1999 Chern Lectures
The 1999 Chern Lectures will be delivered by Yuri Manin on September 15-October 6, 1999.
Department of Mathematics, University of California, Berkeley, presents
The 1999 Chern Lectures
Yuri Manin
Professor of Mathematics
Max-Planck-Institut, Bonn
"Counting rational points and rational curves: from Waring's problem to quantum cohomology"
Thursday, September 16
Sibley Auditorium - 4:10pm-5:00pm
Abstract: Consider a homogeneous equation of degree d with integer coefficients F(x_0,...,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.
Reception in 1015 Evans following the lecture.