Dan Christensen's papers, preprints and talks

Comments and feedback welcome.

Some recent talks:

• Sphere bundles and their invariants, HoTTEST seminar, April 2024, 60 minutes. Video.
• Ext in homotopy type theory, HoTT 2023, CMU, May 2023, 25 minutes.
• An introduction to homotopy type theory, colloquium, Waterloo, March 2023, 55 minutes.
• No set of spaces detects isomorphisms in the homotopy category, Categories and Topology, Mathematical Congress of the Americas, July 2021, 20 minutes. Video. Based on the paper with Arlin, below.
• Non-accessible localizations, 20 minute talk at HoTT/UF 2021, July, 2021, based on the paper below. Two-page abstract. Video. Formalization.
• Reasoning in an ∞-topos with homotopy type theory, colloquium, Topos Institute, April 2021, 60 minutes. Video.
• Non-accessible reflective subuniverses in HoTT, CMU, March 2021, 90 minutes. Based on the paper below.
• The Hurewicz theorem in homotopy type theory, HoTT/UF 2020, 20 minutes. Video. Based on the paper with the same title, below.
• An introduction to homotopy type theory, colloquium, Kyoto University, March 2020, 90 minutes. Includes a brief discussion of localization.
• Localization in homotopy type theory, HoTT 2019, CMU, August 2019, 25 minutes. Based on the paper with the same title, below.
• The interaction between homotopy type theory and mathematics, Workshop on Computation in Scientific Theory and Practice, Munich, June 2019, 45 minutes. Directed to philosophers.
• Higher Toda brackets and the Adams spectral sequence, Cascade Topology Seminar, November 2018, 60 minutes.
• Topological data analysis, a series of four one-hour lectures at the Fields Institute Summer School on Topological Data Analysis and Finance, UWO, July 2018.
• Group theory, one-way functions, spam prevention and digital currency, talk to UWO undergraduates, October 2016.
• Computation in Coq, UWO seminar, June 2016.
• Univalent fibrations in type theory and topology, colloquium, Wayne State University, April 2016.
• A characterization of univalent fibrations, CT2015, June 2015, 30 minutes.

Formal proofs:

• Contributions to Coq-HoTT library.
• Central types, bands and H-spaces. See paper below.
• Non-accessible localizations. See paper below.
• The Hurewicz theorem (in progress).
• A characterization of univalent fibrations.

Papers and preprints:

47. The Beauty of Roots, joint with John C. Baez and Sam Derbyshire. Notices of the AMS 70(9) (2023), 1495-1497. Published version as html, doi:10.1090/noti2789 (pdf), arXiv:2310.00326.

46. Ext groups in homotopy type theory, joint with Jarl G. Taxerås Flaten. Submitted. 29 pages. arXiv:2305.09639.

45. Central H-spaces and banded types, joint with Ulrik Buchholtz, Jarl G. Taxerås Flaten, and Egbert Rijke. Submitted. 25 pages. arXiv:2301.02636. Formalization available in the central-types repository.

44. Non-accessible localizations. Journal of Topology 17(2) (2024), 19 pages. doi:10.1112/topo.12336, arXiv:2109.06670, pdf file.

43. Exterior bundles in diffeology, joint with Enxin Wu. Israel Journal of Mathematics 253 (2023), 673-713. doi:10.1007/s11856-022-2372-9, arXiv:2009.01770, pdf file as published online.

42. On good morphisms of exact triangles, joint with Martin Frankland. Journal of Pure and Applied Algebra 226(3) (2022), 106846, 33 pages. doi:10.1016/j.jpaa.2021.106846, arXiv:2008.03643, pdf file.

41. Characterizations of modalities and lex modalities, joint with Egbert Rijke. Journal of Pure and Applied Algebra 226(3) (2022), 106848, 21 pages. doi:10.1016/j.jpaa.2021.106848, arXiv:2008.03538, pdf file.

40. The Hurewicz theorem in homotopy type theory, joint with Luis Scoccola. Algebraic & Geometric Topology 23(5) (2023), 2107-2140. doi:10.2140/agt.2023.23.2107, arXiv:2007.05833, pdf file.

39. Detecting isomorphisms in the homotopy category, joint with Kevin Arlin. Algebraic & Geometric Topology 23(7) (2023), 2975-2991. doi:10.2140/agt.2023.23.2975, arXiv:1910.04141, pdf file.

38. Smooth classifying spaces, joint with Enxin Wu. Israel Journal of Mathematics 241(2) (2021), 911-954. doi:10.1007/s11856-021-2120-6, arXiv:1709.10517, pdf file.

37. Localization in homotopy type theory, joint with Morgan Opie, Egbert Rijke and Luis Scoccola. Higher Structures 4(1) (2020), 1-32. doi:10.21136/HS.2020.01, arXiv:1807.04155, pdf file. The last two have a minor correction compared to the published version. See the footnote on the first page.

