by Yu I Manin (Max-Planck-Institut für Mathematik, Germany)

The book is a collection of research and review articles in several areas of modern mathematics and mathematical physics published in the span of three decades. The ICM Kyoto talk “Mathematics as Metaphor” summarises the author's view on mathematics as an outgrowth of natural language.

• Algebraic Geometry:
  • The Hasse-Witt Matrix of an Algebraic Curve
  • Rational Points of Algebraic Curves over Functional Fields
  • Correspondences, Motives, and Monoidal Transformations
  • New Directions in Geometry
  • Arrangements of Hyperplanes, Higher Braid Groups and Higher Bruhat Orders
• Modular Forms and Diophantine Equations:
  • The p-Torsion of Elliptic Curves is Uniformly Bounded
  • Parabolic Points and Zeta-Functions of Modular Curves
  • Periods of p-Adic Schottky Groups
  • Rational Points of Bounded Height on Fano Varieties
  • Points of Bounded Height on del Pezzo Surfaces
• Differential Equations and Mathematical Physics:
  • Conservation Laws and Lax Representation of Benney's Long Wave Equations
  • Gelfand–Dikii Hamiltonian Operator and the Coadjoint Representation of the Volterra Group
  • The Twistor Transformation and Algebraic–Geometric Constructions of Solutions of the Equations of Field Theory
  • The Mumford Form and the Polyakov Measure in String Theory
  • and other papers