Bjørn Ian Dundas

Scientific Interests
Preprints and notes
Former (NTNU, UiB) and current students,
Suggestions for master's projects
my face
4A4d Realfagbygget, Allegaten 41, 4 etg
E-mail address:
dundas at
Snail mail address:
Department of Mathematics
University of Bergen
Postboks 7803
5020 Bergen, NORWAY
A. Solheia 7
N-5251 Søreidgrend
(47) 976 22 576 (email is probably a better option)

Scientific Interests

K-theory, (motivic, stable and/or equivariant) homotopy theory, cyclic homology, homotopy type theory/univalent foundations.

If you don't feel that this list tells you very much, you may consult my pages I'm a topologist, Topologi på roterommet, "Romforskning", Goodwillie calculus or The sphere spectrum (some in Norwegian, they are really my notes for some popular lectures) where I try to give a brief and inadequate idea of various aspects of topology. You may also benefit from the "survey articles for the general public" posted on Hopf and the Wikipedia topology, algebraic topology and algebraic K-theory entries.

For what it's worth, my published papers have been classified according to the AMS Mathematics Subject Classification under 01, 11, 13, 14, 16, 18, 19, 55, 57 and 81, with 19 by far the most frequent.


(UiB changes platform ever so often, so many links will be dead)
Course Semester
MAT111 Grunnkurs i matematikk I fall 2020
MAT292 Project in Mathematics spring 2020
MAT111 Grunnkurs i matematikk I fall 2019
MAT243 Manifolds spring 2017
MAT211 Real analysis fall 2016
MAT243 Manifolds spring 2016
MAT244 Algebraic Topology fall 2015
MAT344 Cohomology fall 2015
MAT342 Differential Geometry fall 2015
MAT243 Manifolds spring 2015
MAT229 Algebraic Geometry I fall 2014
MAT343 Calculus/Applied topology fall 2013
MAT229 Algebraic Geometry I spring 2013
MAT211 Real analysis fall 2012
MAT343 Simplicial methods fall 2012
MAT243 Manifolds spring 2012
MAT211 Real analysis fall 2011
Course Semester
MAT121 Linear Algebra spring 2010
MAT343 Simplicial methods spring 2010
MAT341 Algebraic Topology fall 2009
MAT121 Linear Algebra spring 2009
MAT342 Differential forms fall 2008
MAT225 Number theory fall 2008
MAT243 Manifolds spring 2008
MAT224 Commutative algebra fall 2007
MAT243 Manifolds spring 2007
MAT221 Discrete mathematics fall 2006
MAT224 Commutative algebra spring 2006
MAT111 Grunnkurs i matematikk I fall 2005
Algebraic topology spring 2005
MAT111 Grunnkurs i matematikk I fall 2004
MAT242 Topologi fall 2004



A collaborative effort emanating from the HoTT/UF-year 2018/19 at the Centre for Advanced Study at the Norwegian Academy of Sciences and Letters. Joint with, among others, Bezem, Buchholtz, Coquand, Dybjer, Grayson, Huber.

The official goal is that this book will be an undergraduate textbook written in the univalent style, taking advantage of the presence of symmetry in the logic at an early stage.

my face

The Local structure of algebraic K-theory

Bjørn Ian Dundas, Tom G. Goodwillie and Randy McCarthy

Algebra and Applications, Springer, 2012, XV, 435 p. 5 illus. ISBN 978-1-4471-4392-5

Springer has allowed me to keep a copy of the text on my home page. The pdf-file you find here is the next to last version before the galley proof (and so has some very minor differences from the printed edition). To get the correct numberings for exact reference, please refer to the printed version.

