Although more than 20 items in Scott's bibliography are singly-authored,
he often has preferred to work in collaboration. His principal collaborators
are listed alphabetically below, followed by synopses of the collaborations,
indexed to his bibliography.
-
Michael Aschbacher, Professor of Mathematics, California Institute of
Technology (NAS)
-
George Avrunin, Professor of Mathematics, University of Massachusetts
-
Edward Cline, Professor of Mathematics, University of Oklahoma
-
Charles Curtis, Professor Emeritus, University of Oregon
-
Jie Du, Senior Lecturer in Mathematics, University of New South Wales, Sydney
-
Vlastimil Dlab, Distinguished Professor Emeritus, Carleton University,
Ottawa (FRSC)
-
Charles Dunkl, Professor of Mathematics, The University of Virginia
-
Walter Feit, Professor Emeritus (deceased), Yale University (NAS)
-
Wilburd van der Kallen, Department of Mathematics, University of Utrecht
-
David Goldschmidt, Director of the Center for Communications Research,
Princeton,
formerly Professor of Mathematics, University of California at Berkeley
-
Peter Hoefsmit, Industrial mathematician, Airborne Express Company
-
I. Martin Isaacs, Professor of Mathematics, University Wisconsin
-
Roger Lyndon, Professor Emeritus (deceased), University of Michigan
-
Gordon Keller, Professor of Mathematics (deceased), The University of Virginia
-
Peter Neumann, Fellow of Queen's College, and former Assessor, Oxford
University
-
Don Newhart, Mathematician, National Security Agency
-
Michael O'nan, Professor of Mathematics, Rutgers University
-
Brian Parshall, Whyburn Professor of Mathematics, The University of Virginia
-
Frank Raymond, Professor Emeritus, University of Michigan
-
Klaus Roggenkamp, Professor of Mathematics, University of Stuttgart
-
Jian-pan Wang, Professor of Mathematics and President, East China Normal
University, Shanghai
Aschbacher-Scott [35]
obtained a general
theory of maximal subgroups of finite groups, reducing most questions to
groups which were almost simple, and others to irreducible modules for the
latter groups. Some of this had been proposed in the program Scott described
in [29]. The latter paper adapted Dynkin's maximal subgroups program in Lie
theory to finite group theory, and contained the often quoted
Onan-Scott theorem, which described the types of maximal subgroups possible
in the alternating groups. Actually, there were two theorems, and another
gave an early version of the Aschbacher-Scott work. There was no
collaboration between Onan and Scott, but both arrived at the 1979 Santa
Cruz conference with similar results, and agreed to share credit. O'nan
never published separately. "Same damn theorem" was Michael Onan's comment
after seeing the manuscript Scott had brought with him. In spite of this
vote of confidence, that version turned out to have some inaccuracies, and
[35] includes corrections. For a revisiting of the theory of maximal
subgroups in the case of finite groups of Lie type, and its several
connections with linear representation theory, see [74].
Avrunin-Scott [33]
proved Carlson's conjecture that support and
rank varieties defined the same sets, and established a Quillen
stratification theory of cohomology of groups with nontrivial coefficients.
Another main result was that the support variety of the tensor product of
modules was the intersection of their respective support varieties. This is
often inaccurately attributed to Carlson, who had only proved such a result
for rank varieties for elementary groups. To work with cohomology varieties
in the global case, Avrunin-Scott proved the (new) compatibility of support
varieties with the inverse image of cohomological restriction. Another
important method of proof in this paper was to regard the group algebra of
an elementary abelian p-group as the restricted enveloping algebra of a
commutative Lie p-algebra. The purpose was to get the right Hopf
structure. This introduction of such algebras was the beginning of the
support variety theory of Lie p-algebras later developed by
Parshall-Friedlander.
Cline-Parshall-Scott (CPS) [9], [14], [17], [20], [22],
[23], [24], [26], [34], [36], [38], [40], [41], [45], [49], [52],
[55], [58], [59], [60], [66], [67], [75],
[76].
