Title: Algebraic Algorithms: A Personal Perspective
Lecturer: Charles Sims
Lecture I. Why Bother?
The attitudes of algebraists about algorithmic questions vary
widely. For some algebraists, algorithms and computation form a major
focus of their work. Others see little need for explicit calculation
in algebra, particularly machine calculation. The views of computer
scientists about algebraic algorithms differ significantly from those
of algebraists and of mathematicians more generally. This lecture
will attempt to survey the ways mathematicians and computer scientists
look at algebraic algorithms and to discuss the motivations of
individuals who study these algorithms.
Lecture II. Three Fundamental Algorithms
Probably the oldest algebraic algorithm is the method Euclid gives for
computing greatest common divisors of integers. Every student of
mathematics should know this algorithm. There are other algorithms
that are nearly as fundamental, algorithms that should be familiar at
least to all students of algebra. This lecture will give a brief
discussion to three such algorithms: The LLL lattice reduction
algorithm, the Groebner basis algorithm, and the algorithm for finding
the order of a permutation group from a set of generators.
Lecture III. Finitely Presented Groups, Trying to Beat the Odds
The area of finitely presented groups seems at first not to lend
itself to useful work on algorithms. Almost every interesting
computational question about a finitely presented group turns out to
be impossible to answer in general. Nevertheless, many mathematicians
have devoted a great deal of effort to attempts to compute with
finitely presented groups. Why this optimism in the face of almost
certain failure? What has been accomplished?