October 31, 2003:  Prof. Fedor Bogomolov, CIMS

Algebraic Curves and Algebraic Numbers
  

  In algebraic geometry an algebraic curve, or better, an affine algebraic curve, is roughly speaking a subset of solutions of a polynomial equation P(x,y) = 0 where x,y run though elements of some field (rational numbers, complex numbers, real numbers - for example, in the latter case this set is indeed in general a curve in the plane - hence the name).  I will touch on some aspects of the theory which in its most elementary form investigates the relation between geometry of the algebraic curve P(x,y) = 0 and the properties of the set of rational solutions P(x,y) = 0 if P is a polynomial with rational coefficients.