Papers

Preprints

This link gives a list of my papers on the arXiv.

  • A class of continuous non-associative algebras arising from algebraic groups including E8 (arxiv) (code)
    with Maurice Chayet

To appear

  • Minuscule embeddings (arxiv)
    with Benedict Gross
    to appear in Indagationes Mathematicae

Publications: papers & books

  1. Optimal play for multiple lottery wins (journal)
    with Richard Stong
    Electronic Journal of Combinatorics 27 (2020), P3.56.
  2. Generically free representations I: large representations (journal) (preprint)
    with Robert Guralnick
    Algebra & Number Theory 14 (2020), 1577–1611
  3. Generically free representations II: irreducible representations (journal) (preprint)
    with Robert Guralnick
    Transformation Groups 25 (2020), 793-817

    This and the next paper involve some computer calculations. You can download the code and its output from GitHub.

  4. Generically free representations III: extremely bad characteristic (journal) (preprint)
    with Robert Guralnick
    Transformation Groups 25 (2020), 819-841
  5. Globally irreducible Weyl modules for quantum groups (published) (preprint)
    with Robert Guralnick and Daniel Nakano
    in Geometric and topological aspects of group representations, edited by J.F. Carlson et al
    Springer Proceedings in Mathematics & Statistics 242 (2018), 313-326.
  6. The van den Berg-Kesten-Reimer operator and inequality for infinite spaces (journal) (preprint)
    with Richard Arratia and Alfred W. Hales
    Bernoulli 24 (2018), 433-448
  7. Rost invariant on the center, revisited (journal) (preprint)
    with Alexander Merkurjev
    Pacific Journal of Mathematics 291 #2 (2017), 369-3976
  8. Tits p-indexes of semisimple groups (journal) (preprint)
    with Charles De Clercq
    Journal of the London Mathematical Society 95 (2017), 567–585
  9. Spinors and essential dimension (journal) (preprint)
    with Robert Guralnick and an appendix by Alexander Premet
    Compositio Mathematica 153 (2017), 535-556
    The essential dimension of spin groups is in OEIS as Sequence A280191
  10. Globally irreducible Weyl modules
    formerly titled Irreducibility of Weyl modules over fields of arbitrary characteristic
    with Robert Guralnick and Daniel Nakano
    Journal of Algebra 477 (2017), 69-87
  11. E8, the most exceptional group
    Bulletin of the AMS 53 #4 (2016), 643-671
  12. Outer automorphisms of algebraic groups and a Skolem-Noether theorem for Albert algebras
    with Holger P. Petersson
    Documenta Mathematica 21 (2016), 917-954
  13. Essential dimension of algebraic groups, including bad characteristic
    with Bob Guralnick
    Archiv der Mathematik 107 (2016), 101-119
    Selected as a Springer Basel Distinguished Article (and therefore free open access for one year)
  14. Bilinear and quadratic forms on rational modules of split reductive groups (journal)
    with Daniel K. Nakano
    Canadian Journal of Mathematics 68 #2 (2016) 395-421
  15. Asymptotic distribution for the birthday problem with multiple coincidences, via an embedding of the collision process (journal)
    with Richard Arratia and Joe Kilian
    Random Structures & Algorithms 48 #3 (2016), 480-502
  16. Shells of twisted flag varieties and the Rost invariant (journal)
    with Victor Petrov and Nikita Semenov
