Brave New Rings

Introduction

We propose to organize a research year in pure mathematics in the academic year 2005-2006 at the Centre for Advanced Study at the Norwegian Academy of Science and Letters, focused on the study of the objects popularly known as ``brave new rings,'' or more formally, ``S-algebras.'' The program will bring together leading Norwegian and international researchers in geometry and topology to a coordinated effort in this currently very active topic.

1. The field of research

The study of ``brave new rings'' links together ideas and results from several different parts of mathematics. Its technical foundations lie within algebraic topology, or more precisely, within stable homotopy theory. But the structures that are studied are most simply viewed as generalizations of the ring concept from algebra and algebraic geometry, and it is fruitful to consider how known constructions and phenomena in algebra can be enriched to this algebraic-topological situation. The development of the theory for ``brave new rings'' was historically motivated by concrete applications in nearby areas of mathematics, such as the study of topological and differentiable symmetry groups of manifolds in geometric topology (Waldhausen). But the development has subsequently also been justified through new applications to wider areas, such as a construction of the Witten genus for string manifolds (Ando-Hopkins-Strickland), and of conformal quantum-field theories (Segal, Baas-Dundas-Rognes) in mathematical physics/string theory, and an improvement of the theory of (elliptic) modular forms in number theory (Borcherds, Hopkins-Mahowald-Miller). A corresponding development is conceivable for K3-surfaces or higher-dimensional Calabi-Yau varieties over finite fields. Several of these applications are based on techniques from algebraic K-theory. The proposed topic is therefore fully central to modern basic research in geometric and topological topics with an algebraic flavor.

2. Brief historical background

The spectra of algebraic topology are the picture images of topological spaces, as seen by (generalized) cohomology theories mapping the topology to algebra. They naturally behave as modules over the fundamental ring spectrum S called the sphere spectrum, and can therefore be interpreted as S-modules. However, large parts of algebra (including algebraic geometry) is really concerned with the richer structure of modules equipped with products, i.e., algebras. The analogous structure in stable homotopy theory is an S-module equipped with a (good) product, which is called an S-algebra or a ``brave new ring.''

The need to study S-algebras appeared already in F. Waldhausen's work from the 1970's, relating the geometric topology of a high-dimensional manifold M to the algebraic K-theory of modules over the group S-algebra S[G], where G is the loop group of M. At that time the technical foundations for making good sense of the product structure on an S-algebra, with an associated module theory, were not in place, so Waldhausen made use of ad hoc definitions that sufficed for his purposes. A related construction was used by M. Bökstedt to define topological Hochschild homology for an S-algebra, and in a 1988 talk at Northwestern University, Waldhausen coined the term ``brave new ring'' (by reference to Huxley's ``Brave New World'' and Shakespeare's ``The Tempest''), and had also planned to discuss topological André-Quillen (co-)homology in such a framework.

These and many other desired applications of S-algebras justified the search for such good foundations. The critical issue was the construction of a smash product of S-modules with as good formal properties as the tensor product of abelian groups. This was achieved by three different approaches in the mid 1990's, by (1) Elmendorf, Kriz, Mandell and May developing the operadic viewpoint of J.P. May, (2) Hovey, Shipley and Smith developing the symmetric spectra of J. Smith, and (3) Lydakis developing the Gamma-spaces of G. Segal.

These constructions opened up the modern field of ``brave new rings,'' also known as the study of S-algebras, or ``spectral algebra.'' Since three distinct constructions appeared almost simultaneously, some initial work was needed to compare the three definitions. This was soon done by the authors above, and S. Schwede, and is now largely completed. Instead, the emphasis is now turning to the analysis and applications of these new mathematical tools.

3. More recent developments

Since the 1990's, the program to enrich classical algebra to topological S-algebra has achieved a number of successes, both internally within stable homotopy theory, in neighboring areas, and also in seemingly remote parts of mathematics. We can point to at least four important threads in this development, which often are braided together.

