Noncommutative Geometry and Quantum-Group Theory are relatively new (c.a. 20 years old) and rapidly growing branches of mathematics requiring the knowledge of differential geometry, algebraic geometry, group theory, algebra, functional analysis, K-theory. The diversity of prerequisite mathematical tools, short but abundant history and rapid development of these subjects make direct assistance of the experts extremely important for advancing research in this area. The International Banach Mathematical Centre school/conference on quantum geometry is designed to provide such a guidance and inspiration to both newcomers and scientists already active in this field of mathematics. The school topics concern research themes where recent breakthroughs took place, and which are of strong interest to many mathematicians and mathematical physicists.

In the classical geometry, spaces are thought of as collections of points, and functions on such spaces are treated as auxiliary objects. Noncommutative geometry inverts this picture in a Copernican way, placing the abstract concept of functions at the centre of the theory, and making a generalized version of a space but a derived concept. In the spirit of the Gelfand-Naimark theorem establishing the equivalence between commutative C*-algebras and locally compact Hausdorff spaces, these generalized spaces given by noncommutative algebras are often called noncommutative or quantum spaces. This opens up the world of naturally occurring examples of quantum spaces and turns out to be extremely helpful in studying some particularly difficult spaces, e.g., spaces of foliations, where standard classical geometry techniques do not work.