Hilbert schemes, which parametrize subschemes in algebraic varieties, have been extensively studied in algebraic geometry for the last 50 years. The most interesting class of Hilbert schemes are schemes \(X^{[n]}\) of collections of \(n\) points (zero-dimensional subschemes) in a smooth algebraic surface \(X\). Schemes \(X^{[n]}\) turn out to be closely related to many areas of mathematics, such as algebraic combinatorics, integrable systems, representation theory, and mathematical physics, among others.

This book surveys recent developments of the theory of Hilbert schemes of points on complex surfaces and its interplay with infinite dimensional Lie algebras. It starts with the basics of Hilbert schemes of points and presents in detail an example of Hilbert schemes of points on the projective plane. Then the author turns to the study of cohomology of \(X^{[n]}\), including the construction of the action of infinite dimensional Lie algebras on this cohomology, the ring structure of cohomology, equivariant cohomology of \(X^{[n]}\) and the Gromov–Witten correspondence. The last part of the book presents results about quantum cohomology of \(X^{[n]}\) and related questions.

The book is of interest to graduate students and researchers in algebraic geometry, representation theory, combinatorics, topology, number theory, and theoretical physics.