Yu. I. Manin

Linear codes and modular curves

Results of recent investigations at the juncture of coding theory, the theory of computability, and algebraic geometry over finite fields are presented. The basic problems of the asymptotic theory of codes and Goppas construction of codes on the basis of algebraic curves are presented, and a detailed algorithmic analysis is given of the codes arising on the modular curves of elliptic modules of V. G. Drinfeld.

Holomorphic supergeometry and Yang-Mills superfields

The work is devoted to a description of the mathematical structures at the basis of supersymmetry — field theory in which the symmetry groups mix bosons and fermions. The approach developed is based on the theory of supertwistors.

Elements of supergeometry

This paper is devoted to an exposition of the structure theory of supermanifolds and bundles on them, a description of Serre duality on supermanifolds, investigation of inverse sheaves, definition of characteristic classes and proof of the Grothendieck-Riemann-Roch theorem for supermanifolds.

Supercell partitions of flag superspaces

A description is given of the partition of flat superspaces, which correspond to classical simple Lie supergroups, into Schubert supercells. The relative positions of Schubert supervarieties are studied and their singularities are resolved.

Hilbert's tenth problem

A complete account is given of a question which is of great interest to specialists in mathematical logic and number theory.

Geometry of supergravity and Schubert supercells

A generalization of the Penrose twistor model to the case of theN-extended supersymmetry is described. It is shown that the geometry of Schubert supercells of complex flag supermanifolds is the source of invariant constraints, Lagrangians, and dynamical equations of supergravity.

Benney's longwave equations. II. Lax representation and conservation laws

One gives the Lax representation for Benneys longwave equations. One constructs the “universal conserved densities.”

The problem of the continuum

Set theory required a half century to become generally recognized and a half century more to become accepted. The end of the first period was marked by Hilberts famous phrase~ Cantors paradise in which Cantor was explicitly compared to a godlike creator, and his critics, less explicitly, to diabolical tempters. Set theory has now become the language a professional mathematician learns to speak in from the very beginning. He very quickly loses the ability to be bothered about its semantic obscurities. The serious disputes from the beginning of the century about the legitimacy of the axiom of choice is now perceived nearly psychologically. Even the purely negative results of G~del and Cohen on the continnum problem have been sensed as its solution, though Churchs term unsolving conveys quite precisely the essence of these remarkable discoveries. The primary cause for this circumstance is undoubtedly the isolation of problems of infinity from the rest of mathematics in combination with the remarkable universality, flexibility, and effectiveness of set-theoretic language itself as a means of association, discovery, and definition of material.