    Duke Math. Journal 165 (2016), 285-339
  17. Simple groups stabilizing polynomials (journal)
    with Bob Guralnick
    Forum of Mathematics Pi 3 (2015) e3 (41 pages)
    This paper was the main subject of my plenary address at the fall 2014 AMS southeast meeting
    For further results in this direction, see Classifying forms of simple groups via their invariant polynomials by H. Bermudez and A. Ruozzi
  18. Some people have all the luck (journal)
    with Richard Arratia, Lawrence Mower, and Philip B. Stark
    Mathematics Magazine 88 (2015), 196-211.
    My lottery page has links to media coverage of the consequences of our theorems
  19. Lottery spending: a non-parametric analysis
    with Kayla Frisoli, Li Ke, and Melody Lim
    PLoS ONE 10(2): e0115730.
  20. The γ-filtration and the Rost invariant (preprint) (journal)
    with Kirill Zainoulline
    Journal für reine und angewandte Mathematik #696 (2014), 225-244
  21. Linear preservers and representations with 1-dimensional ring of invariants (preprint) (journal)
    with Hernando Bermudez and Victor Larsen
    Transactions of the AMS 366 #9 (2014), 4755-4780
  22. Weakly commensurable S-arithmetic subgroups in almost simple algebraic groups of types B and C (journal)
    with Andrei S. Rapinchuk
    Algebra & Number Theory 7 #5 (2013) 1147-1178
  23. Exceptional collections of line bundles on projective homogeneous varieties (journal)
    with Alexey Ananyevskiy, Asher Auel, and Kirill Zainoulline
    Advances in Mathematics 236 (2013) 111-130
  24. Outer automorphisms of algebraic groups and determining groups by their maximal tori (journal)
    Formerly titled Weak commensurability and forms of Spin4n and also something else starting with Outer automorphisms
    Michigan Mathematical Journal 61 #2 (2012), 227-237.
  25. Did a 1-dimensional magnet detect a 248-dimensional Lie algebra?
    with David Borthwick
    The cover picture on the right was motivated by our paper
    Our paper was partially motivated by a discussion on MathOverflow
    Notices of the American Mathematical Society 58 #8, September 2011, 1055-1066.
  26. Open problems on central simple algebras (journal)
    with Asher Auel, Eric Brussel, and Uzi Vishne
    Transformation Groups 16 #1 (2011), 219-264.
    There is also a slightly older arxiv version
    This paper grew out of the conference “The Brauer group in Israel”, held at Kibbutz Ketura, 3-11 January 2010.
  27. Wild Pfister forms over Henselian fields, K-theory, and conic division algebras (preprint) (journal)
    with Holger P. Petersson
    Journal of Algebra 327 #1 (2011), 386-465.
  28. What is… a linear algebraic group?, Notices of the American Mathematical Society 57 #9, October 2010, 1125-1126.
  29. Quaternion algebras with the same subfields (preprint) (journal)
    with David J. Saltman
    in the book Quadratic forms, linear algebraic groups, and cohomology below, pp. 225-238.
  30. Edited volume: Quadratic forms, linear algebraic groups, and cohomology
    edited with J.-L. Colliot-Thélène, R. Sujatha, and V. Suresh
    Developments in Mathematics, vol. 18, Springer, 2010, ISBN 978-1-4419-6210-2
    Buy: (Springer) (Amazon)
  31. There is no “Theory of Everything” inside E8 (preprint) (journal)
    with Jacques Distler
    An extended abstract appeared in Oberwolfach Reports 6 #2 (2009), 1412-1413 (journal)

