2020s
Book: Albert algebras over commutative rings
with Holger P. Petersson and Michel L. Racine
Cambridge University Press New Mathematical Monographs vol. 48 (2024)
- 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)
- Current list of errata
- 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 scratch-off tickets (journal)
College Mathematics Journal 25 (2024), 117-125 - 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This paper refers to unpublished notes by George Seligman, “On the split exceptional Lie algebra E7“, from 1962 - Generic stabilizers for simple algebraic groups (arxiv) (journal)
with Bob Guralnick
Michigan Math. J. 72 (2022), 343-387. - Minuscule embeddings (arxiv) (journal)
with Benedict Gross
Indagationes Mathematicae 32 (2021), 987-1004. - 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 - 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), 793-817
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), 819-841 - 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
- 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. - 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 - Rost invariant on the center, revisited (journal) (preprint)
with Alexander Merkurjev
Pacific Journal of Mathematics 291 #2 (2017), 369-397 - Tits p-indexes 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), 535-556
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), 69-87
If you like this article, check out this later article by J.C. Jantzen - E8, the most exceptional group
Bulletin of the AMS 53 #4 (2016), 643-671 - 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) - 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) - 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) - 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 - Shells of twisted flag varieties and the Rost invariant (journal)
with Victor Petrov and Nikita Semenov
Duke Math. Journal 165 (2016), 285-339 - 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), 196-211.
My lottery page has links to media coverage of the consequences of our theorems - Lottery spending: a non-parametric analysis
with Kayla Frisoli, Li Ke, and Melody Lim
PLoS ONE 10(2): e0115730. - The γ-filtration and the Rost invariant (preprint) (journal)
with Kirill Zainoulline
Journal für reine und angewandte Mathematik #696 (2014), 225-244 - 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 - 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 - 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 - 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. - 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. - 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. - 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. - What is… a linear algebraic group?, Notices of the American Mathematical Society 57 #9, October 2010, 1125-1126.
- 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. - 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
- 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 - Degree 5 invariant of E8 (preprint) (journal)
with Nikita Semenov
International Mathematics Research Notices “rnq018” (2010), no. 19, 3746-3762. - 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), 543-566 - Algebraic groups with few subgroups (preprint) (journal)
with Philippe Gille
Journal of the London Math. Society 80 (2009) 405-430 - 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. - 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) - Pfister’s theorem for orthogonal involutions of degree 12 (journal)
with Anne Quéguiner-Mathieu
Proceedings of the American Mathematical Society 137 (2009), 1215-1222 - 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) - 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 - 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 - 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] - Groups of outer type E6 with trivial Tits algebras (journal)
with Holger P. Petersson
Transformation Groups 12 #3 (2007), 443-474. - Kneser-Tits for a rank 1 form of E6 (after Veldkamp) (journal)
Compositio Mathematica 143 #1 (2007), 191-200 - Geometries, the principle of duality, and algebraic groups (journal)
with Michael P. Carr
Expositiones Mathematicae 24 #3 (2006) 195-234 - 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 - Unramified cohomology of classifying varieties for exceptional simply connected groups (journal)
Transactions of the American Mathematical Society 358 #1 (2006), 359-371 - The characteristic polynomial and determinant are not ad hoc constructions (journal)
American Mathematical Monthly 111 #9 (2004), 761-778 - 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) - The Rost invariant has trivial kernel for quasi-split groups of low rank (journal)
Commentarii Mathematici Helvetici 76 #4 (2001), 684-711 - 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 - Clifford algebras of hyperbolic involutions (journal)
Math. Zeit. 236 #2 (2001), 321-349 - Groups of type E7 over arbitrary fields (journal)
Comm. Algebra 29 #6 (2001), 2689-2710 - Structurable algebras and groups of type E6 and E7 (journal)
J. Algebra 236 #2 (2001), 651-691
1990s
- Twisted flag varieties of trialitarian groups (journal)
Comm. Algebra 27 #2 (1999), 841-856 - 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. - Isotropic trialitarian algebraic groups (journal)
J. Algebra 210 #2 (1998), 385-418 - Galois cohomology of special orthogonal groups (journal)
with Jean-Pierre Tignol and Adrian Wadsworth
manuscripta mathematica 93 #2 (1997), 247-266
Miscellaneous items
- 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.
- Supporting faculty in mentoring students for careers beyond academia, Notices of the AMS 70, October 2023, 1442-1446
- 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.