36. Diffeological vector spaces, joint with Enxin Wu. Pacific Journal of Mathematics 303(1) (2019), 73-92. doi:10.2140/pjm.2019.303.73, arXiv:1703.07564, pdf file.

35. Counterexamples to conjectures about Subset Takeaway and counting linear extensions of a Boolean lattice, joint with Andries E. Brouwer. Order 35(2) (2018), 275-281. doi:10.1007/s11083-017-9431-6, arXiv:1702.03018, pdf file.

34. Higher Toda brackets and the Adams spectra sequence in triangulated categories, joint with Martin Frankland. Algebraic & Geometric Topology 17(5) (2017), 2687-2735. doi:10.2140/agt.2017.17.2687, arXiv:1510.09216, pdf file.

33. Tangent spaces of bundles and of filtered diffeological spaces, joint with Enxin Wu. Proceedings of the AMS 145(5) (2017), 2255-2270. doi:10.1090/proc/13334, arXiv:1510.09182, pdf file.

32. Tangent spaces and tangent bundles for diffeological spaces, joint with Enxin Wu. Cahiers de Topologie et Geométrie Différentielle Catégoriques 57(1) (2016), 3-50. Published version, arXiv:1411.5425. The arXiv version has better typesetting.

31. The homotopy theory of diffeological spaces, joint with Enxin Wu. New York J. Math. 20 (2014), 1269-1303. Published version, arXiv:1311.6394.

30. Ghost numbers of group algebras II, joint with Gaohong Wang. Algebras and Representation Theory 18(3) (2015), 849-880. doi:10.1007/s10468-015-9519-x, arXiv:1310.5682.

29. The D-topology for diffeological spaces, joint with Gord Sinnamon and Enxin Wu. Pacific Journal of Mathematics 272(1) (2014), 87-110. doi:10.2140/pjm.2014.272.87, arXiv:1302.2935.

28. Ghost numbers of group algebras, joint with Gaohong Wang. Algebras and Representation Theory, 18(1) (2015), 1-33. doi:10.1007/s10468-014-9476-9, arXiv:1301.5740. The arXiv version has better typesetting.

27. Freyd's generating hypothesis for groups with periodic cohomology, joint with Sunil Chebolu and Ján Mináč. Canadian Mathematical Bulletin, 55(1) (2012), 48-59. CMB version (April 2012), doi:10.4153/CMB-2011-090-5, arXiv:0710.3356.

26. The sub-leading asymptotic behaviour of area correlations in the Barrett-Crane model, joint with Igor Khavkine, Etera Livine and Simone Speziale. Class. Quantum Grav. 27 (2010) 035012, 19 pages. doi:10.1088/0264-9381/27/3/035012, arXiv:0908.4476 [gr-qc], pdf file.

25. A Dual Non-abelian Yang-Mills Amplitude in Four Dimensions, joint with Wade Cherrington. Nuclear Physics B 813(3) (2009) 370-382. doi:10.1016/j.nuclphysb.2008.12.016, arXiv:0808.3624, pdf file.

24. Numerical evidence of regularized correlations in spin foam gravity, joint with Etera Livine and Simone Speziale. Physics Letters B 670(4-5) (2009), 403-406. doi:10.1016/j.physletb.2008.09.065, arXiv:0710.0617 [gr-qc], pdf file. Some plots are available here.

23. Eigentheory of Cayley-Dickson algebras, joint with Daniel Biss, Dan Dugger and Daniel Isaksen. Forum Mathematicum 21(5) (2009), 833-851. doi:10.1515/FORUM.2009.041, arXiv:0905.2987 [math.RA], pdf file.

22. Ghosts in modular representation theory, joint with Sunil Chebolu and Ján Mináč. Advances in Mathematics 217(6) (2008), 2782-2799. doi:10.1016/j.aim.2007.11.008, arXiv:math.RT/0609699.

21. Groups which do not admit ghosts, joint with Sunil Chebolu and Ján Mináč. Proceedings of the AMS 136 (2008), 1171-1179. doi:10.1090/S0002-9939-07-09058-2, arXiv:math.RT/0610423.

20. Dual computations of non-abelian Yang-Mills on the lattice, joint with Wade Cherrington and Igor Khavkine. Physics Review D 76 094503 (2007), 17 pages. doi:10.1103/PhysRevD.76.094503, arXiv:0705.2629 [hep-lat].

19. Large annihilators in Cayley-Dickson algebras II, joint with Daniel K. Biss, Daniel Dugger and Daniel C. Isaksen. Boletin de la Sociedad Matematica Mexicana (3) 13(2) (2007), 269-292. arXiv:math.RA/0702075, pdf file.

18. q-deformed spin foam models of quantum gravity, joint with Igor Khavkine. Classical and Quantum Gravity 24 (2007), 3271-3290. doi:10.1088/0264-9381/24/13/009, arXiv:0704.0278 [gr-qc].

17. The generating hypothesis for the stable module category of a p-group, joint with Dave Benson, Sunil Chebolu and Ján Mináč. J. Algebra 310(1) (2007), 428-433. doi:10.1016/j.jalgebra.2006.12.013, arXiv:math.RT/0611403.

