Trends in noncommutative geometry

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AbstractAward: DMS-0728322Principal Investigator: Boris Tsygan, Dmitry TamarkinThis award provides partial support for a conference at the endof the emphasis year on noncommutative geometry at NorthwesternUniversity. Its aim was to explore the interaction betweenseveral directions in noncommutative geometry: index theorems,noncommutative calculus and operads, relations to mathematicalphysics, to number theory, and to the geometric Langlandsprogram. An important objective was to highlight the subjectsthat are strong in the Chicago area (algebraic geometry, motives,microlocal analysis, representation theory, geometric Langlandsprogram, mirror symmetry) and investigate their links to theworks in what is more traditionally considered noncommutativegeometry, and vice versa. For example, an algebraic approach toloop spaces that is used in the geometric Langlands program andin representation theory of real groups was compared to thetopological string theory which is related to algebraic topologyand mathematical physics, as well as to an approach to therepresentation theory of p-adic groups. A relation wasestablished between these subjects, in particular stringtopology, and current research on motives. Several approaches toa conjectural relationship between noncommutative geometry andmirror symmetry were presented.Noncommutative geometry is an extension of the classical geometryto "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 theHeisenberg uncertainty principle in quantum mechanics: if xstands for a position of a particle and y for its momentum, youcannot know precise values of both x and y. In a moremathematical context, noncommutativity is an expression of thefact that, if you apply two transformations to an object, theresult depends on the order in which you do it. Mathematicalsituations where noncommutativity occurs have been known for along time but the idea to treat them as "noncommutative spaces"and to study them by geometric methods is relatively recent. Oneof the successes of this approach is a radical simplification ofthe standard model in quantum field theory by means of allowingthe space to be noncommutative. The aim of the conference was tocompare several approaches to noncommutative geometry, as well asits links to other areas of mathematics and physics. You can findmore information on the conference website athttp://www.math.northwestern.edu/~tsygan/conf