3.1. Hopkins-Miller theory and topological modular forms

Around 1995 Mike Hopkins and Haynes Miller developed an obstruction theory that in many cases can determine when a diagram of ring spectra comes from a diagram of S-algebras. It was presented in a paper by Charles Rezk, and a corresponding theory for commutative ring spectra and commutative S-algebras is being presented by Paul Goerss and Hopkins. A great advantage of having a diagram of S-algebras is that this has a well-defined limit, while such a limit does usually not exist for a diagram of ring spectra. The limit of such a diagram produces a new S-algebra, and several exceptionally interesting such ``brave new rings'' have only been constructed in this fashion. This is a very general principle, but has been seen to be applicable in many very interesting situations.

For example, Hopkins and Miller have constructed a variant of the ring of (elliptic) modular forms that is a commutative S-algebra called topological modular forms, denoted tmf. Modular forms are central objects in number theory, and occur e.g. in the Taniyama-Shimura-Weil conjecture that A. Wiles partially confirmed to prove Fermat's last theorem. The topological modular forms differ somewhat from the classical ones at the primes 2 and 3, and there they are in a sense better than the classical ones. For example, Richard Borcherds (Fields medal 1998) has shown that the modular forms that appear as theta-functions of even unimodular lattices satisfy certain congruences, which appear as mysterious from the classical perspective, but which from the ``brave new'' perspective have the simple interpretation that these theta-functions precisely occur as topological modular forms. The S-algebra tmf can thus shed light on deep and currently interesting number-theoretic information.

Simultaneously, the spectrum tmf is a very elegant and useful tool within stable homotopy theory. Hopkins and Mark Mahowald showed in 1998 that most of what is known about the 2- and 3-primary stable homotopy groups of spheres can comparatively easily be read off from known information about tmf. The topology research groups in Norway have been involved in this development from an early stage; the proposer gave lecture series on this matter in Oslo in 1999 and in Trondheim in 2001, and Mahowald gave a series of talks on this work in Oslo in 1999. Topological modular forms is currently the subject of much international activity, including a conference at the Isaac Newton institute, Cambridge, England in December 2002 and at Universität Münster, Germany in October 2003.

The example with topological modular forms arises by studying elliptic curves over finite fields. There are other more complex algebro-geometric objects that can be studie in a similar fashion, and which may be expected to provide even more subtle S-algebras than tmf. An elliptic curve can be identified with its Jacobian, which is an abelian variety and especially a 1-dimensional algebraic group. This can in turn be completed to a 1-dimensional formal group of height 1 or 2. Going up one geometric dimension, from curves to surfaces, the class of K3-surfaces has associated Brauer-groups that deformation-theoretically give rise to an enveloping 1-dimensional formal group, this time of height up to 10 (Artin 1975, Artin-Mazur 1977). Particularly symmetric K3-surfaces (and such exist, with large simple Mathieu-groups as automorphism-groups (Mukai 1988)) give rise to diagrams of such formal groups, and it is reasonable to expect that the universal (Lubin-Tate) deformations of these formal groups give diagrams of commutative (Landweber exact) ring spectra. The Goerss-Hopkins-Miller obstruction theory should be able to decide whether such a diagram exists as a diagram of commutative S-algebras. Such commutative S-algebras could then be called K3-cohomology theories, and the limit of the diagram would be a commutative S-algebra representing a universal K3-cohomology. Such a theory can probably (only) be constructed and analyzed by topological methods, which by now are well developed, and should contain extremely subtle information about ``chromatic'' periodic phenomena in stable homotopy theory that correspond to heights up to 10. Topological modular forms has given (nearly) complete such information up to height 2, and some things are known by other methods for heights 3 and 4, but K3-cohomology would appear to be able to revolutionize the detailed mathematical knowledge about spectra. This is naturally a possible working area for the researchers that are planning to participate in this program in 2005-2006.

