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The first two parts of this document, dealing with Scott's more immediate goals, are somewhat technical in nature. The last part, entitled "Future directions and applications," contains expository material and is suitable for both mathematical and non-mathematical readers.

Scott has, often in collaboration, provided an answer or breakthrough for a number of open mathematical questions of some standing, cf. [25], [39], [56], [62], as well as [19], [20], [22], and [33]. Currently he aims, again in collaborations, toward the a proof of a main conjecture in group representation theory, due to George Lusztig at MIT. Nineteen years ago Scott explained [29], using an analogy with continuous group theory, how the main obstacles to understanding finite group actions would eventually reduce to hard problems in the linear case, cf. also [74]. The Lusztig conjecture is such a problem, perhaps the most central.

Suppose G(q) is a finite group of Lie type (that is, an analog of a continuous group G, but parameterized by a finite coefficient field with q elements, which must be a power of a prime p). Then the conjecture provides a capsule description, or character formula, for the most important irreducible representations, provided p satisfies a modest, though serious, size requirement. The conjecture is known to be true for very large values of p, depending on G (but it is a pure existence proof, and no one knows how large p must be). Very recently, using computer programs developed over the last five years with the assistance of Mike Konikoff and Chris McDowell, two Virginia undergraduates, Scott showed empirically that the conjecture held as stated, and even in a stronger form proposed by Kato, in the first open cases, with G the group of 5 x 5 matrices of determinant 1, and p = 5 and 7. Scott and his collaborators, especially Ed Cline and Brian Parshall, have been working on the Lusztig conjecture for some time (each of the papers [40], [41], [44], [45], [47], [48], [49], [55], [57], [58], [59], [60], [63], [66], [67], [68], [77] is relevant, though many develop general theories of broader significance). Though the problem has proved difficult, Scott is optimistic that it can be solved within a few years and that much more can be done.

Beyond the Lusztig conjecture

Most immediate prospects for a proof involve a reduction to previous known geometric calculations (on so-called perverse sheaves). However, it remains a possibility that there is a purely algebraic proof. The same should be true of related, already proved conjectures due to Kazhdan and Lusztig in the continuous case. Such a proof would have a considerable impact on all of Lie theory, where currently geometric methods are perhaps over-rated, in Scott's view. Though Scott has an appreciation of geometry, and some expertize with it, he believes that, instead, finite-dimensional algebras could be the principal organizational theme, with geometry appearing only in the inspiration of some of the algebra.

In addition, a purely algebraic proof would make it easier to extend the theory to the nondescribing characteristic (or nondefining characteristic) representation theory of finite groups of Lie type. Scott has already begun, with Jie Du as well as Cline and Parshall, the study of algebras appropriate to these nondefining characteristic representations [68], [69], [70], [71], [72], [73], [75], [76]. For a history of this representation theory, see [74]. Here one is studying representations of these groups over finite fields of coefficients, but the coefficient systems are quite different from those used to define the group. Still, these representations must be understood if one is to understand all the linear representations. As a bonus, one would get information, likely decisive, for nonlinear representations (permutation group actions). The relationship of defining, nondefining, and nonlinear representations was the subject of Scott's article [74], the lead survey article for the Proceedings of the 1997 Newton Institute program on representations of algebraic groups and related finite groups.

It remains problematic what to do with the smaller coefficient fields, both in defining and nondefining characteristics, those not large enough even for the original formulation of the Lusztig conjecture to apply. If the Lusztig conjecture and similar results in nondefining characteristic could be established, it would at least be possible to handle some of the smaller characteristics fairly immediately with computer calculations, and have complete tables of character formulas for modest size ranks.

Future directions and applications

Though Scott is a pure mathematician, at least in his published work, he has also participated in many interdisciplinary seminars and believes that mathematics ultimately should be applied. He hopes to publish applied articles himself some day, and intends to be active in research for many years to come. Here is a view of how the theoretical world of group representations relates to more applied areas, and a glimpse of what might become future research directions after today's current problems are solved.

Finite group actions are the fundamental building blocks of all finite symmetries and dynamical systems: Abstract finite groups, when equipped with a selected set of generators and a selected action of these generators on a set, can produce all possible patterns of symmetry, even in spaces of almost unimaginably high dimensions. This large number of dimensions is critical for applications, if one is thinking of more than just producing pretty pictures: Each dimension might be viewed as describing a numerical parameter of a complicated physical system, so that the number of dimensions would correspond to the number of parameters required to describe the system. One can then think of the system as a point or particle in a very large dimensional space. Such a particle, constrained to move by only following the movements of a sequence of generators for a group, may follow a path that may be remarkably confined and understandable--that is, the production of symmetry by the group translates into constraints in the behavior of the system. There are other ways group actions can impose constraints, like the spinning of a top imposes an axis of rotation--any directional motion which is compatible with the spinning (commutes with it) must proceed along the axis. Many past applications have been in the domain of physics, e.g., the constraints of the Lorentz group on relativistic space-time, or of the Gell-Mann eight-fold way on fundamental particle interactions. But, in the information age, group actions have been used in communication codes and disk drive error correction. Here all the symmetry is in bit patterns of 0's and 1's. Theoretical results in automata theory, such as the Krohn-Rhodes theorem, suggest applications to computing will continue.

Roughly, the Krohn-Rhodes theorem says that every finite-state dynamical system may be understood in terms of a combinatorial arrangement of finite groups and their actions, the latter intimately related to the reversible movements of particles or sets of particles allowed by the system. To repeat, the particle terminology is suggested by physics, but it is meant as a purely abstract term. Indeed, it might well be more appropriate to think of the rapidly changing bit pattern in a computing or communication device, rather than a particle moving in a traditional physical space.

Scott regularly lectures on the Krohn-Rhodes theorem in Mathematics 355, and it is part of his future research plans (as well as one of the long-term justifications of his current work on group actions, since the latter play the role of fundamental building blocks in Krohn-Rhodes theory). He would like to first reformulate the theory so that there are more perfect and precise analogies with basic concepts in continuous and differential geometry. Such a program was already begun by the deceased topologist S. Eilenberg, in a great treatise, "Automata, languages, and machines," though Eilenberg's analogies remain only suggestive. However, the constraint of a particle moving on a smooth geometric surface should be interpretable precisely in the same way as the constraints imposed by a finite group on a particle in a finite dynamic system. After addressing this issue, Scott would then like to extend the theory so that it would give a satisfactory theoretical framework for modern analog and probabilistic computing devices, such as neural networks. Probably most practical benefits will be would be for artificial, rather than biological, computing devices, but one should not underestimate the power in any context of having a correct conceptual framework for analyzing data processing.