Though multiply-authored papers are common in many scientific
disciplines, most papers in mathematics have only one or two authors. CPS
is by all accounts the longest continuously productive three-person
collaboration in the history of mathematics. Their achievements
include the proof [20] of Green's conjecture on exact induction and affine
quotients, the demonstration [22] with van der Kallen of generic cohomology,
the creation [45] of the theory of highest weight categories and
quasihereditary algebras ([44], [47], [45]), the best purely algebraic
reductions [55] of the Lusztig conjecture, and, recently, an expansion of
many ideas of the defining characteristic theory of finite groups of Lie
type into the nondefining arena, cf. [76]. Their older 1975 IHES paper [9]
on 1-cohomology calculated many more results, more quickly than ever before,
and is quoted in Andrew Wiles' proof of Fermat's Last Theorem (Annals of
Math., 1995).
Cline-Parshall-Scott-van der Kallen [22]
See above.
Curtis-Scott
produced manuscripts in 1974 with the first generic
block theory for Hecke algebras (including a complete defect 0 theory) and
finite groups of Lie type in nondefining characteristics. There work was not
published, but influenced work of Curtis's student R. Boyce and work of
Hoefsmit-Scott described below.
Du-Scott [54], [57], [69] (see also Du-Parshall-Scott
below)
developed in [57] some of the first integral theory of quantum
enveloping algebras, and applied it to both the defining and nondefining
characteristic representation theory of finite groups of Lie type. They
discussed in the same framework conjectures of Lusztig and James (proving
special cases), and introduced the notion of a generalized q-Schur
algebra. Later, they introduced the q-Schur2 algebra, a
decomposition-number preserving enlargement of an endomorphism algebra of
type B important in the nondefining theory. This enlargement was
quasihereditary in all cases, unlike the endomorphism algebra.
Du-Parshall-Scott [70], [71], [72]
undertook a broader
enlargement project (see above) for nondefining characteristic
endomorphism algebras in all root system types, using Kazhdan-Lusztig cells
and the new CPS notion of a stratified algebra. As a by-product, they
obtained in [72] [the most complete version (the full integral quantum case)
of the Schur-Weyl double centralizer theorem].
Dlab-Scott [61]
edited a volume of invited papers from the 1993
Canadian Mathemtics Society conference on Representations of Algebras,
especially devoted to finite dimensional algebras and Lie theory, and
marking a new level of interaction between these areas.
Dunkl-Scott
is not so well known as a collaboration, but Dunkl and
Scott were jointly responsible, cf. Scott's abstract [8], for the Krein
condition, or rather, the realization that there was a positivity condition
in finite permutation groups and combinatorics, perfectly analogous to a
condition obtained in harmonic analysis by the Soviet mathematician M. G.
Krein. An especially elementary proof, using observations by D. Higman on
work of Schur, was presented by Scott in [12], who also presented further
results. The impact of the Krein condition in combinatorics was substantial,
and many (former) Soviets, especially the associates of Krein, were very
grateful for the choice of name.
Feit-Lyndon-Scott [15]
gave a one-page elementary combinatorial
proof of a result on permutations that had previously been obtained only
through the theory of Riemann surfaces. Later, in his 1979 Annals paper,
Scott [21] generalized the theory from permutations to matrices,
incorporating the Riemann surface geometry into some inequalities based on
algebraic 1-cohomology. This led, in the same paper of Scott, to a
generalization, now widely applied, of a famous method of Brauer (the
Brauer trick) for constructing nonobvious subgroups of finite groups.
Goldschmidt-Scott [25]
gave a counterexample to an open question of
W. A. Manning, the leading expert in his day on permutation groups, of more
than fifty years standing. The coauthors had exchanged ideas on previous
occasions as well, and were office mates at Yale in 1970-71. Also,
Goldschmidt had earlier been a graduate student while Scott was an
instructor at the University of Chicago.