More generally, Calabi-Yau n-folds, by way of a 1-dimensional formal group derived from degree n cohomology with coefficients in the multiplicative group, might give even further such S-algebras. The case n=1 corresponds to elliptic curves, and n=2 to K3-surfaces, but at the present stage it is certainly most reasonable to understand these cases more fully before eventually going on to higher dimensional Calabi-Yau varieties over finite fields.

Topological modular forms are also linked to string theory, via works in mathematical physics by Ed Witten (Fields medal 1990) and about S-algebras by Matthew Ando, Hopkins and Neil Strickland. In ``classical'' quantum mechanics the Dirac operator on sections in a spinor bundle plays a key role. In string theory Witten defined a corresponding operator for imagined spinor bundles over the free loop space of a so-called string manifold (spin with c_2=0), and associated an (elliptic) modular form to this operator, called the Witten genus of the string manifold. This heuristic construction is important for string theory, and was given a mathematically precise and very natural form in the framework of S-algebras by Ando, Hopkins and Strickland (2001).

3.2. Topological André-Quillen cohomology

The theory of brave new rings generalizes the classical theory for rings, in that each ring R in the algebraic sense gives rise to a brave new ring HR (which represents singular cohomology with coefficients in R). It is also possible to go the other way, from a brave new ring A to its homotopy groups, perceived as a graded ring in the algebraic sense, but this reverse process loses some essential information. A tool for keeping track of the lost information is provided by the k-invariants in the Postnikov tower for A, and these live in certain cohomology groups. For spectra or S-modules A these cohomology groups are the classical singular cohomology groups from algebraic topology, but for S-algebras A the correct cohomology groups are given in terms of topological Hochschild cohomology, and for commutative S-algebras A the k-invariants live in a theory called topological André-Quillen cohomology.

This last theory was envisioned by Waldhausen in 1988, and partially developed in a preprint by Kriz addressing the existence of a commutative S-algebra model for the Brown-Peterson spectrum BP. With the new frameworks for brave new rings in place, the May students Maria Basterra and Mike Mandell, as well as Randy McCarthy and Vaughn Minasian, have promoted topological André-Quillen cohomology to an effective, well-defined tool. Alan Robinson has worked on an obstruction theory that is parallel to the Hopkins-Miller theory from 3.1, which led to what he calls Gamma-cohomology, and which is also closely related to topological André-Quillen cohomology. His coauthor Sarah Whitehouse, as well as Teimuraz Pirashvili, Birgit Richter and Andy Baker have developed these tools further, and can now in a more systematic ``synthetic'' fashion achieve results that are similar to the more isolated ``natural'' outcomes of the Hopkins-Miller theory. These methods were e.g. the subject of a conference organized by Baker and Richter at Glasgow University in January 2002. The potential for these ``synthetic'' constructions is far from exhausted, and the range of effective techniques for using e.g. the natural operations in topological André-Quillen cohomology have so far not been fully mapped out. Further work concerning the construction and classification of commutative S-algebras should probably also account for their k-invariants in this cohomology theory. A famous and long unsolved problem, which could be attacked in this context, is whether the Brown-Peterson spectra BP and BP<n> can be realized as commutative S-algebras. A positive answer to this problem would simplify a couple of conceptual ideas in stable homotopy theory, such as the chromatic red-shift phenomenon explained in 3.3 below, and will be a natural target for the research during this planned program.

Topological André-Quillen cohomology can naturally be used to define étale maps and smooth maps in the category of commutative S-algebras, extending the corresponding notions in algebraic geometry. Randy McCarthy and the proposer have independently developed such theories, in part motivated by ideas concerning the implications of brave new rings for étale and Galois-descent in algebraic K-theory. See 3.3 below. Similar ideas were studied in the associative case by Stephan Schwede and Jeff Smith, and also by Andrei Lazarev.

