2020s
 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
 Invariant derivations and trace bounds (arxiv) (code)
with Robert M. Guralnick and Eric M. Rains  The luckiest strategy on earth? A better way to buy lottery scratchoff tickets (journal)
College Mathematics Journal 25 (2024), 117125  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  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This paper refers to unpublished notes by George Seligman, “On the split exceptional Lie algebra E_{7}“, from 1962  Generic stabilizers for simple algebraic groups (arxiv) (journal)
with Bob Guralnick
Michigan Math. J. 72 (2022), 343387.  Minuscule embeddings (arxiv) (journal)
with Benedict Gross
Indagationes Mathematicae 32 (2021), 9871004.  A class of continuous nonassociative algebras arising from algebraic groups including E_{8} (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  Optimal play for multiple lottery wins (journal)
with Richard Stong
Electronic Journal of Combinatorics 27 (2020), P3.56.  Generically free representations I: large representations (journal) (preprint)
with Robert Guralnick
Algebra & Number Theory 14 (2020), 1577–1611  Generically free representations II: irreducible representations (journal) (preprint)
with Robert Guralnick
Transformation Groups 25 (2020), 793817
This and the next paper involve some computer calculations. You can download the code and its output from GitHub.
 Generically free representations III: extremely bad characteristic (journal) (preprint)
with Robert Guralnick
Transformation Groups 25 (2020), 819841  Appendix to the paper Quasiisometric embeddings of nonuniform lattices (journal)(arxiv) by David Fisher and Thang Nguyen
with D.B. McReynolds, Nicholas Miller, and Dave Witte Morris
Commentarii Mathematici Helvetici 95 (2020), 3778
2010s
 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), 313326.  The van den BergKestenReimer operator and inequality for infinite spaces (journal) (preprint)
with Richard Arratia and Alfred W. Hales
Bernoulli 24 (2018), 433448  Rost invariant on the center, revisited (journal) (preprint)
with Alexander Merkurjev
Pacific Journal of Mathematics 291 #2 (2017), 369397  Tits pindexes of semisimple groups (journal) (preprint)
with Charles De Clercq
Journal of the London Mathematical Society 95 (2017), 567–585  Spinors and essential dimension (journal) (preprint)
with Robert Guralnick and an appendix by Alexander Premet
Compositio Mathematica 153 (2017), 535556
The essential dimension of spin groups is in OEIS as Sequence A280191  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), 6987
If you like this article, check out this later article by J.C. Jantzen  E_{8}, the most exceptional group
Bulletin of the AMS 53 #4 (2016), 643671  Outer automorphisms of algebraic groups and a SkolemNoether theorem for Albert algebras
with Holger P. Petersson
Documenta Mathematica 21 (2016), 917954
For the exciting resolution to Conjecture 1.1.2, see the paper by Queguiner and Tignol (arxiv)(journal)  Essential dimension of algebraic groups, including bad characteristic
with Bob Guralnick
Archiv der Mathematik 107 (2016), 101119
Selected as a Springer Basel Distinguished Article (and therefore free open access for one year)  Bilinear and quadratic forms on rational modules of split reductive groups (journal)
with Daniel K. Nakano
Canadian Journal of Mathematics 68 #2 (2016) 395421
If you like this article, check out Mikko Korhonen’s 2018 article on irreducible representations (arxiv) (journal)  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), 480502  Shells of twisted flag varieties and the Rost invariant (journal)
with Victor Petrov and Nikita Semenov
Duke Math. Journal 165 (2016), 285339  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  Some people have all the luck (journal)
with Richard Arratia, Lawrence Mower, and Philip B. Stark
Mathematics Magazine 88 (2015), 196211.