"Algebraic K-theory encodes important invariants for several mathematical disciplines, spanning from geometric topology and functional analysis to number theory and algebraic geometry. As is commonly encountered, this powerful mathematical object is very hard to calculate. Apart from Quillen's calculations of finite fields and Suslin's calculation of algebraically closed fields, few complete calculations were available before the discovery of homological invariants offered by motivic cohomology and topological cyclic homology. This book covers the connection between algebraic K-theory and Bökstedt, Hsiang and Madsen's topological cyclic homology and proves that the difference between the theories are "locally constant". The usefulness of this theorem stems from being more accessible for calculations than K-theory, and hence a single calculation of K-theory can be used with homological calculations to obtain a host of "nearby" calculations in K-theory. For instance, Quillen's calculation of the K-theory of finite fields gives rise to Hesselholt and Madsen's calculations for local fields, and Voevodsky's calculations for the integers give insight into the diffeomorphisms of manifolds. In addition to the proof of the full integral version of the local correspondence between K-theory and topological cyclic homology, the book provides an introduction to the necessary background in algebraic K-theory and highly structured homotopy theory; collecting all necessary tools into one common framework. It relies on simplicial techniques, and contains an appendix summarizing the methods widely used in the field. The book is intended for graduate students and scientists interested in algebraic K-theory, and presupposes a basic knowledge of algebraic topology."

A Short Course in Differential Topology

differential topology differential topology Cambridge Mathematical Textbooks (2018)

A Short Course in Differential Topology is the official Cambridge University Press version (available June 28, 2018) of the original notes which continues to be freely available from this address: To get the correct numberings for exact reference, please refer to the printed version.

According to the promotional material:
"Manifolds abound in mathematics and physics, and increasingly in cybernetics and visualization where they often reflect properties of complex systems and their configurations. Differential topology gives us the tools to study these spaces and extract information about the underlying systems. This book offers a concise and modern introduction to the core topics of differential topology for advanced undergraduates and beginning graduate students. It covers the basics on smooth manifolds and their tangent spaces before moving on to regular values and transversality, smooth flows and differential equations on manifolds, and the theory of vector bundles and locally trivial fibrations. The final chapter gives examples of local-to-global properties, a short introduction to Morse theory and a proof of Ehresmann's fibration theorem. The treatment is hands-on, including many concrete examples and exercises woven into the text, with hints provided to guide the student."

I am very happy for any feedback. The irony with claiming that "the earth is (fairly) flat" at the same time as admitting that the revisions have been undertaken in an "inspiring environment" (combined with climbing the local mountains and fishing in the occasionally rough seas of Northern Norway) has not escaped me; it is not to be considered a typo. The Frontispiece (with the happy fish) is a T-shirt design by Vår Iren Hjorth Dundas and the cover is of Per Karlsa peak which is a nice hike close to where I have made some of the revisions.


Motivic Homotopy Theory

Dundas, B.I., Levine, M., Østvær, P.A., Röndigs, O., Voevodsky,
Unversitext/Springer 2007. X 226p, Jahren (ed). ISBN 978-3-540-45897-5

"This book is based on lectures given at a summer school on motivic homotopy theory at the Sophus Lie Centre in Nordfjordeid, Norway, in August 2002. Aimed at graduate students in algebraic topology and algebraic geometry, it contains background material from both of these fields, as well as the foundations of motivic homotopy theory. It will serve as a good introduction as well as a convenient reference for a broad group of mathematicians to this important and fascinating new subject. Vladimir Voevodsky is one of the founders of the theory and received the Fields medal for his work, and the other authors have all done important work in the subject."

«Gummistrikkgeometri» - fra matematikkens magiske skattekiste

joint with Nils Kristian Rossing. Vitensenteret i Trondheim, juli 2003, ISBN 82-92088-19-9. A popular exposition of some topological features (in Norwegian) written for teachers and other interested laymen. Contains some typos. It can be ordered from Vitensenteret by sending a mail to postkasse at


Dundas, Bjørn Ian. Fibrations and homology sphere bordism. Math. Scand. 72 (1993), no. 1, 20--28. 57Q20 (55R05)

Dundas, Bjørn Ian; McCarthy, Randy. Stable $K$-theory and topological Hochschild homology. Ann. of Math. (2) 140 (1994), no. 3, 685--701. 19D55 (18G60 19D06)

Dundas, Bjørn Ian; McCarthy, Randy. Topological Hochschild homology of ring functors and exact categories. J. Pure Appl. Algebra 109 (1996), no. 3, 231--294. 19D55 (16E40 18G30 18G60)

Dundas, Bjørn Ian. Relative K-theory and topological cyclic homology. Acta Math. 179 (1997), no. 2, 223--242. 19D55

Dundas, Bjørn Ian. K-theory theorems in topological cyclic homology. J. Pure Appl. Algebra 129 (1998), no. 1, 23--33. 19-XX (16Exx)

