This book is based on lectures given at a summer school for undergraduate students at the Park City Mathematical Institute in 2001. The aim of the lectures was quite ambitious – to explain to (a group of selected) undergraduate students classical problems of enumerative algebraic geometry, together with the recent approach to these problems based on Gromov-Witten invariants and quantum cohomology and to describe how these notions were created using ideas coming from string theory! It seems impossible and it was necessary to adopt quite an unusual style of writing to reach the goal. The author had to introduce a long list of concepts (at least definitions and main examples) starting from basic ones (categories of topological spaces and continuous maps, groups and their homomorphisms) to more advanced ones (topological, smooth and complex manifolds, differential forms and the de Rham cohomology, various types of homological and cohomology groups and their relations, Poincaré duality and cohomology rings, smooth and holomorphic vector bundles) as well as pieces of modern physics education (classical and quantum mechanics, path integrals, supersymmetry, a pair of dual models in string theory).
He provides precise definitions as much as he can and adds basic examples and comments, helping to develop the intuition of the reader, and a lot of references for additional study. The book contains a lot of extra material that was not included in the original fifteen lectures. It is a nicely and intuitively written remarkable little booklet (only about 200 pages of A5 format) covering a huge amount of interesting material describing a beautiful area, where modern mathematics and theoretical physics meet. It can give inspiration to teachers for a lecture series on the topic as well as a chance for self-study by students.