Hoefsmit-Scott
produced only a manuscript (1975), but it influenced
later work of Scott's Ph.D. student, Leonard Jones, and Scott's postdoctoral
student Jie Du. The manuscript contained, with some inaccuracies, the first
effort at proving the Nakayama conjecture for Hecke algebras, as well as
results on vertex and source theory, and a Mackey induction theory in the
Hecke algebra case. Before he decided on a career in industry, Hoefsmit was
Scott's postdoctoral student, following a well-known but never published
Ph.D. dissertation at UBC, Vancouver.
Issacs-Scott [5]
proved some concrete results about blocks and
restriction of characters. They also exchanged many ideas on permutation
groups and subnormal subgroups as postdoctoral office mates at the
University of Chicago in 1968-70.
Keller-Scott
produced only an abstract [2] from their common
interests in permutation groups and representation, yet the collaboration
was the major catalyst in Scott's decision to move from Yale to Virginia in
1971. Ed Cline (the 'C' in CPS above) had earlier been Keller's collaborator
and colleague at the University of Minnesota, and Keller and Scott jointly
sponsored Cline's visit in 1973-75 to the University of Virginia Mathematics
Department.
Neumann-Scott [11], [18]
found several applications of character
theory, especially modular theory, to finite permutation groups. They are
unpublished, except for the abstract [18] by Scott. Nevertheless, the
collaboration began Scott's long association with Oxford University,
occasioning two visits there instigated by Neumann. In turn, Neumann met
with Scott on two visits to the US, once as a short term visitor to the
University Virginia. In addition, Neumann sent his Ph.D. student, Cheryl
Praeger, to study as a postdoctoral student with Keller and Scott at
Virginia, and his Ph.D. student, Jan Saxl, for a shorter visit.
Newhart-Scott [30]
are coauthors of a virtually complete
handwritten manuscript applying the representation theory of permutation
module endomorphism rings to combinatorial issues of interest in coding
theory. Newhardt was the Ph.D. student of Scott's colleague, H. N. Ward, at
the University of Virginia, and is currently employed by the National
Security Agency.
O'Nan-Scott
See the discussion of Aschbacher-Scott above.
Parshall-Scott [28], [47], [63] (see also Cline-Parshall-Scott
and Du-Parshall-Scott above, and Parshall-Scott-Wang below)
in their
lectures (detailed, with additions, in the Moose notes [47]) at the 1989
Ottawa-Moosonee conference introduced, with considerable effect, the finite
dimensional algebras community to the new notions of quasihereditary
algebras and highest weight categories, born in the CPS research on
algebraic group representations, cf. [41], [44], and [45]. In [63], combining
methods of classical algebra and of etale cohomology, they produced a new
proof (and, actually, the first into print) of the very strong Koszulity
property for categories of perverse sheaves important in Lie representaion
theory.
Parshall-Scott-Wang [77]
presented a complete revision of the
theory of Borel subalgebras, and many new results, especially in the
infinitesimal and quantum cases. In the process they gave replacements for
an incorrect result in Scott's appendix [65], and alternative connections to
Kazhdan-Lusztig theory, in the spirit of proposed results of S. Koenig.
Raymond-Scott [19],
using 3-cohomology, answered negatively (for
higher dimensions) the question as to whether or not a well-known result of
Nielsen in differential topology (on homotopy-periodic diffeomorphisms)
generalized beyond 2-manifolds.
Roggenkamp-Scott [31], [32], [37], [39], [42], [50], [53],
in a ten-year collaboration produced unprecedented progress in
understanding the famous finite group ring isomorphism problem (Graham
Higman, Oxford thesis, 1940). Their main positive result [39] was the
validity of the problem for finite nilpotent groups, and in an especially
strong form for finite p-groups, which Higman and other experts had
considered the worst case. However, in subsequent work, cf. [50], [56],
Roggenkamp and Scott found a counterexample to a conjecture posed by Hans
Zassenhaus, a stronger version of the problem, considered by Zassenhaus to
be the key issue. Indeed, recently, M. Hertweck, partly building on the
techniques of that counterexample, has now set forth details to a
counterexample to the original problem of Higman. Hertweck is a Ph.D.
student at U. Stuttgart of a former University of Virginia postdoctoral
visitor, Wolfgang Kimmerle.