3.3. Algebraic K-theory and topological cyclic homology

As recalled above, Waldhausen's theorems about the connection from topological and differentiable symmetry groups of high-dimensional manifolds, via higher simple homotopy theory, to the algebraic K-theory of spaces, was an important motivation for the development of the theoretical foundations for brave new rings, alias S-algebras. In this area the point of view of such generalized rings has been long known, and has motivated or led to many important results about algebraic K-theory. Waldhausen sketched in the 1980's an S-algebraic argument leading to the conjecture that stable K-theory is equivalent to topological Hochschild homology. This was attempted worked into a precise proof by Roland Schwänzl, Ross Staffeldt and Waldhausen, but the technical foundations then present made this very difficult. Instead the conjecture was proven by Bjørn Dundas and Randy McCarthy in 1992.

Parallel to this, Marcel Bökstedt, Wu-Chung Hsiang and Ib Madsen developed a topological cyclic homology-theory, which is related to topological Hochschild homology in essentially the same way as Connes' cyclic homology (Fields medal 1982) is to classical Hochschild homology. Bökstedt, Hsiang and Madsen (1993) used their topological cyclic homology to prove a K-theoretic form of the well-known Novikov conjecture, which concerns Waldhausen's algebraic K-theory of spaces. Later, McCarthy and Dundas (1997) could prove a conjecture of Tom G. Goodwillie saying that for suitable ``nilpotent'' extensions of S-algebras, the relative form of algebraic K-theory is equivalent to the relative form of topological cyclic homology. Their arguments were motivated both by the ideas of brave new rings, and Goodwillie's ``calculus'' for homotopy functors.

These results set the stage for completely new and precise calculations of algebraic K-theory, both for classical rings and for S-algebras, which previously had been basically inaccessible. For example, Bökstedt and Madsen (1995) proved the Lichtenbaum-Quillen conjecture about the algebraic K-theory of p-adic number fields by explicit computation, for odd primes p. The proposer (1996) showed that the result also holds for p=2, before Vladimir Voevodsky (Fields medal 2002), Chuck Weibel and the proposer (2000) established the Lichtenbaum-Quillen conjecture for algebraic K-theory of all global and local number fields at p=2. (Also this last argument uses S-algebras in place of classical rings in one critical place.) Thanks to the S-algebra point of view, these computations in algebraic K-theory of rings can also be fed back to Waldhausen's theory, and have given explicit new knowledge about the previously mentioned symmetry groups of high-dimensional manifolds, such as spheres and discs (Rognes 2002, 2003).

Lars Hesselholt and Madsen (1999) have taken these calculations much further in the algebraic context, by developing a Grothendieck/Berthelot crystalline cohomology, or more precisely, a deRham-Witt complex with logarithmic poles, which allows them to compute algebraic K-theory for local fields of mixed characteristic. Thomas Geisser and Hesselholt (2003) extend this approach to Henselian local rings. This empirically now well-founded connection between topological cyclic homology and crystalline cohomology, as recently studied by e.g. Gerd Faltings (Fields medal 1986), should be of considerable interest, also within (arithmetic) algebraic geometry.

Christian Ausoni and the proposer have carried the computations of Madsen et al further in the topological direction, i.e., to compute topological cyclic homology and algebraic K-theory of S-algebras A that are not of the form HR for a classical ring R. Under the hypothesis that the Brown-Peterson spectrum BP<n> is a commutative S-algebra (which is true for n=0 and n=1, but an open conjecture for larger n, as mentioned in 3.2), they computed the algebraic K-theory of this brave new ring. The result demonstrates a fascinating new phenomenon that ties algebraic K-theory to the so-called chromatic picture of stable homotopy theory, where homotopy theoretic phenomena are ordered into periodic families of increasing ``wavelengths'' that are tied to the height concept for formal groups, as in 3.1. (The phenomena that are detected by cohomology theories related to a formal group of height n occur in periodic families where the periodicity is given by multiplication by powers of a certain element v_n.) The phenomenon is a kind of red-shift, somewhat like Hubble's discovery in cosmology, in that algebraic K-theory takes an S-algebra in the n-th periodic family to an S-algebra in the next, (n+1)-periodic family. That the wavelength increases does of course correspond to a red-shift in the optical sense.

