Are you looking for information about the lottery or E₈?

Other work

For a description of my research, see my research profile.

Preprints (intended for publication)

• Shells of twisted flag varieties and non-decomposability of the Rost invariant (October 2012)
  with Victor Petrov and Nikita Semenov
  There is also an older version on the arxiv

To appear

• Linear preservers and representations with 1-dimensional ring of invariants (20 October 2012)
  with Hernando Bermudez and Victor Larsen
  There is also an older version on the arxiv
  to appear in Transactions of the AMS

• The gamma-filtration and the Rost invariant (28 September 2012) (published version)
  with Kirill Zainoulline
  There is also an older version on the arxiv
  to appear in Journal für reine und angewandte Mathematik

• Weakly commensurable S-arithmetic subgroups in almost simple algebraic groups of types B and C (7 May 2012)
  with Andrei S. Rapinchuk
  arxiv:1205.1458
  to appear in Algebra & Number Theory

Publications (MathSciNet) (Google Scholar)

1. Exceptional collections of line bundles on projective homogeneous varieties (published version)
   with Alexey Ananyevskiy, Asher Auel, and Kirill Zainoulline
   There is also an older version on the arxiv
   Advances in Mathematics 236 (2013) 111-130

2. Outer automorphisms of algebraic groups and determining groups by their maximal tori (published version)
   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.

3. Did a 1-dimensional magnet detect a 248-dimensional Lie algebra? with David Borthwick The cover picture on the left 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.

4. Open problems on central simple algebras (published version)
   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.

5. Wild Pfister forms over Henselian fields, K-theory, and conic division algebras (preprint version) (published version)
   with Holger P. Petersson
   Journal of Algebra 327 #1 (2011), 386-465.

6. What is... a linear algebraic group?
   Notices of the American Mathematical Society 57 #9, October 2010, 1125-1126.

7. Quaternion algebras with the same subfields (preprint version) (published version)
   with David J. Saltman
   in the book Quadratic forms, linear algebraic groups, and cohomology below, pp. 225-238.

8. 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)

9. There is no "Theory of Everything" inside E₈ (preprint version) (published version)
   with Jacques Distler
   An extended abstract appeared in Oberwolfach Reports 6 #2 (2009), 1412-1413 (preliminary version) (published version)
   You can see my slides from the Banff conference in July 2010
   or 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

10. Degree 5 invariant of E₈ (preprint version) (published version)
    with Nikita Semenov
    International Mathematics Research Notices "rnq018" (2010), no. 19, 3746-3762.

11. 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 at this link.

12. Vanishing of trace forms in low characteristics (preprint version) (published version)
    with an appendix by Alexander Premet
    Algebra & Number Theory 3 #5 (2009), 543-566

13. Algebraic groups with few subgroups (preprint version) (published version)
    with Philippe Gille
    Journal of the London Math. Society 80 (2009) 405-430

14. Orthogonal involutions on algebras of degree 16 and the Killing form of E₈ (preprint version)
    with an appendix by Kirill Zainoulline
    Slides from my talk at the Chile conference in December 07
    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.

15. 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)

16. Pfister's theorem for orthogonal involutions of degree 12 (preprint version) (published version)
    with Anne Quéguiner-Mathieu
    Proceedings of the American Mathematical Society 137 (2009), 1215-1222

17. Discriminant of symplectic involutions (published version)
    with R. Parimala and Jean-Pierre Tignol
    Pure & Applied Mathematics Quarterly 5 #1 (2009), 349-374 (Serre special issue, peer-reviewed)

18. Orthogonal representations of twisted forms of SL₂ (preprint version) (published version)
    Representation Theory 12 (2008), 435-446
    

19. Somewhat more than governors need to know about trigonometry (published version)
    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

20. Restricting the Rost invariant to the center (preprint version) (published version)
    with Anne Quéguiner-Mathieu
    St. Petersburg Math. J. 19 (2008) #2, 197-213 (MR 2333896)
    [Original: Algebra i Analiz 19 (2007) #2, 52-73]

21. Groups of outer type E₆ with trivial Tits algebras (preprint version) (published version)
    with Holger P. Petersson
    Transformation Groups 12 #3 (2007), 443-474.

22. Kneser-Tits for a rank 1 form of E₆ (after Veldkamp) (preprint version) (published version)
    Compositio Mathematica 143 #1 (2007), 191-200 (MR 2295201)

23. Geometries, the principle of duality, and algebraic groups (preprint version) (published version)
    with Michael P. Carr
    Expositiones Mathematicae 24 #3 (2006) 195-234

24. Totaro's question on zero-cycles on G₂, F₄, and E₆ torsors (preprint version) (published version)
    with Detlev W. Hoffmann
    Journal of the London Mathematical Society 73 #2 (2006) 325-338 (MR 2225489)

25. Unramified cohomology of classifying varieties for exceptional simply connected groups (preprint version) (published version)
    Transactions of the American Mathematical Society 358 #1 (2006), 359-371 (MR 2171237 / 2006g:11080)

26. The characteristic polynomial and determinant are not ad hoc constructions (preprint version) (published version)
    American Mathematical Monthly 111 #9 (2004), 761-778 (MR 2104048 / 2006d:15012)

27. Book: Cohomological invariants in Galois cohomology with Alexander Merkurjev and J-P. Serre American Mathematical Society University Lecture Series, volume 28 ISBN: 0-8218-3287-5 Reviews: (Bull. AMS) (MR 2004f:11034) Buy: (AMS) (Amazon) List of typos Summary table from my talk in Besancon, 2 July 2003

28. The Rost invariant has trivial kernel for quasi-split groups of low rank (published version)
    Commentarii Mathematici Helvetici 76 #4 (2001), 684-711 (MR 2003g:20079)

29. Involutions and trace forms on exterior powers of a central simple algebra (published version)
    with Anne Quéguiner-Mathieu and Jean-Pierre Tignol
    Documenta Mathematica 6 (2001), 97-118 (MR 2002k:16034)

30. Clifford algebras of hyperbolic involutions (published version)
    Math. Zeit. 236 #2 (2001), 321-349 (MR 2002d:16024)

31. Groups of type E₇ over arbitrary fields (published version)
    Comm. Algebra 29 #6 (2001), 2689-2710 (MR 2002f:17003)

32. Structurable algebras and groups of type E₆ and E₇ (published version)
    J. Algebra 236 #2 (2001), 651-691 (MR 2001m:20069)

33. Twisted flag varieties of trialitarian groups (published version)
    Comm. Algebra 27 #2 (1999), 841-856 (MR 2000a:20103)

34. Trialitarian groups and the Hasse principle
    manuscripta mathematica 98 #1 (1999), 97-113 (MR 99m:11049)
    Erratum, 113 #2 (2004), p.267
    Go to the published version of the article or the erratum.

35. Isotropic trialitarian algebraic groups (published version)
    J. Algebra 210 #2 (1998), 385-418 (MR 99k:20092)

36. Galois cohomology of special orthogonal groups (published version)
    with Jean-Pierre Tignol and Adrian Wadsworth
    manuscripta mathematica 93 #2 (1997), 247-266 (MR 98k:11039)

    (If you are curious about such things: this paper makes my Erdös number 4, via Erdös -> Ernst Straus -> Murray Schacher -> Adrian Wadsworth. The MathSciNet "Collaboration Distance" function reports 3, going through a paper collection credited to S. Chowla, Serre, and others.)

Miscellaneous items

• 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

• 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

• The note Twisted flag varieties of exceptional algebraic groups (August 1997) has been replaced.