### 2020s

Book: **Albert algebras over commutative rings**

with Holger P. Petersson and Michel L. Racine

Incomplete draft 14 November 2023

Solutions to exercises*We want to hear your comments!**The luckiest strategy on earth? A better way to buy lottery scratch-off tickets*(preprint) (journal)

To appear in*College Mathematics Journal*- 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*(journal)**Z**and other rings

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This paper refers to unpublished notes by George Seligman, “On the split exceptional Lie algebra E_{7}“, from 1962*Generic stabilizers for simple algebraic groups*(arxiv) (journal)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), 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 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), 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 - E
_{8}, 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 Spin*and also something else starting with_{4n}*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 E*(preprint) (journal)_{8}

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 E*(preprint) (journal)_{8}

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 E*(preprint)_{8}

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 SL*(journal)_{2}*Representation Theory***12**(2008), 435-446

See MathOverflow for a direct proof of Theorem B*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 E*(journal)_{6}with trivial Tits algebras

with Holger P. Petersson*Transformation Groups***12**#3 (2007), 443-474.*Kneser-Tits for a rank 1 form of E*(journal)_{6}(after Veldkamp)*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 G*, and_{2}, F_{4}*E*torsors (journal)_{6}

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 E*(journal)_{7}over arbitrary fields*Comm. Algebra***29**#6 (2001), 2689-2710*Structurable algebras and groups of type E*(journal)_{6}and E_{7}*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

- 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,
- 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 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 (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.