Symmetries and Computations
J. Rognes, P. A. Østvær
Professors John Rognes and Paul Arne Østvær in the Department of Mathematics at the
University of Oslo propose to lead a research program in pure mathematics titled “Symmetries
and Computations” at the Centre for Advanced Study in the academic year 2013/2014. The
subject matter of the program will concern the part of geometry known as topology, or the
study of spatial shapes and forms in a wide sense. It will create advances in central parts of
homotopy theory, concentrating on its presently active connections with algebraic K-theory and
algebraic geometry. These have implications for and applications to theoretical physics, number
theory and geometric topology.
The program will create a collaborative research environment where both established and
emerging specialists can work together on a focused set of problems in an exciting subject area.
The format of the CAS research programs, with the possibility of bringing together selected
experts for an extended period of time under creative working conditions, is very well suited for
such a collaborative effort. To set the stage, and to disseminate the results, it will be natural
to organize international conferences within the program, one in each half of its duration.
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Leader Candidates
John Rognes, born April 28th 1966, was called to the faculty of the Department of Mathematics
at the University of Oslo in 1994, and promoted to Full Professor in 1998.
He led the Norwegian Research Council (NFR) strategic university program “Suprema”
from 2003 to 2006, as well as the NFR Yngre Fremragende Forskere (YFF) project “Brave New
Rings” from 2005 to 2009, with a combined budget of 21 million NOK. He participates in the
NFR national project “Topology” from 2008 to 2011.
He gave the public presentation of the work of the Abel Prize winners Atiyah and Singer in
2004. He chaired the local organizing committee for the Abel Symposium 2007 on “Algebraic
Topology”, on behalf of the Abel Board, and has organized 12 other international symposia and
conferences since 1998.
He is editor of Acta Mathematica, one of the four leading international journals in mathematics. He is a member of the Royal Norwegian Society of Sciences and Letters, and the Norwegian
Academy of Science and Letters. He has supervised 12 master students, 5 PhD students and
13 postdocs, and given about 100 invited conference and colloquium talks abroad.
John Rognes has published 28 academic papers and 1 monograph since 1992, with 2 more
papers accepted for publication, and 7 papers and 1 monograph submitted for publication, for a
total of 1340 pages. Here are four selected papers relevant to the subject matter of the proposed
program, cited a total of 88 times:
1. J. Rognes and C. Weibel: “Two-primary algebraic K-theory of rings of integers in number
fields.” J. Amer. Math. Soc. 13 (2000), no. 1, 1–54.
2. C. Ausoni and J. Rognes: “Algebraic K-theory of topological K-theory.” Acta Math. 188
(2002), no. 1, 1–39.
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3. N. A. Baas, B. I. Dundas and J. Rognes: “Two-vector bundles and forms of elliptic
cohomology.” Topology, geometry and quantum field theory, 18–45, London Math. Soc.
Lecture Note Ser., 308, Cambridge Univ. Press, Cambridge, 2004.
4. J. Rognes: “Galois extensions of structured ring spectra. Stably dualizable groups.” Mem.
Amer. Math. Soc. 192 (2008), no. 898, viii+137 pp.
Of the papers submitted for publication or in preparation, the four most relevant are:
1. S. Lunøe–Nielsen and J. Rognes: “The topological Singer construction.” 31 pages, submitted to J. of Topology.
2. S. Lunøe–Nielsen and J. Rognes: “The Segal conjecture for topological Hochschild homology of complex cobordism.” 24 pages, submitted to J. of Topology.
3. R. R. Bruner and J. Rognes: “Topological Hochschild homology of topological modular
forms.” 23 pages, in preparation.
4. G. Carlsson, B. I. Dundas and J. Rognes: “Higher red-shift.” 20 pages, in preparation.
See folk.uio.no/rognes/cv.html for a longer CV, and folk.uio.no/rognes/publications.html
for a full list of publications.
Paul Arne Østvær, born August 15th 1973, was hired by the Department of Mathematics
at the University of Oslo in 2005, and promoted to Full Professor in 2008. He is currently the
chair of the Mathematics Section at the Department of Mathematics.
He was awarded the Gold Medal of His Majesty the King of Norway for his PhD thesis in
2000, a stipend from the Rector of the University of Oslo in 2006 and the Abel Extraordinary
Chair in 2010. His research has been supported by the NFR, Niels Henrik Abel and C. M.