My lottery page has links to media coverage of the consequences of our theorems  Lottery spending: a nonparametric analysis
with Kayla Frisoli, Li Ke, and Melody Lim
PLoS ONE 10(2): e0115730.  The γfiltration and the Rost invariant (preprint) (journal)
with Kirill Zainoulline
Journal für reine und angewandte Mathematik #696 (2014), 225244  Linear preservers and representations with 1dimensional ring of invariants (preprint) (journal)
with Hernando Bermudez and Victor Larsen
Transactions of the AMS 366 #9 (2014), 47554780  Weakly commensurable Sarithmetic subgroups in almost simple algebraic groups of types B and C (journal)
with Andrei S. Rapinchuk
Algebra & Number Theory 7 #5 (2013) 11471178  Exceptional collections of line bundles on projective homogeneous varieties (journal)
with Alexey Ananyevskiy, Asher Auel, and Kirill Zainoulline
Advances in Mathematics 236 (2013) 111130  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), 227237.  Did a 1dimensional magnet detect a 248dimensional 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, 10551066.  Open problems on central simple algebras (journal)
with Asher Auel, Eric Brussel, and Uzi Vishne
Transformation Groups 16 #1 (2011), 219264.
There is also a slightly older arxiv version
This paper grew out of the conference “The Brauer group in Israel”, held at Kibbutz Ketura, 311 January 2010.  Wild Pfister forms over Henselian fields, Ktheory, and conic division algebras (preprint) (journal)
with Holger P. Petersson
Journal of Algebra 327 #1 (2011), 386465.  What is… a linear algebraic group?, Notices of the American Mathematical Society 57 #9, October 2010, 11251126.
 Quaternion algebras with the same subfields (preprint) (journal)
with David J. Saltman
in the book Quadratic forms, linear algebraic groups, and cohomology below, pp. 225238.  Edited volume: Quadratic forms, linear algebraic groups, and cohomology
edited with J.L. ColliotThélène, R. Sujatha, and V. Suresh
Developments in Mathematics, vol. 18, Springer, 2010, ISBN 9781441962102
Buy: (Springer) (Amazon)
2000s
 There is no “Theory of Everything” inside E_{8} (preprint) (journal)
with Jacques Distler
An extended abstract appeared in Oberwolfach Reports 6 #2 (2009), 14121413 (journal)
You can see the Ars Technica article about this paper
For some context, see Section 4 of the paper String and Mtheory: answering the critics
Communications in Mathematical Physics 298 (2010), 419436  Degree 5 invariant of E_{8} (preprint) (journal)
with Nikita Semenov
International Mathematics Research Notices “rnq018” (2010), no. 19, 37463762.  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.  Vanishing of trace forms in low characteristics (preprint) (journal)
with an appendix by Alexander Premet
Algebra & Number Theory 3 #5 (2009), 543566  Algebraic groups with few subgroups (preprint) (journal)
with Philippe Gille
Journal of the London Math. Society 80 (2009) 405430  Orthogonal involutions on algebras of degree 16 and the Killing form of E_{8} (preprint)
with an appendix by Kirill Zainoulline
in “Quadratic forms–algebra, arithmetic, and geometry” (R. Baeza, W.K. Chan, D.W. Hoffmann, and R. SchulzePillot, eds.), Contemporary Mathematics 493 (2009), 131162.  Cohomological invariants: exceptional groups and spin groups (notes for the Lens minicours)
with an appendix by Detlev W. Hoffmann
Memoirs of the American Mathematical Society 200 (2009), no. 937
ISBN10: 0821844040 / ISBN13: 9780821844045
Buy: (AMS) (Amazon)  Pfister’s theorem for orthogonal involutions of degree 12 (journal)
with Anne QuéguinerMathieu
Proceedings of the American Mathematical Society 137 (2009), 12151222  Discriminant of symplectic involutions (journal)
with R. Parimala and JeanPierre Tignol
Pure & Applied Mathematics Quarterly 5 #1 (2009), 349374 (Serre special issue, peerreviewed)  Orthogonal representations of twisted forms of SL_{2} (journal)
