Courant Lecture Notes in Mathematics. 3. New York, NY: Courant Institute of Mathematical Sciences. Providence, RI: American Mathematical Society (AMS). ix, 261 p. $ 24.00 (2000).
Apart of certain additional preparatory material, this book is a pedagogic illustration of the general methods and results of the author with his collaborators in 1997-1999, in a special case (see chapters 7-8) in which the technical difficulties are at a minimum. The above general methods/results are presented in the long papers of P. Deift, K. T. R. McLaughlin, T. Kriecherbauer, S. Venakides and X. Zhou [Int. Math. Res. Not. 1997, No. 16, 759-782 (1997; Zbl 0897.42015), Comm. Pure Appl. Math. 52, No. 11, 1335-1425 (1999; Zbl 0944.42013), ibid. 52, No. 12, 1491-1552 (1999; Zbl 1026.42024)]. The above particular case means in particular that the author considers in this book only contours which are a finite union of (finite or infinite) smooth curves in , i.e. the curves intersect at most at a finite number of points and all intersections are transversal.
Chapter 1 devotes the definition of the -dimensional Riemann-Hilbert (R-H) problem and some applied examples. Let points of self-intersection of . Suppose in addition that there exists a map which is smooth on . The R-H problem consists in seeking an matrix-valued function which is: analytic in , , as where denote the limits of as from the positive (resp. negative) side of .
In Chapters 2 and 3 the author presents with proofs a number of basic and well-known facts from the classical theory of orthogonal polynomials and Jacobi matrices/operators in the way of relating this theory to the theory of R-H problems.
Chapter 4 devotes to some aspects of the beautiful relationship between continuous fractions and orthogonal polynomials. Chapter 5 devotes basic concepts of random matrix theory. Chapter 6 devotes the principal properties of equilibrium measures.
The crucial Chapters 7-8 devote the above results for asymptotics of orthogonal polynomials and their applications to universality questions in random matrix theory via the corresponding R-H problems.