Selected Topics in Algebra and its Interrelations with Logic, Number Theory and Algebraic Geometry

Approach your problems from the right It isn't that they can't see the solution. end and begin with the answers. Then It is that they can't see the problem. one day, perhaps you will find the final G. K. Chesterton. The Scandal of question. Father Brown 'The Point of a Pin'. 'The Hermit Clad in Crane Feathers' in R. van Gulik'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 addi tion to this there are such new emerging subdisciplines as "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.

I. Quadratic Forms and Arithmetical Symbols.- I. Elements of Quadratic Forms Theory Quadratic Forms over a Field k of Characteristic ? 2.- II. Symbols. Elements of the Theory of the Group K2(A).- II. Methods of Logic and Algorithmic Methods in Algebra.- §1. Countable Sets.- §2. The Notion of Lattice.- §3. The Algebra of Propositions.- §4. Formal Systems.- §5. Primitive Recursive Functions.- §6. Recursive Functions.- §7. Turing Machines.- §8. The Concept of Recursive (Algorithmic) Decidability.- §9. On the Formalist and the Constructive Point of View in Mathematics.- §10. Derived Terms in the Theory of Formal Systems.- §11. The Constructive Approach to Set Theory.- III. Introduction to Modern Algebraic Geometry.- §1. Generalities.- §2. The Spectrum of a Commutative Ring.- §3. Schemes. Chevalley Schemes.- §4. Elements of Projective Geometry.- §5. The Chevalley Scheme Associated to an Integral and Irreducible Scheme.- IV. Theory of Topoi.- I. General Theory of Topoi.- II. Theory of Lawvere-Tierney Topoi.- V. Elements of the Theory of Elliptic Curves.- I. Elliptic Curves Defined over the Complex Field.- II. Divisors.- VI. Algebraic Varieties over a Finite Field.- §1. The Zeta Function.- §2. Cohomology Theories.- §3. Weil’s Diophantine Conjectures.- VII. Elementary Theory of Non-Standard Real Numbers.- Appendix. On Local Rings with the Approximation Property.