Consider a homogeneous equation of degree d with integer coefficients F(x₀,...,x_r)=0, and ask the following question: how many integer solutions are there with all |x_i| at most B, when B is large? A similar question can be asked about the variety of solutions whose coordinates are homogeneous polynomials in two variables of bounded degree. An elementary heuristic argument relying upon a probabilistic reasoning in the first case, and count of constants in the second case, suggests that the answer must depend on the sign of the number k=r+1-d: there must be `many' solutions for positive k, `few' for negative k, and some interesting boundary effects might take place for k=0.
            In fact, k is simply the degree of the first Chern class of the projective manifold F=0 (if it is nonsingular), and it turns out that many results of number theory and algebraic geometry nicely fit into this crude heuristic scheme, if one makes some subtle changes in basic definitions and questions.
            The first part of the colloquium talk will discuss the number-theoretical program, which can be considered as an extension of the classical work using the circle method.
            The second part of the colloquium talk will be dedicated to the counting of rational curves. Motivated by quantum string theory, this subject has developed into a rich and beautiful theory centered around quantum cohomology and the mirror conjecture. In the introductory lecture, the physical context of quantum cohomology will be described. The following minicourse will contain a more detailed review of the relevant mathematical constructions and results.