Skip's preprints, publications, etc.

I study linear algebraic groups, their representations (20G05), and their cohomology (11E72), mostly over arbitrary fields (20G15). In some sense, this encompasses the classical (11E57) and exceptional groups (17B25?); varieties on which those group act (14Mxx); and traditional algebraic objects like quadratic and hermitian forms (11Exx), algebras with involution (16K20), and Lie algebras (17B10, 17B45). Currently, my research centers on adapting techniques from classical representation theory over the complex numbers to prove new results about algebraic groups over arbitrary fields, including anisotropic groups and fields of bad characteristic.

Preprints (intended for publication)

• Wild Pfister forms over Henselian fields, K-theory, and conic division algebras (29 January 2010)
  with Holger P. Petersson

To appear

• Degree 5 invariant of E₈ (30 December 2009)
  with Nikita Semenov
  to appear in International Mathematics Research Notices

• Quaternion algebras with the same subfields (22 October 2009)
  with David J. Saltman
  to appear in the book Quadratic forms, linear algebraic groups, and cohomology, Springer, 2010.

• There is no "Theory of Everything" inside E₈ (preprint version)
  with Jacques Distler
  Extended abstract to appear in Oberwolfach Reports (not peer-reviewed); preliminary version
  to appear in Communications in Mathematical Physics

• Edited volume: Quadratic forms, linear algebraic groups, and cohomology
  edited with J.-L. Colliot-Thelene, R. Sujatha, and V. Suresh
  To be published by Springer in the series Developments in Mathematics
  Expected publication date: 3rd quarter 2010

Publications

1. Expository article: Finding good bets in the lottery, and why you shouldn't take them (preprint version) (published version)
   with Aaron Abrams
   American Mathematical Monthly 117 #1 (2010), 3--26

2. 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

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

4. 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.

5. 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)

6. 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

7. 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)

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

9. Expository article: Somewhat more than governors need to know about trigonometry
   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

10. 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]

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

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

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

14. 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)

15. 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)

16. Expository article: 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)

17. 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

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

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

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

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

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

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

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

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

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

    (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 (not for publication)

• The Rost invariant has zero kernel for quasi-split trialitarian groups (9 February 2009)
  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.

• Posters summarizing properties of exceptional groups, their roots, their representations and their Tits indexes:
  • Type E₆ (June 06, revised November 06)
  • Type G₂ (June 06)

• 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.