network
provides the nodes with post docs.
EAGER is the acronym for the
European Algebraic Geometry training and research network funded over 4 years
by the European Commission under the fifth framework program, starting October
2000 with members in almost all European countries. University of Oslo is one
of the twelve geographical nodes, responsible for the management of joint
research projects and for the training of young researchers through exchange
between the EAGER groups.
NORDAG is a Nordic network
within algebraic geometry funded over 3 years by NorFA (Nordisk Forskerutdanningsakademi), starting January
1, 2000. The purpose of the network is to promote mobility and co-operation
between algebraic geometers in the Nordic countries. About half of the funding
is for workshops, the rest is for mobility scholarships
NOG (acronym for
Noncommutative Geometry) is a European Science Foundation (ESF) based mobility
program for researchers in noncommutative geometry. There are 11 contributing
organisations from 10 European countries and the network organises workshops
and awards money for exchange visits. The duration of the program is five years,
2000-2004. The NOG network will be responsible for the noncommutative geometry
year at the Institute Mittag-Leffler during the academic year 2003-2004.
GAIA is the acronym for a
FET-Open project, “Application of approximate algebraic geometry in industrial
computer aided geometry”, co-ordinated by SINTEF Applied Mathematics. The
University of Oslo is one of three partners.
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
· 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.
Application
to the Research Council of Norway
(all figures are in MNOK)
|
|
2003 |
2004 |
2005 |
2006 |
SUM |
|
Pre doc. positions |
0,5 |
1,0 |
1,0 |
0,5 |
3,0 |
|
Post doc. positions |
1,2 |
1,8 |
1,8 |
2,4 |
7,2 |
|
Long-term guest researchers |
0,3 |
0,3 |
0,3 |
0,3 |
1,2 |
|
Guest researchers, Univ.
Col. |
0,2 |
0,2 |
0,2 |
0,2 |
0,8 |
|
Seminar leadership |
0,1 |
0,1 |
0,1 |
0,1 |
0,4 |
|
Seminar, conferences |
0,3 |
0,3 |
0,3 |
0,3 |
1,2 |
|
Short-term guests |
0,1 |
0,1 |
0,1 |
0,1 |
0,4 |
|
Travelling |
0,2 |
0,2 |
0,2 |
0,2 |
0,8 |
|
SUM |
2,9 |
4,0 |
4,0 |
4,1 |
15,0 |
The research activities of the
applicants and a significant part of the research activities of the
collaboraters will be dedicated to the questions sketched in the program. The
Department of Mathematics will contribute with necessary infrastructural and
administrative support, such as accounting services, personnel management,
libarary facilities and computer service. The estimated value of this
contribution is MNOK 3,0 annually.
John Rognes, professor, born
1966. Ph.D. from Princeton, 1990. Working in algebraic K-theory. Rognes
presents a long and impressive list of publications, he has been an invited
speaker at several universities and conferences. He will be the project leader
and responsible for the scientific program of the SUP. Detailed information at http://www.math.uio.no/~rognes
Ola Bratteli, professor, born
1946. Dr.Philos. from Oslo, 1974. Working in operator algebras. Bratteli is an
outstanding mathematician who has given deep and important contributions to his
mathematical field of interest. He has published more than 100 papers,
including four books. Detailed information at http://www.math.uio.no/~bratteli
Kristian Ranestad, professor, born
1960. Dr.Philos. from Oslo, 1988. Ranestad works in algebraic geometry.
Ranestad has a strong scientific production and has over the last decade
supervised a number of master and doctoral
students. He has been the organiser of conferences and research summer
schools for young mathematicians.
Detailed information at http://www.math.uio.no/~ranestad
Arne B. Sletsjøe, associated
professor, born 1960. Dr.Scient. from Oslo, 1989. Sletsjøe works in
non-commutative geometry. In addition to his scientific work he has been head
of the Department of Mathematics and he was one of the organisers of the Abel
Bicentennial Conference 2002. He has also been central in the communication
program of the Department of Mathematics and will be project coordinator for
the SUP. Detailed information at http://www.math.uio.no/~arnebs
The applicants represents
three groups of research at the Department of Mathematics. The research group
in Algebraic geometry/non-commutative geometry consists of, in addition to the
above mentioned, Inger Christin
Borge, Jan Christophersen, Geir Ellingsrud, Arnfinn Laudal and Ragni Piene. The Operator
algebras group has four more members; Erik Alfsen, Erik Bedos, Sergey
Neshveyev and Erling Størmer and the other members of the
Topology group are Hans Brodersen, Per Holm, Bjørn Jahren, Jon Reed and Per Tomter.
The collaborating researchers at the university colleges are Trond
Gustavsen (algebraic geometry) at the University College of Agder, Arvid Siqveland (non-commutative
geometry) at the University College of Buskerud, Runar Ile (algebraic
geometry) at the University College of Hedmark, Jan Kleppe (algebraic geometry) and Lars Tueset (operator
algebras) at the University College of Oslo