Dundas, Bjørn Ian. Continuity of K-theory: an example in equal characteristics. Proc. Amer. Math. Soc. 126 (1998), no. 5, 1287--1291. 11Sxx (13Jxx 19D45 19D55)

Dundas, Bjørn Ian. On K-Theory of Simplicial Rings and Degreewise Constructions. K-theory 18 (1999), 77--92. 19D06, 18G30, 19D25, 16N20

Dundas, Bjørn Ian. The Cyclotomic trace for symmetric monoidal categories. in "Geometry and Topology: Aarhus (1998)", 121--143, Contemp. Math., 258, Amer. Math. Soc., Providence, RI, 2000. 19D23 (18D20 19D55 55P43)

Dundas, Bjørn Ian. Localization of V-categories. Theory Appl. Cat. 8 (2001), pp. 284-312. 18D20 (18G55)

Dundas, Bjørn Ian; Röndigs, Oliver and Østvær, Paul Arne, Enriched functors and stable homotopy theory. Documenta Mathematica, Vol. 8 (2003), 409-488. 55P42 (18D20 55P91 55U35)

Dundas, Bjørn Ian; Röndigs, Oliver and Østvær, Paul Arne, Motivic Functors, Documenta Mathematica, Vol. 8 (2003), 489-525. 55P42 (14F42)

Baas, Nils A.; Dundas, Bjørn Ian and Rognes, John Two-vector bundles and forms of elliptic cohomology. In Topology, Geometry and Quantum Field Theory, LMS Lecture note series 308, Cambridge University Press. Ed. Ulrike Tillmann (2004), p. 18--45. 55N34 (18D05 57T30)

Dundas, Bjørn Ian, The cyclotomic trace for S-algebras. J. London Math. Soc. (2) 70 (2004), no. 3, 659--677. 19D23 (18D20 19D55 55P43)

Dundas, Bjørn Ian, Prerequisites in algebraic topology in "Motivic Homotopy Theory" Unversitext/Springer 2007, Dundas/Levine/Østvær/Röndigs/Voevodsky. Jahren (ed).

Dundas, Bjørn Ian and Kittang, Harald Excision for K-theory of connective ring spectra, Homology, Homotopy and Applications, vol. 10(1), 2008, pp. 29 - 39

Ausoni, Christian; Dundas, Bjørn Ian; Rognes, John Divisibility of the Dirac magnetic monopole as a two-vector bundle over the three-sphere. Doc. Math. 13 (2008), 795--801.

Brun, Morten; Carlsson, Gunnar and Dundas, Bjørn Ian Covering homology. Adv. Math. 225 (2010), 3166-3213.

Carlsson, Gunnar; Douglas, Cristopher L. and Dundas, Bjørn Ian Higher topological cyclic homology and the Segal conjecture for tori. Adv. Math. 226 (2011), 1823-1874.

Baas, Nils A.; Dundas, Bjørn Ian; Richter, Birgit and Rognes, John Stable bundles over rig categories. J. Topology 4 (2011), 623-640.

Dundas, Bjørn Ian and Kittang, Harald Integral excision for K-theory, Homology, Homotopy and Applications, vol. 15(1), (2013), 1 - 25.

Baas, Nils A.; Dundas, Bjørn Ian; Richter, Birgit and Rognes, John Ring completion of rig categories. J. Reine Angew. Math. 674 (2013), 43 - 80

Dundas, Bjørn Ian and Morrow, Matthew, Finite generation and continuity of topological Hochschild and cyclic homology, Ann. Sci. Ec. Norm. Super. (4) 50 (2017), no. 1, 201 - 238.

Dundas, Bjørn Ian and Tenti, Andrea, Higher Hochschild homology is not a stable invariant, Math. Zeit. (2017)

Dundas, Bjørn Ian; Lindenstrauss, Ayelet and Richter, Birgit, On higher topological Hochschild homology of rings of integers, Math. Z. 290 (2018), no. 1-2, 145-154.

Dundas, Bjørn Ian and Rognes, John, Cubical and cosimplicial descent, J. Lond. Math. Soc. (2) 98 (2018), no. 2, 439-460.

