55. Book: Albert algebras over commutative rings
    with Holger P. Petersson and Michel L. Racine
    Appearing November 21, 2024 as Cambridge University Press New Mathematical Monographs vol. 48
    
    • The book at the publisher’s website or Amazon
    • Download a draft version (dated 20 June 2024)
      Theorems, lemmas, etc. are numbered the same in the draft and the published version, but page numbers are different.
    • Get solutions to the exercises from the arxiv (2406.02933)
    • Blog post describing the book
    If you want to read more about LG rings, check out this paper by Gille and Neher
    
    
56. Invariant derivations and trace bounds (arxiv) (code)
    with Robert M. Guralnick and Eric M. Rains
57. The luckiest strategy on earth? A better way to buy lottery scratch-off tickets (journal)
    College Mathematics Journal 25 (2024), 117-125
58. Illustrating the impact of the mathematical sciences
    with T. G. Kolda, R. E. Caflisch, I. Fonseca, M. Fuentes, T. Grandine, T. Jackson, X.-L. Meng, J. Pipher, T. Tao
    poster series with additional materials
    Released January 2023
59. Albert algebras over Z and other rings (journal)
    with Holger P. Petersson and Michel L. Racine
    Forum of Mathematics, Sigma 11 (2023), e18
    This paper refers to unpublished notes by George Seligman, “On the split exceptional Lie algebra E₇“, from 1962
60. Generic stabilizers for simple algebraic groups (arxiv) (journal)
    with Bob Guralnick
    Michigan Math. J. 72 (2022), 343-387.
61. Minuscule embeddings (arxiv) (journal)
    with Benedict Gross
    Indagationes Mathematicae 32 (2021), 987-1004.
62. A class of continuous non-associative algebras arising from algebraic groups including E₈ (journal) (arxiv) (code)
    with Maurice Chayet
    Forum of Mathematics, Sigma 9 (2021), e6
    Check out also this paper by De Medts and Van Couwenberghe for complementary results

63. Optimal play for multiple lottery wins (journal)
    with Richard Stong
    Electronic Journal of Combinatorics 27 (2020), P3.56.
64. Generically free representations I: large representations (journal) (preprint)
    with Robert Guralnick
    Algebra & Number Theory 14 (2020), 1577–1611
65. 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.

66. Generically free representations III: extremely bad characteristic (journal) (preprint)
    with Robert Guralnick
    Transformation Groups 25 (2020), 819-841
67. Appendix to the paper Quasi-isometric embeddings of non-uniform lattices (journal)(arxiv) by David Fisher and Thang Nguyen
    with D.B. McReynolds, Nicholas Miller, and Dave Witte Morris
    Commentarii Mathematici Helvetici 95 (2020), 37-78

2010s

29. 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.
30. 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
31. Rost invariant on the center, revisited (journal) (preprint)
    with Alexander Merkurjev
    Pacific Journal of Mathematics 291 #2 (2017), 369-397
32. Tits p-indexes of semisimple groups (journal) (preprint)
    with Charles De Clercq
    Journal of the London Mathematical Society 95 (2017), 567–585
33. 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
34. 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
    If you like this article, check out this later article by J.C. Jantzen
35. E₈, the most exceptional group
    Bulletin of the AMS 53 #4 (2016), 643-671
36. Outer automorphisms of algebraic groups and a Skolem-Noether theorem for Albert algebras
    with Holger P. Petersson
    Documenta Mathematica 21 (2016), 917-954
    For the exciting resolution to Conjecture 1.1.2, see the paper by Queguiner and Tignol (arxiv)(journal)
37. 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)
38. 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
    If you like this article, check out Mikko Korhonen’s 2018 article on irreducible representations (arxiv) (journal)
39. 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
40. Shells of twisted flag varieties and the Rost invariant (journal)
    with Victor Petrov and Nikita Semenov
    Duke Math. Journal 165 (2016), 285-339
41. 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
42. 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
43. Lottery spending: a non-parametric analysis
    with Kayla Frisoli, Li Ke, and Melody Lim
    PLoS ONE 10(2): e0115730.
44. The γ-filtration and the Rost invariant (preprint) (journal)
    with Kirill Zainoulline
    Journal für reine und angewandte Mathematik #696 (2014), 225-244
45. 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
46. 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
47. 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
48. Outer automorphisms of algebraic groups and determining groups by their maximal tori (journal)
    Formerly titled Weak commensurability and forms of Spin_4n and also something else starting with Outer automorphisms
    Michigan Mathematical Journal 61 #2 (2012), 227-237.
49. 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.
50. 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.
51. 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.
52. What is… a linear algebraic group?, Notices of the American Mathematical Society 57 #9, October 2010, 1125-1126.
53. 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.
54. 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)

