The seminal work of A.A. Markoff in 1872 described the degree of approximability of badly approximable real numbers in terms of a sequence of quadratic forms and their homogeneous minima. The study of this - the first Markoff Chain - and of the corresponding theory for other real numbers and forms has made considerable progress over the last twenty years. One direction has been to study the structure of the spectrum itself. A second direction has been to increase the understanding of the relevance of hyperbolic geometry and the connection with the algebra of the free group on two generators. Much of the detailed technical calculations rely on the cutting sequences of the geodesics which in turn are related to the so-called two-distance sequences arising in the study of quasi-crystals. This grant will partially support a six day workshop on the Markoff spectrum and higher dimensional approximation constants to be held at Brigham Young University in May, 1991. The workshop will facilitate the exchange of information and will encourage the research of about thirty mathematicians working in several related areas of mathematics connected with Diophantine Approximation. Recent results in this area make such a workshop most timely.
