Bjørn Ian Dundas
Scientific Interests
Ktheory, (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
Ktheory 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.
Classes
(UiB changes platform ever so often, so many links will be dead)
Books

Symmetry
A collaborative effort emanating from the
HoTT/UFyear 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.

The Local structure of algebraic Ktheory
Bjørn Ian Dundas, Tom G. Goodwillie and Randy
McCarthy
Algebra and Applications,
Springer, 2012, XV, 435 p. 5 illus. ISBN 9781447143925
Springer has allowed me to keep a copy of the text on my home page. The pdffile 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 Ktheory 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 Ktheory 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 Ktheory, and hence a single calculation of Ktheory can be used with homological calculations to obtain a host of "nearby" calculations in Ktheory. For instance, Quillen's calculation of the Ktheory 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 Ktheory and topological cyclic homology, the book provides an introduction to the necessary background in algebraic Ktheory 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 Ktheory, and presupposes a basic knowledge of algebraic 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: http://folk.uib.no/nmabd/dt/dtcurrent.pdf. 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 localtoglobal properties, a short introduction to Morse theory and a proof of Ehresmann's fibration theorem. The treatment is handson, 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 Tshirt 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.

Dundas, B.I., Levine, M., Østvær, P.A., Röndigs, O., Voevodsky,
Unversitext/Springer 2007. X 226p, Jahren (ed). ISBN 9783540458975
"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."