    You can see the Ars Technica article about this paper
    For some context, see Section 4 of the paper String and M-theory: answering the critics
    Communications in Mathematical Physics 298 (2010), 419-436
  32. Degree 5 invariant of E8 (preprint) (journal)
    with Nikita Semenov
    International Mathematics Research Notices “rnq018” (2010), no. 19, 3746-3762.
  33. Finding good bets in the lottery, and why you shouldn’t take them
    with Aaron Abrams
    American Mathematical Monthly 117 #1 (2010), 3–26
    This paper and other lottery links are available here.
  34. Vanishing of trace forms in low characteristics (preprint) (journal)
    with an appendix by Alexander Premet
    Algebra & Number Theory 3 #5 (2009), 543-566
  35. Algebraic groups with few subgroups (preprint) (journal)
    with Philippe Gille
    Journal of the London Math. Society 80 (2009) 405-430
  36. Orthogonal involutions on algebras of degree 16 and the Killing form of E8 (preprint)
    with an appendix by Kirill Zainoulline
    in “Quadratic forms–algebra, arithmetic, and geometry” (R. Baeza, W.K. Chan, D.W. Hoffmann, and R. Schulze-Pillot, eds.), Contemporary Mathematics 493 (2009), 131-162.
  37. Cohomological invariants: exceptional groups and spin groups (notes for the Lens mini-cours)
    with an appendix by Detlev W. Hoffmann
    Memoirs of the American Mathematical Society 200 (2009), no. 937
    ISBN-10: 0-8218-4404-0 / ISBN-13: 978-0-8218-4404-5
    Buy: (AMS) (Amazon)
  38. Pfister’s theorem for orthogonal involutions of degree 12 (journal)
    with Anne Quéguiner-Mathieu
    Proceedings of the American Mathematical Society 137 (2009), 1215-1222
  39. Discriminant of symplectic involutions (journal)
    with R. Parimala and Jean-Pierre Tignol
    Pure & Applied Mathematics Quarterly 5 #1 (2009), 349-374 (Serre special issue, peer-reviewed)
  40. Orthogonal representations of twisted forms of SL2 (journal)
    Representation Theory 12 (2008), 435-446
  41. Somewhat more than governors need to know about trigonometry (journal)
    Mathematics Magazine 81 (2008) #3, 191-200.This was approximately the content of my talk at the 2006 MAA State Dinner for Georgia at Mercer University
  42. Restricting the Rost invariant to the center (journal)
    with Anne Quéguiner-Mathieu
    St. Petersburg Math. J. 19 (2008) #2, 197-213
    [Original: Algebra i Analiz 19 (2007) #2, 52-73]
  43. Groups of outer type E6 with trivial Tits algebras (journal)
    with Holger P. Petersson
    Transformation Groups 12 #3 (2007), 443-474.
  44. Kneser-Tits for a rank 1 form of E6 (after Veldkamp) (journal)
    Compositio Mathematica 143 #1 (2007), 191-200
  45. Geometries, the principle of duality, and algebraic groups (journal)
    with Michael P. Carr
    Expositiones Mathematicae 24 #3 (2006) 195-234
  46. Totaro’s question on zero-cycles on G2, F4, and E6 torsors (journal)
    with Detlev W. Hoffmann
    Journal of the London Mathematical Society 73 #2 (2006) 325-338
  47. Unramified cohomology of classifying varieties for exceptional simply connected groups (journal)
    Transactions of the American Mathematical Society 358 #1 (2006), 359-371
  48. The characteristic polynomial and determinant are not ad hoc constructions (journal)
    American Mathematical Monthly 111 #9 (2004), 761-778
  49. Book: Cohomological invariants in Galois cohomology
    with Alexander Merkurjev and J-P. Serre
    American Mathematical Society University Lecture Series, volume 28, 2003
    ISBN: 0-8218-3287-5
    Reviews: (Bull. AMS) (MR 2004f:11034)
    Buy: (AMS) (Amazon)
  50. The Rost invariant has trivial kernel for quasi-split groups of low rank (journal)
    Commentarii Mathematici Helvetici 76 #4 (2001), 684-711
  51. Involutions and trace forms on exterior powers of a central simple algebra (journal)
    with Anne Quéguiner-Mathieu and Jean-Pierre Tignol
    Documenta Mathematica 6 (2001), 97-118
  52. Clifford algebras of hyperbolic involutions (journal)
    Math. Zeit. 236 #2 (2001), 321-349
  53. Groups of type E7 over arbitrary fields (journal)
    Comm. Algebra 29 #6 (2001), 2689-2710
  54. Structurable algebras and groups of type E6 and E7 (journal)
    J. Algebra 236 #2 (2001), 651-691
  55. Twisted flag varieties of trialitarian groups (journal)
    Comm. Algebra 27 #2 (1999), 841-856
  56. Trialitarian groups and the Hasse principle
    manuscripta mathematica 98 #1 (1999), 97-113
    Erratum, 113 #2 (2004), p.267
    Go to the published version of the article or the erratum.
  57. Isotropic trialitarian algebraic groups (journal)
    J. Algebra 210 #2 (1998), 385-418
  58. Galois cohomology of special orthogonal groups (journal)
    with Jean-Pierre Tignol and Adrian Wadsworth
    manuscripta mathematica 93 #2 (1997), 247-266

Miscellaneous items