It would be very illuminating if the deRham-Witt complex with logarithmic poles of Hesselholt and Madsen, which has been used to keep track of the bookkeeping in the computation of algebraic K-theory in many algebraic cases, also could be transported to the topological situation. A test case would be to systematize and generalize the computations made by Ausoni and the proposer, where for the topological cyclic homology of BP<1> an element occurs that formally behaves as v_2 dlog v_1, i.e., with a ``logarithmic pole at v_1,'' in an analogous fashion to how for the topological cyclic homology of BP<0> = HZ an element v_1 dlog p arises from the logarithmic pole at p = v_0. Such a study would correspond to the development of a topological deRham-Witt complex, which again is an instance of the program to enrich important algebraic constructions to a topological situation. A conference is planned in October 2003 at Universität Münster, where Hesselholt, Holger Reich and the proposer will give lecture series that collect together much of the present knowledge about this problem, but the work with such a topological crystalline cohomology will most certainly also be a current topic in 2005-2006.

In connection with algebraic K-theory, where the Lichtenbaum-Quillen conjecture postulates how algebraic K-theory behaves in relation to Galois extensions of fields, or more generally, in relation to étale covers of commutative rings or schemes, it is also natural to analyze how the concepts Galois extension and étale cover naturally can be extended to commutative S-algebras. These commutative notions are relevant also to the geometric topological applications of S-algebras through Waldhausen's algebraic K-theory of spaces, since by the work of Farrell and Jones (1991), the algebraic K-theory of non-positively curved manifolds is controlled by the algebraic K-theory of points and circles (closed geodesics), and the S-algebras corresponding to these spaces are commutative S-algebras. The proposer has worked on extending these concepts, and has made precise how the famous Lichtenbaum-Quillen conjecture can be extended to the topological situation. This gives a framework that encompasses many of the results of the Hopkins-Miller theory, e.g. by Jack Morava, Ethan Devinatz and Hopkins. It also suggests how algebraic K-theory of commutative S-algebras may best be understood by way of a topological generalization of schemes that corresponds to the generalizations from rings to brave new rings. They might be called brave new schemes, or S-schemes.

An interesting phenomenon tied to Galois extensions of commutative S-algebras is that while one in classical arithmetic is concerned with global fields and local fields, where local means concentrated at a single prime p, for S-algebras one can localize infinitely much more finely, through the already mentioned chromatic filtration. For each prime p and for each natural number n it is possible to concentrate only on the n-th kind of periodic families at p, which amounts to studying the Morava K(n)-local category at p. Detailed information about this topologically most local situation has been given e.g. by Mark Hovey and Neil Strickland. It is reasonable to expect that the classical class field theory, which describes all abelian Galois extensions of local or global fields, will have a counterpart in a class field theory for such abelian extensions of K(n)-local commutative S-algebras. Tilman Bauer, Strickland and the proposer have studied some aspects of such a ``K(n)-local class field theory.'' Just as the local class field theories assemble through the adèle ring to control the global class field theory, a similar assembly may serve to govern the abelian extensions of p-local or integral commutative S-algebras. But again these results belong to the future.

3.4. Elliptic cohomology and string theory

Between the sphere spectrum S and the integers Z there are many other brave new rings, including the one called ku that represents connective complex topological K-theory. It has chromatic complexity 1, and as such it a first example of an S-algebra that does not come from classical algebra. Recall that the algebraic K-theory of S is a case of Waldhausen's algebraic K-theory of spaces, which is linked to geometric topology, while the algebraic K-theory of Z is linked to the Bernoulli numbers and algebraic number theory. Similarly, Nils Baas, Dundas and the proposer have shown that the algebraic K-theory of ku is also connected to another mathematical topic, namely elliptic cohomology, elliptic objects and conformal quantum field theories, which belong to string theory and mathematical physics. This relies on a case of the calculations of Ausoni and the proposer from 3.3, which essentially says that algebraic K-theory of topological K-theory detects the homotopy theoretic phenomena connected to the n-th kind of periodic families for n=2. This is a previously known property of the so-called elliptic cohomology theories, so algebraic K-theory of topological K-theory is a ``form'' of elliptic cohomology. These elliptic theories were recently the topic of a conference at the Isaac Newton institute, Cambridge, England in December 2002.

