Noncommutative geometry, microlocal analysis, index theorems and symplectic geometry

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AbstractAward: DMS-0906391Principal Investigator: Boris TsyganThis project is devoted to studying microlocal methods in indextheory and in symplectic geometry. A part of the project dealswith noncommutative differential calculus which is one of thecentral tools in the above. By noncommutative differentialcalculus we mean the study of algebraic structures arising fromthe standard differential calculus on manifolds, defined in termsin such a way that is valid of the algebra of functions on amanifold and in such a way that is valid if this algebra isreplaced by any algebra, commutative or not. This leads to thestudy of algebraic structures on Hochschild chain and cochaincomplexes of an algebra; those complexes play the role ofdifferential forms and multivector fields, a development of worksof Dolgushev-Tamarkin-Tsygan, Kontsevich-Soibelman, Costello,Lurie and other authors. The techniques of noncommutativecalculus will be applied to obtain new index theorems andtheorems about the determinant of the cohomology of ellipticsystems, generalizing recent results of Beilinson. Also, theproject will study a new object on a symplectic manifold that wecall an oscillatory module. These objects are modules overdeformed algebras of functions on a manifold, but with extrastructures that make their category much closer to the Fukayacategory. This extra structure is largely motivated by microlocalmethods in partial differential equations, in particular the WKBmethod. This is a part of a series of works that try tounderstand mirror symmetry microlocally (Bressler-Soibelman,Kapustin-Witten, Nadler-Zaslow, Gukov-Witten, Tamarkin).There are three related parts in the project: noncommutativecalculus, index theory, and oscillatory modules. Noncommutativecalculus is a theory that describes parts of the standarddifferential calculus in such a way that it can be applied insituations more general than that of an ordinary space, inparticular when, for coordinates on our imaginary "space", theequation xy=yx is no longer required. Index theory is a part ofthe theory of differential equations that expresses the number ofsolutions of an equation in terms of the topology of theunderlying space. Oscillatory modules are objects on geometricspaces that describe the motion of quantum particles on thesespaces (in other words, they treat the space in question as thephase space of a quantum mechanical system). These objects areintended to be applied to study other invariants of these spaces,also physically motivated and much more complicated; they arecalled A-branes and describe the motion of a quantum string onour space. They are called A-branes and are subject of muchcurrent research in mathematical physics and geometry. Let usfinish by explaining why noncommutative calculus is natural forapplications both in differential equations and in quantummechanics. Indeed,if x and y are basic operations withfunctions, for example multiplication by some function anddifferentiation, then xy differs from yx, in the sense that, ifyou apply them in different orders, you get differentresults. Similarly, in quantum mechanics, if x is the position ofa particle and y is its momentum, then xy differs from yx; thisis a mathematical expression of the Heisenberg uncertaintyprinciple.