joint with Nils Kristian Rossing. Vitensenteret i
Trondheim, juli 2003, ISBN 8292088199. 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 viten.ntnu.no.
Papers
 Dundas, Bjørn Ian. Fibrations and homology sphere bordism.
Math. Scand. 72 (1993), no. 1, 2028. 57Q20 (55R05)
 Dundas, Bjørn Ian; McCarthy, Randy. Stable $K$theory and topological
Hochschild homology. Ann. of Math. (2) 140 (1994),
no. 3, 685701. 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, 231294. 19D55 (16E40 18G30 18G60)
 Dundas, Bjørn Ian. Relative Ktheory and topological cyclic
homology. Acta Math. 179 (1997), no. 2, 223242.
19D55
 Dundas, Bjørn Ian. Ktheory theorems in topological cyclic
homology. J. Pure Appl. Algebra 129 (1998), no. 1,
2333. 19XX (16Exx)
 Dundas, Bjørn Ian. Continuity of Ktheory: an example in equal
characteristics. Proc. Amer. Math. Soc. 126 (1998),
no. 5, 12871291. 11Sxx (13Jxx 19D45 19D55)
 Dundas, Bjørn Ian. On KTheory of Simplicial Rings and Degreewise Constructions. Ktheory 18 (1999),
7792. 19D06, 18G30, 19D25, 16N20
 Dundas, Bjørn Ian. The Cyclotomic trace for symmetric monoidal
categories. in "Geometry and Topology: Aarhus
(1998)", 121143, Contemp. Math., 258, Amer. Math. Soc.,
Providence, RI, 2000. 19D23 (18D20 19D55 55P43)
 Dundas, Bjørn Ian. Localization of Vcategories.
Theory Appl. Cat. 8 (2001), pp. 284312. 18D20 (18G55)
 Dundas, Bjørn Ian; Röndigs, Oliver and
Østvær, Paul Arne, Enriched functors and stable homotopy
theory.
Documenta
Mathematica, Vol. 8 (2003), 409488. 55P42 (18D20 55P91
55U35)
 Dundas, Bjørn Ian; Röndigs, Oliver and
Østvær, Paul Arne, Motivic Functors, Documenta
Mathematica, Vol. 8 (2003), 489525. 55P42 (14F42)
 Baas, Nils A.; Dundas, Bjørn Ian and Rognes, John
Twovector 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. 1845. 55N34 (18D05 57T30)
 Dundas, Bjørn Ian, The cyclotomic trace for
Salgebras. J. London Math. Soc. (2) 70 (2004),
no. 3, 659677. 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 Ktheory 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 twovector bundle over the threesphere. Doc. Math. 13 (2008), 795801.
 Brun, Morten; Carlsson, Gunnar and Dundas, Bjørn
Ian Covering homology. Adv. Math. 225 (2010), 31663213.
 Carlsson, Gunnar; Douglas, Cristopher L. and Dundas, Bjørn
Ian Higher topological cyclic homology and the Segal
conjecture for tori. Adv. Math. 226 (2011), 18231874.
 Baas, Nils A.; Dundas, Bjørn Ian;
Richter, Birgit and Rognes, John
Stable bundles over rig categories.
J. Topology 4 (2011), 623640.
 Dundas, Bjørn Ian and Kittang, Harald Integral excision
for Ktheory, 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) https://doi.org/10.1007/s002090172012y
 Dundas, Bjørn Ian; Lindenstrauss, Ayelet and Richter, Birgit, On higher topological Hochschild homology of rings of integers, Math. Z. 290 (2018), no. 12, 145154.
 Dundas, Bjørn Ian and Rognes, John, Cubical and cosimplicial descent, J. Lond. Math. Soc. (2) 98 (2018), no. 2, 439460.
 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, 435454.
Interviews
 Dundas, Bjørn Ian and Skau, Christian, Interview with Abel Laureate Yves Meyer, Eur. Math. Soc. Newsl. No. 105 (2017) 1422, Notices AMS, 65 (2018) No. 5 520529 and Mathematical Advances in Translation 37 (2018) 226240.
 Dundas, Bjørn Ian and Skau, Christian, Interview with Abel Laureate Robert P. Langlands, Eur. Math. Soc. Newsl. No. 109 (2018) 1927, Notices AMS 66 (2019), no. 4, 494503 and Mathematical Advances in Translation.
 Dundas, Bjørn Ian and Skau, Christian, Interview with Abel Laureate Karen Uhlenbeck, Eur. Math. Soc. Newsl. No. 113 (2019) 2129 and Notices AMS 67 (2020), no. 3, 393403.
Preprints.
Equivariant Structure on Smash Powers, Brun, Dundas, Stolz, 2016
[old version: contact me if you need an update]
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, KTheory and trace invariants in Oslo, March
26th28th 1998.
 The schedule for the mini symposium in algebraic
Ktheory and homotopy theory in Trondhjem, November 5th6th
1998.
 The schedule for the mini symposium on
algebraic KTheory and topology in Oslo, May 25th26th
1999.
 The schedule for the mini symposium on topology in
Trondhjem, November 16th17th 1999.
 The workshop on mathematics education for
engineering students in Trondhjem, May 2930, 2000.
 The schedule for the mini symposium on topology in
Trondhjem, November 9th10th 2000.
 The ski and mathematics meeting in Oppdal January
4th7th 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
9th12th 2003.
 The schedule for the topology meeting in Oslo,
May 15th16th 2003.
 The schedule
for the topology symposium in Trondhjem, November
25th26th 2004.
 Homological methods in algebra and topology, session at the
24th Nordic and 1st FrancoNordic Congress of
Mathematicians, Reykjavik, Iceland, January 6th  9th 2005
 The schedule for the topology symposium in Oslo, June
2nd3rd 2005.
 The Nordic Conference in Topology in Trondhjem, November
24th25th 2006.
 The topology symposium in Bergen, June
11th12th 2007.
 The Abel symposium in Oslo, August
5th10th 2007.
 Spring symposium, May 21st23rd 2008 at Thorbjørnrud near Oslo
 Nordic topology
meeting, November 27  28 2008, Trondheim
 The algebraic
topology session at the 25th Nordic and 1st
BritishNordic congress of Mathematicians, 2009.
 The Nordic topology symposium in Bergen, June
1011 2010.
 The Algebra, Topology and Fjords!
Summer School,
Nordfjordeid, Norway June 3rd  11th 2011.
 The Advances in Ktheory West coast algebraic topology
summer school 2012, Stanford University, July 1621 2012
 The Topology symposium in Bergen, June
1314 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.
Projects
Homotopy Type Theory and Univalent Foundations (20182019 at CAS/the Norwegian Academy of Science and Letters in Oslo, PI Marc Bezem, BID)
Computational aspects of Univalence (RCN FRIPRO 20162019, PI Marc Bezem (Dept. of Computer Science, UiB) and BID)
Topology, (RCN
project 20082012, PI BID)
TiN,
(RCN
project "Topology in Norway", 20042007, PI BID)
Bjørn Ian Dundas
Last modified: Last modified: Thurs Dec 22 13:48:12 CET 2016