Elliptic cohomology theories were introduced by Peter Landweber, Doug Ravenel and Bob Stong in 1986, and were quickly connected to string theory and elliptic modular forms. Graeme Segal accounted for this development in a 1988 Bourbaki seminar, where he also foresaw the existence of ``elliptic objects,'' which were to be geometrically defined objects associated to a space X, which on one hand should define a conformal quantum field theory for strings in X, and on the other hand should correspond to cycles in an elliptic cohomology theory at X. The works of Ausoni, Baas, Dundas and the proposer show that the so-called ``2-vector bundles'' over X have many, and perhaps all, of the properties that Segal expected. These constructions are essentially linked to the technology underpinning the modern notion of S-algebras, and it is reasonable to expect that this string-theoretic connection can evolve in the proposed project. For further details, see the preprint at http://www.math.uio.no/~rognes/papers/segal60.dvi .

4. Participants

The proposed topic has ties to a large part of modern mathematics. In a somewhat limited program such as this proposal concerns it is not possible to cover everything, so we must concentrate on some central areas and realistic aims.

We aim to have 8 researchers present in each semester, to make full use of the Centre's capacity. The most strongly involved Norwegian researchers are professors Nils Baas (3.4) and Bjørn Dundas (3.3 and 3.4) at NTNU in Trondheim, and professors Bjørn Jahren (3.3) and John Rognes (the proposer/all topics) at UiO in Oslo. We expect that Dundas and the proposer participate for 2 semesters, while Baas and Jahren take part in 1 semester each, with extraordinary research leave funded from their home institutions. This leaves room for 5 international researchers in each semester, which is also commensurable with the expected budget of 2.5 million Norwegian kroner from the Centre for Advanced Study.

A preliminary international query about the interest for such a research year in Oslo gave a very positive response from a number of established and active researchers in the subject. The most central in the current plan will be professors Andy Baker (Glasgow University, 3.1 and 3.2), Bjørn Dundas (NTNU, 3.3 and 3.4), Lars Hesselholt (MIT, 3.3) and Neil Strickland (Sheffield University, 3.1 and 3.3). Letters of intent to participate and short CVs are attached for these researchers. Other researchers that have stated their interest in participating include Marcel Bökstedt (Århus), John Greenlees (Sheffield), Ib Madsen (Århus), Haynes Miller (MIT), Birgit Richter (Bonn) and Stefan Schwede (Bonn), but other specialists may also become involved. We can expect to invite 3 of these for one semester each, so arranged as to have working groups of ca. 4 people focused on each of the topics 3.1-3.4, with two groups in each semester.

In addition, we will organize one week-long conference in each semester, with the project partners and other international guests. These two conferences can naturally be tied to the Norwegian research council financed Strategic university program in pure mathematics (SUPREMA) that the proposer leads at the University of Oslo, which in the academic year 2005-2006 can accommodate the costs for these conferences, estimated at 100,000-150,000 Norwegian kroner.

In addition to the researchers that are included as participants in this project, there are also several other Norwegian and Scandinavian researchers that work in adjacent areas of mathematics, which will be relevant for the project and may benefit from its expected results. In algebraic geometry this includes e.g. Arnfinn Laudal, Arne B. Sletsjøe and Christin Borge in deformation theory, Kristian Ranestad (elliptic curves and modular forms), Geir Ellingsrud and Torsten Ekedal (algebraic surfaces) and Ragni Piene (Calabi-Yau varieties). In topology this includes Paul Arne Østvær in algebraic K-theory, as well as several post docs that will be supported by the Strategic university program in Oslo. Likewise for the current interaction between the mathematics and theoretical physics groups in Oslo (Carsten Lütken, Håkon Enger) on topics in string theory. The results of the research will, as usual, be disseminated through publications and conferences, as well as through increased expertise both internationally and nationally.