Guldbergs mindefond, Centre de Recerca Mathematica, Alexander von Humboldt Association,
Mathematisches Forschungsinstitut Oberwolfach, Fields Institute, NILS mobility project and
the Institute for Advanced Study. He was a finalist in the European Young Investigator Award
(EURYI) in 2004 and participates in the NFR national project “Topology” from 2008 to 2011.
He has visited research institutes and universities in Barcelona, Bielefeld, Cambridge, Essen–
Duisburg, Genoa, Mumbai, Munich, Paris, Princeton and Toronto. He is a co-editor of the Abel
Symposium 2007 proceedings on “Algebraic Topology” and the forthcoming proceedings in the
Journal of K-Theory on “Motivic Homotopy Theory.” He will be appointed as an editor of
Advances in Mathematical Sciences Journal. He co-organized the international conference on
“Motivic Homotopy Theory” in Münster in 2009. He has supervised 6 master students, 2 PhD
students and 2 postdocs, and given about 40 invited conference and colloquium talks abroad.
Paul Arne Østvær has published 35 academic papers and 1 research monograph since 1999,
with 1 more paper accepted for publication, and 8 papers in preparation. Here are 4 selected
papers relevant to the subject matter of the proposed program:
1. B. I. Dundas, O. Röndigs and P. A. Østvær: “Motivic functors.” Doc. Math. 8 (2003),
489-525.
2. A. Rosenschon and P. A. Østvær: “Rigidity for pseudo pretheories.” Invent. Math. 166
(2006), no. 1, 95-102.
3. O. Röndigs and P. A. Østvær: “Modules over motivic cohomology.” Adv. Math. 219
(2008), no. 2, 689-727.
4. N. Naumann, M. Spitzweck and P. A. Østvær: “Motivic Landweber exactness.” Doc.
Math. 14 (2009), 551-593.
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Of the papers in preparation, the four most relevant ones are:
1. K. Ormsby and P. A. Østvær: “Motivic invariants of the rationals at 2.” 21 pages.
2. O. Röndigs and P. A. Østvær: “Slices of hermitian K-theory.” 57 pages.
3. M. Spitzweck and P. A. Østvær: “Motivic twisted K-theory.” 35 pages.
4. J. J. Gutiérrez, O. Röndigs, M. Spitzweck and P. A. Østvær. “Motivic slices and colored
operads.” 30 pages.
See folk.uio.no/paularne for more detailed information.
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Project Description
The Ansatz of the program is the possibility of realizing classical mathematical objects, such as
rings of integers, quadratic forms or modular forms, as the homotopy groups of a ring spectrum,
also known as a “brave new ring.” Such promotion of structure from classical rings to ring
spectra has proved to be a powerful strategy in topology, and the program leaders propose to
bring the successes of this idea further, especially in the modern and active areas of chromatic
homotopy theory and motivic homotopy theory.
A precursor of this program is found in geometric topology, in the Browder–Novikov–
Sullivan–Wall surgery theory (ca. 1970) for the classification of manifolds in terms of linear
algebra over classical integral group rings Z[π], as promoted to the Hatcher–Waldhausen concordance theory (ca. 1980) for the parametrized classification of manifolds, including their groups
of symmetries, in terms of algebraic K-theory over the brave new spherical group rings S[ΩM ].
The full theory will appear in a monograph by Waldhausen–Jahren–Rognes.
The idea was explicitly formulated by Hopkins–Miller (late 1990’s), in the context of modular forms of elliptic curves (the subject of Wiles’ work leading to the proof of Fermat’s last
theorem). They promoted the classical ring of modular forms to the topological modular forms
ring spectrum T M F , whose construction has been at the center of attention in stable homotopy
theory and higher category theory during the last decade. In particular, this theory explains
the Witten genus in the string theory of mathematical physics.