2000s

5. 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
6. Degree 5 invariant of E₈ (preprint) (journal)
   with Nikita Semenov
   International Mathematics Research Notices “rnq018” (2010), no. 19, 3746-3762.
7. 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.
8. Vanishing of trace forms in low characteristics (preprint) (journal)
   with an appendix by Alexander Premet
   Algebra & Number Theory 3 #5 (2009), 543-566
9. Algebraic groups with few subgroups (preprint) (journal)
   with Philippe Gille
   Journal of the London Math. Society 80 (2009) 405-430
10. 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.
11. 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)
12. Pfister’s theorem for orthogonal involutions of degree 12 (journal)
    with Anne Quéguiner-Mathieu
    Proceedings of the American Mathematical Society 137 (2009), 1215-1222
13. 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)
14. Orthogonal representations of twisted forms of SL₂ (journal)
    Representation Theory 12 (2008), 435-446
    See MathOverflow for a direct proof of Theorem B
    See Braun-Nebe 2024 for analogous results in the case of a finite field
15. 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
16. 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]
17. Groups of outer type E₆ with trivial Tits algebras (journal)
    with Holger P. Petersson
    Transformation Groups 12 #3 (2007), 443-474.
18. Kneser-Tits for a rank 1 form of E₆ (after Veldkamp) (journal)
    Compositio Mathematica 143 #1 (2007), 191-200
19. Geometries, the principle of duality, and algebraic groups (journal)
    with Michael P. Carr
    Expositiones Mathematicae 24 #3 (2006) 195-234
20. 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
21. Unramified cohomology of classifying varieties for exceptional simply connected groups (journal)
    Transactions of the American Mathematical Society 358 #1 (2006), 359-371
22. The characteristic polynomial and determinant are not ad hoc constructions (journal)
    American Mathematical Monthly 111 #9 (2004), 761-778
23. 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)
24. The Rost invariant has trivial kernel for quasi-split groups of low rank (journal)
    Commentarii Mathematici Helvetici 76 #4 (2001), 684-711
25. 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
26. Clifford algebras of hyperbolic involutions (journal)
    Math. Zeit. 236 #2 (2001), 321-349
27. Groups of type E₇ over arbitrary fields (journal)
    Comm. Algebra 29 #6 (2001), 2689-2710
28. Structurable algebras and groups of type E₆ and E₇ (journal)
    J. Algebra 236 #2 (2001), 651-691

1990s

1. Twisted flag varieties of trialitarian groups (journal)
   Comm. Algebra 27 #2 (1999), 841-856
2. 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.
3. Isotropic trialitarian algebraic groups (journal)
   J. Algebra 210 #2 (1998), 385-418
4. Galois cohomology of special orthogonal groups (journal)
   with Jean-Pierre Tignol and Adrian Wadsworth
   manuscripta mathematica 93 #2 (1997), 247-266

• The Defense Science Study Group: another way to help the nation
  Notices of the AMS 71, August 2024, 952-953
  with Mark Taylor
• Poster series Illustrating the Impact of the Mathematical Sciences
  The production of this series was organized by the National Academies’ Board on Mathematical Sciences and Analytics
• From the Early Career column in the Notices of the AMS:
  • Supporting faculty in mentoring students for careers beyond academia, Notices of the AMS 70, October 2023, 1442-1446
    with Lee DeVille, Tegan Emerson, Mary Lynn Reed, Talitha M. Washington and Suzanne L. Weekes
  • How to proceed in mathematics, Notices of the AMS, May 2021, 748-749
    with Dan Gordon
  • Finding your reward, Notices of the AMS, January 2020, 57-58.
• Pictures of compact Lie groups (after Serre)
  Tobias Schwaiger has made pictures for the three rank 3 cases in GeoGebra, which you can see here: A₃, B₃, C₃
• The Rost invariant has zero kernel for quasi-split trialitarian groups
  The Book of Involutions essentially includes a proof of the theorem in the title of this note, but some mechanical details are omitted. This note (not for publication) writes down those details.
• Undergraduate textbook: Optimization in microeconomics
  with Christopher Curran
  Published by University Readers, ISBN-13: 978-1-60927-735-2
  Revised edition published by Cognella (2016), ISBN-13: 978-1-5165-0551-7
• GAP program to calculate a sorted list of dimensions of irreducible dimensions of a simple complex Lie algebra. This is the program used to produce sequences A104599 (G₂), A121214 (A₃), A121732 (E₈), A121736 (E₇), A121737 (E₆), A121738 (F₄), and A121739 (D₄) in the Online Encyclopedia of Integer Sequences
• My Erdős number is 3, due to my paper with Aaron Abrams.