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)

• Vanishing of trace forms in low characteristics (16 July 2008)
  Formerly titled E₈ has no quotient trace form in characteristic 5

• Algebraic groups with few subgroups (13 May 2008)
  with Philippe Gille

• Orthogonal involutions on algebras of degree 16 and the Killing form of E₈ (3 March 08)
  with an appendix by Kirill Zainoulline
  Slides from my talk at the Chile conference in December 07

• Orthogonal representations of twisted forms of SL₂ (9 June 08)

To appear

• Pfister's theorem for orthogonal involutions of degree 12
  with Anne Quéguiner-Mathieu
  to appear in Proceedings of the American Mathematical Society

• Discriminant of symplectic involutions (13 July 07)
  with R. Parimala and Jean-Pierre Tignol
  to appear in Pure & Applied Mathematics Quarterly (Serre special issue, peer-reviewed)
  Possibly: vol. 5, #1 (2009)

• Cohomological invariants: exceptional groups and spin groups (notes for the Lens mini-cours)
  with an appendix by Detlev W. Hoffmann
  to appear in Memoirs of the American Mathematical Society

Publications

1. 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

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

3. Groups of outer type E₆ with trivial Tits algebras (24 July 06)
   with Holger P. Petersson
   Transformation Groups 12 #3 (2007), 443-474.
   Go to the published version

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

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

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

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

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

9. 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) Order: (AMS) (Amazon) List of typos Summary table from my talk in Besancon, 2 July 2003

10. 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

11. 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

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

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

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

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

16. 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.

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

18. 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

• 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.