Principal Investigator: Boris Tsygan, Dmitry Tamarkin
This award provides partial support for a conference at the end of the emphasis year on noncommutative geometry at Northwestern University. Its aim was to explore the interaction between several directions in noncommutative geometry: index theorems, noncommutative calculus and operads, relations to mathematical physics, to number theory, and to the geometric Langlands program. An important objective was to highlight the subjects that are strong in the Chicago area (algebraic geometry, motives, microlocal analysis, representation theory, geometric Langlands program, mirror symmetry) and investigate their links to the works in what is more traditionally considered noncommutative geometry, and vice versa. For example, an algebraic approach to loop spaces that is used in the geometric Langlands program and in representation theory of real groups was compared to the topological string theory which is related to algebraic topology and mathematical physics, as well as to an approach to the representation theory of p-adic groups. A relation was established between these subjects, in particular string topology, and current research on motives. Several approaches to a conjectural relationship between noncommutative geometry and mirror symmetry were presented.
Noncommutative geometry is an extension of the classical geometry to "spaces" in which coordinates x, y, etc. no longer commute, i.e. where the identity xy=yx is not necessarily true. Such "spaces" abound in mathematics and physics. For example, noncommutativity is the mathematical manifestation of the Heisenberg uncertainty principle in quantum mechanics: if x stands for a position of a particle and y for its momentum, you cannot know precise values of both x and y. In a more mathematical context, noncommutativity is an expression of the fact that, if you apply two transformations to an object, the result depends on the order in which you do it. Mathematical situations where noncommutativity occurs have been known for a long time but the idea to treat them as "noncommutative spaces" and to study them by geometric methods is relatively recent. One of the successes of this approach is a radical simplification of the standard model in quantum field theory by means of allowing the space to be noncommutative. The aim of the conference was to compare several approaches to noncommutative geometry, as well as its links to other areas of mathematics and physics. You can find more information on the conference website at www.math.northwestern.edu/~tsygan/conf