Grothendieck’s vision of motivic cohomology in algebraic geometry is largely realized by
Morel–Voevodsky’s motivic homotopy theory (2001). One of the fundamental ideas is to soften
geometric objects in some intuitive sense so as to allow for a good notion of homotopy in
the algebro-geometric setup. This allows for the promotion of geometric constructions and
questions to more subtle forms with arithmetic content. For example, while a quadratic form
ax2 + bxy + cy 2 corresponds to an ellipse, a parabola or a hyperbola in real geometry, these
shapes are all essentially the same in complex geometry, whereas in arithmetic geometry their
classification is a subtle problem going back to Gauss (1801). Morel (2004) proved that the 0-th
homotopy group of the motivic sphere spectrum S is the Grothendieck–Witt ring of quadratic
forms. In other words, the brave new rings in motivic homotopy theory carry precise information
about subtle but central mathematical objects from number theory and related subjects.
The central problem addressed in the research program will be the computation of the
homotopy groups of brave new rings, such as the topological and motivic sphere spectra. To
achieve this goal, the research will explore useful approximations to the sphere spectrum, such
as hermitian K-theory in the motivic setting, and algebraic K-theory of topological modular
forms in the topological setting. The research in this area is a highly international enterprise
and the proposed program will bring together some of the world’s leading experts.
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These computations have progressed the farthest in the topological setting of homotopy
theory, where the use of real K-theory (KO) and topological modular forms (T M F ) has led
to a nearly complete understanding of the first two kinds of periodic behavior in the homotopy
groups of the sphere spectrum, called the α- and β-families. Related techniques have lead to
significant computational advances, e.g. the recent proof by Hopkins, Hill and Ravenel of the
longstanding Kervaire invariant problem reviewed in Nature and Scientific American (2009).
The chromatic viewpoint of Ravenel and others tells us that the next systematic behavior
to understand are the v3 -periodic γ-families. At small primes this will take us well beyond
the present understanding of stable homotopy theory, and would constitute an international
breakthrough. According to the evidence supporting Rognes’ red-shift conjecture, an incisive
but computable tool for the study of v3 -periodicity can be obtained by merging algebraic Ktheory with topological modular forms. The detailed computation of the homotopy groups of
the resulting brave new ring K(T M F ) is an example of a very concrete goal of the program.
In the subtler setting of motivic homotopy theory, the progress may in outline follow the
strategies in the topological setting. Hermitian K-theory (KQ), which is the motivic analog
of real K-theory, is currently being explored by Østvær and his coworkers. Computations of
hermitian K-groups, extracted from the slice spectral sequence, will provide new information
about motivic stable stems and quadratic forms in examples of arithmetic interest, e.g. number
fields such as the rational numbers. The systematic construction of infinite families of elements
in the motivic stable stems, generalizing the α-family in topology, will be an integral part
of the future computations of symmetries in motivic homotopy theory. Motivic versions of
the Adams and Adams–Novikov spectral sequences in topology will be pivotal tools in these
investigations. A concrete goal of the program is to understand these spectral sequences over
the rational numbers, and more generally over higher local fields and global number fields.
The mathematics underlying these computations is a consequence of the quest for a theory of
motives; this remains a potent driving force in algebraic geometry, automorphic representation
theory, the study of L-functions and arithmetic.
A key tool in the computation of homotopy groups of spectra is the efficient usage of symmetry. In particular, the symmetric permutation groups encode how commutativity for the
multiplication in a classical ring (xy = yx) must be promoted to structured commutativity for
ring spectra. This technical fact has been exploited at the topological side to give the most
efficient techniques for computations of the homotopy groups of the sphere spectrum (Mahowald
et al., Bruner), but have so far not been applied for motivic spectra.
Along similar lines, the Adams spectral sequence transforming cohomology into homotopy
is based on the Steenrod algebra of symmetries (operations) in the cohomology theory. The
passage from the topological to the motivic setting is largely controlled by the Galois symmetries
of fields. The subtle congruences seen by brave new rings, but invisible to algebra, originate in
the symmetries of algebraic or formal groups.
An important benefit of the CAS program will be the opportunity to bring together experts
from homotopy theory and motivic theory, to cooperate on transferring the efficient usage of
structured commutativity for ring spectra to the motivic setting. In the opposite direction,
experts in motivic theory would be able to shed light on the calculations of v3 -periodic families,
in terms of moduli of varieties or crystals.
The chromatic side of the program is a natural development from Rognes’ former Yngre
Fremragende Forskere (YFF) project on “Brave New Rings”. The interplay between homotopy
theory, algebraic K-theory and motivic theory is central in the national Norwegian Research
Council project on “Topology”, and the program is closely related to the research interests of
the topology groups in Bergen, Trondheim and Oslo.
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