5. Relation to other activities

5.1. Strategic university program

The proposer is leader for the Norwegian research council's Strategic university program in pure mathematics (SUPREMA) at the Section for pure mathematics in the Department of mathematics at the University of Oslo, which runs in the period 2003-2006 with a total budget of 12.5 million Norwegian kroner for doctoral- and postdoctoral stipends, as well as guest researchers, conferences and working seminar activities, within algebra, analysis and topology. The proposed project at the Centre for Advanced Study concerns a clearly delimited field of research within the broader span of the Strategic university program, which is labeled ``New Contexts for Arithmetic and Geometry.'' The project is also more clearly focused on research work at the border of current knowledge, with a list of precise problems within currently active basic research. It is reasonable to expect that the increased number of younger and active researchers that will be based at the Department of mathematics in Oslo during the period of the Strategic university program, also will add to the interest in the proposed research year at the Centre. The proposer's leadership tasks for the Strategic university program are compatible with a year's research stay at the Norwegian Academy of Science and Letters.

5.2. Marie Curie Center of Excellence

The proposer also participates in the group at the Department of Mathematics in Oslo that constitutes an EU-financed Marie Curie Center of Excellence, or Training Site (OMATS) for European doctoral students in algebraic geometry, operator algebras or algebraic K-theory/topology. The OMATS funding runs in the period 2000-2004, but is currently being planned continued in a renewed form during the European Union's 6th framework program, with a continued emphasis on geometry. There does not appear to be any conflict between these overlapping academic interest.

5.3. The Mittag-Leffler institute

Kathryn Hess (Lausanne), Bjørn Jahren (UiO) and Bob Oliver (Paris) are applying to the Mittag-Leffler institute in Stockholm to organize a year in algebraic topology in the same time period as this project, i.e., in the academic year 2005-2006. Their application spans a wider field of research than this project, with topics in unstable homotopy theory and algebraic homotopy theory, in addition to topics from stable homotopy theory and algebraic K-theory similar to those discussed above. It is clear that it would be unfortunate if these two research projects, in Oslo and Stockholm, were to take place at the same time. The situation is complicated somewhat by the fact that the board for the Mittag-Leffler institute will not decide on who shall organize their academic year 2005-2006 before May/June 2003, which is very close to the date when also the board for the Centre wishes to reach such a decision. Since the project at the Mittag-Leffler institute is wider, and involves more people, it seems best that if a collision arises, then the presently proposed project at the Centre for Advanced Study should be postponed one year, to the academic year 2006-2007. As long as it is unknown when there will again be a year in algebraic topology in Stockholm, the proposer will nonetheless apply to the Centre for the year 2005-2006, in spite of this element of uncertainty.

6. Summary

The project will bring together internationally leading and most active researchers in the study of ``brave new rings,'' also known as ``S-algebras,'' for a research year at the Centre for Advanced Study in Oslo. The research will be concentrated on current topics of interest, tied to (3.1) Hopkins-Miller theory and topological modular forms, aiming to construct K3-cohomology, (3.2) topological André-Quillen cohomology, with a view to construction and classification of commutative S-algebras, (3.3) algebraic K-theory and topological cyclic homology, aiming to develop a topological deRham-Witt complex, and (3.4) elliptic cohomology and string theory, relating elliptic objects to S-algebraic constructions. These topics are in turn linked to number theory, stable homotopy theory, geometric topology, algebraic geometry and mathematical physics. In the project period 5 international and 3 Norwegian researchers will work at the Centre each semester. The financing can be strengthened by a cooperation with the Norwegian research council's Strategic university program in pure mathematics at the University of Oslo. If the Mittag-Leffler institute selects the algebraic topology proposal for the same period (5.3), it is desirable that the present program is postponed by one year, to 2006-2007.

Professor John Rognes
Department of Mathematics
University of Oslo
March 14th, 2003