16. Finiteness of Lorentzian 10j symbols and partitions functions. Classical and Quantum Gravity 23 (2006), 1679-1687. doi:10.1088/0264-9381/23/5/013, arXiv:gr-qc/0512004.

15. Positivity in Lorentzian Barrett-Crane Models of Quantum Gravity, joint with Wade Cherrington. Classical and Quantum Gravity 23 (2006), 721-736. doi:10.1088/0264-9381/23/3/011, arXiv:gr-qc/0509080.

14. Causal sites as quantum geometry, joint with Louis Crane. J. Math. Physics 46 (2005), 122502, 17 pages. doi:10.1063/1.2138043, arXiv:gr-qc/0410104.

13. Duality and pro-spectra, joint with Dan Isaksen. Algebraic & Geometric Topology 4 (2004), 781-812. doi:10.2140/agt.2004.4.781, arXiv:math.AT/0403451.

12. Obstruction theory in model categories, joint with Bill Dwyer and Dan Isaksen. Advances in Mathematics 181 (2004), 396-416. doi:10.1016/S0001-8708(03)00070-7, arXiv:math.AT/0109170, pdf file.

11. Asymptotics of 10j symbols, joint with John Baez and Greg Egan. Classical and Quantum Gravity 19 (2002), 6489-6513. doi:10.1088/0264-9381/19/24/315, arXiv:gr-qc/0208010, pdf file.

10. Spin foam models of Riemannian quantum gravity, joint with John Baez, Tom Halford and David Tsang. Classical and Quantum Gravity 19 (2002), 4627-4648. doi:10.1088/0264-9381/19/18/301, arXiv:gr-qc/0202017, pdf file.

9. Positivity of spin foam amplitudes, joint with John Baez. Classical and Quantum Gravity 19 (2002), 2291-2305. doi:10.1088/0264-9381/19/8/316, arXiv:gr-qc/0110044, pdf file.

8. An efficient algorithm for the Riemannian 10j symbols, joint with Greg Egan. Classical and Quantum Gravity 19 (2002), 1184-1193. doi:10.1088/0264-9381/19/6/310, arXiv:gr-qc/0110045, pdf file.

7. Quillen model structures for relative homological algebra, joint with Mark Hovey. Math. Proc. Cambridge Philos. Soc. 133 Part 2 (2002), 261-293. doi:10.1017/S0305004102006126, arXiv:math.KT/0011216, pdf file. The preprint versions linked to here contain corrections compared to the published version. See the last page for details.

6. Failure of Brown representability in derived categories, joint with Bernhard Keller and Amnon Neeman. Topology 40 (2001), 1339-1361. doi:10.1016/S0040-9383(00)00015-X, arXiv:math.AT/0001056, pdf file.

5. Phantom maps and chromatic phantom maps, joint with Mark Hovey. American Journal of Mathematics 122 (2000), 275-293. doi:10.1353/ajm.2000.0011, arXiv:math.AT/9811027, pdf file.

4. Ideals in triangulated categories: phantoms, ghosts and skeleta. Advances in Mathematics 136(2) (1998), 284-339. My thesis. doi:10.1006/aima.1998.1735, arXiv:math.AT/9807071, pdf file. The preprint versions linked to here contain corrections compared to the published version. See the last page for details.

3. Phantom Maps and Homology Theories, joint with Neil Strickland. Topology 37 (1998), 339-364. doi:10.1016/S0040-9383(97)00031-1, arXiv:math.AT/9802008, pdf file.

2. David Gale's subset take-away game, joint with Mark Tilford. American Mathematical Monthly 104 (1997), 762-766. doi:10.2307/2975243, arXiv:math.CO/0207185, pdf file. Subset take-away is the topic of Ian Stewart's Scientific American Mathematical Recreations column from March 2000. A longer version of the SciAm column is Triangulation Takeaway, which summarizes the results of my paper with Tilford.

1. The causal structure of two-dimensional spacetimes, joint with Robert B. Mann. Classical and Quantum Gravity 9 (1992), 1769-1786. doi:10.1088/0264-9381/9/7/010, arXiv:hep-th/9203050 (no figures), pdf file with figures at end.

Volumes edited:

2. Complex cobordism in homotopy theory: its impact and prospects (Baltimore 2007), J. Michael Boardman, J. Daniel Christensen, Donald M. Davis, Haynes R. Miller and Jack Morava, eds. Homology, Homotopy and Applications 10(3) (2008), 368+xix pp. International Press, Somerville, MA.

1. Interactions between homotopy theory and algebra (Chicago 2004), Luchezar L. Avramov, J. Daniel Christensen, William G. Dwyer, Michael A. Mandell and Brooke E. Shipley, eds. Contemporary Math. 436 (2007), 334pp. Amer. Math. Soc., Providence, RI. Conference home page, with links to preprints and notes based on the talks.