Representation Theory 12 (2008), 435446
See MathOverflow for a direct proof of Theorem B
See BraunNebe 2024 for analogous results in the case of a finite field  Somewhat more than governors need to know about trigonometry (journal)
Mathematics Magazine 81 (2008) #3, 191200.This was approximately the content of my talk at the 2006 MAA State Dinner for Georgia at Mercer University  Restricting the Rost invariant to the center (journal)
with Anne QuéguinerMathieu
St. Petersburg Math. J. 19 (2008) #2, 197213
[Original: Algebra i Analiz 19 (2007) #2, 5273]  Groups of outer type E_{6} with trivial Tits algebras (journal)
with Holger P. Petersson
Transformation Groups 12 #3 (2007), 443474.  KneserTits for a rank 1 form of E_{6} (after Veldkamp) (journal)
Compositio Mathematica 143 #1 (2007), 191200  Geometries, the principle of duality, and algebraic groups (journal)
with Michael P. Carr
Expositiones Mathematicae 24 #3 (2006) 195234  Totaro’s question on zerocycles on G_{2}, F_{4}, and E_{6} torsors (journal)
with Detlev W. Hoffmann
Journal of the London Mathematical Society 73 #2 (2006) 325338  Unramified cohomology of classifying varieties for exceptional simply connected groups (journal)
Transactions of the American Mathematical Society 358 #1 (2006), 359371  The characteristic polynomial and determinant are not ad hoc constructions (journal)
American Mathematical Monthly 111 #9 (2004), 761778  Book: Cohomological invariants in Galois cohomology
with Alexander Merkurjev and JP. Serre
American Mathematical Society University Lecture Series, volume 28, 2003
ISBN: 0821832875
Reviews: (Bull. AMS) (MR 2004f:11034)
Buy: (AMS) (Amazon)  The Rost invariant has trivial kernel for quasisplit groups of low rank (journal)
Commentarii Mathematici Helvetici 76 #4 (2001), 684711  Involutions and trace forms on exterior powers of a central simple algebra (journal)
with Anne QuéguinerMathieu and JeanPierre Tignol
Documenta Mathematica 6 (2001), 97118  Clifford algebras of hyperbolic involutions (journal)
Math. Zeit. 236 #2 (2001), 321349  Groups of type E_{7} over arbitrary fields (journal)
Comm. Algebra 29 #6 (2001), 26892710  Structurable algebras and groups of type E_{6} and E_{7} (journal)
J. Algebra 236 #2 (2001), 651691
1990s
 Twisted flag varieties of trialitarian groups (journal)
Comm. Algebra 27 #2 (1999), 841856  Trialitarian groups and the Hasse principle
manuscripta mathematica 98 #1 (1999), 97113
Erratum, 113 #2 (2004), p.267
Go to the published version of the article or the erratum.  Isotropic trialitarian algebraic groups (journal)
J. Algebra 210 #2 (1998), 385418  Galois cohomology of special orthogonal groups (journal)
with JeanPierre Tignol and Adrian Wadsworth
manuscripta mathematica 93 #2 (1997), 247266
Miscellaneous items
 The Defense Science Study Group: another way to help the nation
Notices of the AMS 71, August 2024, 952953
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, 14421446
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, 748749
with Dan Gordon  Finding your reward, Notices of the AMS, January 2020, 5758.
 Supporting faculty in mentoring students for careers beyond academia, Notices of the AMS 70, October 2023, 14421446
 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: A_{3}, B_{3}, C_{3}  The Rost invariant has zero kernel for quasisplit 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, ISBN13: 9781609277352
Revised edition published by Cognella (2016), ISBN13: 9781516505517  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 (G_{2}), A121214 (A_{3}), A121732 (E_{8}), A121736 (E_{7}), A121737 (E_{6}), A121738 (F_{4}), and A121739 (D_{4}) in the Online Encyclopedia of Integer Sequences
 My Erdős number is 3, due to my paper with Aaron Abrams.