The school on noncommutative geometry will take place in Buenos Aires, Argentina, on July 26-August 7, 2010. It will include eight lecture courses by leading experts covering the basics of noncommutative geometry (K-theory, index theory, deformation quantization, quantum groups) as well as most recent developments in the subject and in its connections to other fields, such as topological quantum field theory, symplectic geometry, vertex operator algebras and other topics of mathematical physics.
Noncommutative geometry is the subject of mathematics that generalizes the classical methods of studying spaces to the noncommutative case, i.e. to the situation where the identity xy=yx is no longer valid. This generalization is needed for many applications: quantum physics (where noncomutativity is a mathematical manifestation of the Heisenberg uncertainty principle); geometry, theory of differential equations, topology, etc. (where symmetries of a space or of another system do not commute, i.e. one gets different results if one applies two symmetries in different orders). Perhaps less intuitively, to develop the methods of noncommutative geometry themselves, one needs advanced techniques of algebra, geometry, and topology that are inspired to a large extent by mathematical physics. The school will bring together the leading experts in the field, as well as in the adjacent subjects, with hmany young researchers from the US, the Americas, and Europe.
The project consisted of a summer school in noncommutative geometry that was held in Buenos Aires on July 26-August 7, 2010. Noncommutative geometry is a mathematical discipline that generalizes the classical calculus and geometry to the situations where the commutativity property xy=yx is no longer true. Examples of such situations are numerous, including quantum mechanics where the falure of commutativity is a mathematical expression of the Heisenberg uncertainty principle. Noncommutative geometry has many other applications, including topology and geometry. These disciplines study geometric shapes and spaces, but their main tool is to describe a geometric object in terms of an algebraic system which has no reason to be commutative. This algebraic system is then studied by means of noncommutative geometry. On the other hand, development of noncommutative geometry itself relies on methods of geometry, topology, and mathematical physics. All this makes necessary a robust exchange of ideas and techniques between noncommutative geometry and various other subjects. Intellectual merit. The summer school consisted of eight lecture courses in noncommutative geometry and related topics given by leading experts. In the courses, core methods of noncommutative geometry, foundations and specific topics of the related subjects, and interaction between the two were presented. A volume of proceedings was published by the Clay Mathematical Institute. Broader impact. The audience of the school and the readership of the Proceedings included a large number of students and other young mathematicians from the US, Europe, and Latin America. The lectures and the book, as well as numerous discussions and other contacts, introduced these young mathematicians into the subject and its applications
