The Riemann Boundary Problem on Riemann Surfaces

Approach your problems from the right end It isn't that they can't see the solution. It is and begin with the answers. Then one day, that they can't see the problem. perhaps you will find the final question. G. K. Chesterton. The Scandal of Father 'The Hermit Clad in Crane Feathers' in R. Brown 'The point of a Pin'. van GuIik's The Chinese Maze Murders. Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as "experimental mathematics", "CFD", "completely integrable systems", "chaos, synergetics and large-scale order", which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics.

1. The Riemann Boundary Problem on Closed Riemann Surfaces.- §1. Riemann Surfaces.- §2. Functions and Differential Forms. Abelian Integrals and Differentials.- §3. Riemann Bilinear Relations. The Riemann — Roch Theorem.- §4. Cauchy-type Integrals.- §5. The Riemann Problem. Number of Solutions.- §6. Inversion of Abelian Integrals and Abel’s Theorem. Solvability of the Riemann Problem.- §7. Riemann Theta-Functions. Solvability of the Riemann Boundary Problem.- §8. Explicit Formulae for Solutions of the Riemann Problem.- 2. Complex Vector Bundles over Compact Riemann Surfaces.- §9. De Rham and Dolbeault Theorems.- §10. Divisors. Complex Vector Bundles. Serre and Riemann Theorems.- §11. The Riemann — Roch Theorem. The Riemann Problem.- §12. The Second Cousin Problem. Solvability of the Riemann Problem.- 3. The Riemann Boundary Problem for Vectors on Compact Riemann Surfaces.- §13. The Riemann Boundary Problem for Vector Functions.- 4. The Riemann Boundary Problem on Open Riemann Surfaces.- §14. Open Riemann Surfaces.- §15. D-Cohomologies.- §16. D-Divisors. The Second Cousin Problem.- §17. The Riemann Problem. Solvability.- §18. The Solving of the Riemann Problem in the Explicit Form.- 5. Generalized Analytic Functions.- §19. Bers — Vekua Integral Representations.- §20. The Riemann — Roch Theorem.- §21. Nonlinear Aspects of the Generalized Analytic Function Theory.- 6. Integrable Systems.- §22. The Schrödinger Equation.- §23. The Landau — Lifschitz Equation.- §24. Riemann — Hilbert and Related Problems.- Appendix 1 Hyperelliptic Surfaces.- Appendix 2 The Matrix Riemann Problem on the Plane.- Appendix 3 One Approximate Method of Solving the Matrix Riemann Problem.- Appendix 4 The Riemann — Hilbert Boundary Problem.- Notations.-References.