Dundas, Bjørn Ian; Lindenstrauss, Ayelet and Richter, Birgit, Towards an understanding of ramified extensions of structured ring spectra, Math. Proc. Cambridge Philos. Soc. 168 (2020), no. 3, 435-454.


Dundas, Bjørn Ian and Skau, Christian, Interview with Abel Laureate Yves Meyer, Eur. Math. Soc. Newsl. No. 105 (2017) 14-22, Notices AMS, 65 (2018) No. 5 520-529 and Mathematical Advances in Translation 37 (2018) 226-240.

Dundas, Bjørn Ian and Skau, Christian, Interview with Abel Laureate Robert P. Langlands, Eur. Math. Soc. Newsl. No. 109 (2018) 19-27, Notices AMS 66 (2019), no. 4, 494-503 and Mathematical Advances in Translation.

Dundas, Bjørn Ian and Skau, Christian, Interview with Abel Laureate Karen Uhlenbeck, Eur. Math. Soc. Newsl. No. 113 (2019) 21-29 and Notices AMS 67 (2020), no. 3, 393-403.


Equivariant Structure on Smash Powers, Brun, Dundas, Stolz, 2016

[old version: contact me if you need an update]

Various unofficial notes and talks.

Norwegian topology symposia, and some other conferences I have (co)organized or given lecture series in.

The schedule for the mini symposium on ring spectra, K-Theory and trace invariants in Oslo, March 26th-28th 1998.
The schedule for the mini symposium in algebraic K-theory and homotopy theory in Trondhjem, November 5th-6th 1998.
The schedule for the mini symposium on algebraic K-Theory and topology in Oslo, May 25th-26th 1999.
The schedule for the mini symposium on topology in Trondhjem, November 16th-17th 1999.
The workshop on mathematics education for engineering students in Trondhjem, May 29-30, 2000.
The schedule for the mini symposium on topology in Trondhjem, November 9th-10th 2000.
The ski and mathematics meeting in Oppdal January 4th-7th 2001.
The summer school Motivic homotopy theory Sophus Lie Conference Center, Nordfjordeid, Norway, 12.-16. August 2002
The ski and mathematics meeting at Rondablikk January 9th-12th 2003.
The schedule for the topology meeting in Oslo, May 15th-16th 2003.
The schedule for the topology symposium in Trondhjem, November 25th-26th 2004.
Homological methods in algebra and topology, session at the 24th Nordic and 1st Franco-Nordic Congress of Mathematicians, Reykjavik, Iceland, January 6th - 9th 2005
The schedule for the topology symposium in Oslo, June 2nd-3rd 2005.
The Nordic Conference in Topology in Trondhjem, November 24th-25th 2006.
The topology symposium in Bergen, June 11th-12th 2007.
The Abel symposium in Oslo, August 5th-10th 2007.
Spring symposium, May 21st-23rd 2008 at Thorbjørnrud near Oslo
Nordic topology meeting, November 27 - 28 2008, Trondheim
The algebraic topology session at the 25th Nordic and 1st British-Nordic congress of Mathematicians, 2009.
The Nordic topology symposium in Bergen, June 10-11 2010.
The Algebra, Topology and Fjords! Summer School, Nordfjordeid, Norway June 3rd - 11th 2011.
The Advances in K-theory West coast algebraic topology summer school 2012, Stanford University, July 16-21 2012
The Topology symposium in Bergen, June 13-14 2013.
The Invariants of Structured Ring Spectra European Talbot Workshop 2015, 28 June - 4 July, 2015 Klosters, Switzerland
The Algebraic Topology - Summer School, Lisbon, July 24 - 28, 2017.

Current PhD students

Kristian Alfsvåg
Anders Husebø
Stefano Piceghello

Former students at UiB.

List of old projects at NTNU.


Homotopy Type Theory and Univalent Foundations (2018-2019 at CAS/the Norwegian Academy of Science and Letters in Oslo, PI Marc Bezem, BID)

Computational aspects of Univalence (RCN FRIPRO 2016-2019, PI Marc Bezem (Dept. of Computer Science, UiB) and BID)

Topology, (RCN project 2008-2012, PI BID)

TiN, (RCN project "Topology in Norway", 2004-2007, PI BID)

Bjørn Ian Dundas
Last modified: Last modified: Thurs Dec 22 13:48:12 CET 2016