Hopf Cyclic Cohomology, Characteristic Classes of Foliations, and Quantum Invariant of Knots.

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Noncommutative Geometry (NCG) is an area of mathematics which applies algebras, including non-commutative ones, in solving geometric problems. Expressions in non-commutative algebras are sensitive to the order of their elements. For example, if you shuffle a deck of cards and then cut it, the result is not the same if you cut it and then shuffle it. Manifolds look locally like points, lines, planes, etc. For example, balloons and tire tubes are manifolds of dimension 2 because you may construct them by stitching together a few plane-like pieces. On the other hand, a non commutative manifold usually comes with two parts which are a manifold and a relation on it that identifies some of its points. It is obvious that any manifold is a non-commutative one but not vice versa. A foliation on a manifold slices it into manifolds with a lower dimension. For example, an onion which is a three dimensional manifold is decomposed into its leaves which are all sphere-like and hence two dimensional. Similarly a cabbage is also decomposed into its leaves. Mathematically, onion and cabbage are the same but their foliations are not equal. By identifying all points of each leaf we obtain a non-commutative manifold. Foliations have applications in many areas of mathematics and physics. Study of foliations is very old and their characterization has been open since 1960s. There have been different theories applied in this area with partial successes. However there are still many fundamental open problems. We plan to solve these open problems via methods of non-commutative geometry. Our primary tools are Hopf algebras and Hopf cyclic cohomology. Hopf algebras are the non-commutative counterpart of symmetries. Hopf cyclic cohomology defines invariants for symmetric non-commutative manifolds. This study not only will benefit the theory of foliations but also enables us to apply foliations and their characteristic classes in many other areas of mathematics such as number theory, quantum field theory, knot theory, and deformations theory. For example, symmetry of a round table is all rotations around the origin in the plane. However the symmetry of a rectangle table is reflections to the coordinate axes in the plane. For example the realizablity conjecture, which speculates that for each class in the Gelfand-Fuks cohomology there exists a distinguished foliation, is still far from being solved. Symmetry in classical geometry is governed by groups, while to study noncommutative spaces one needs Hopf algebras(quantum groups) as a generalization of groups. Hopf cyclic cohomology is a theory which provides invariants of algebras under symmetry of Hopf algebras. The theory was invented by Connes-Moscovic in 1998 and since then has been developed by the proposer and his collaborators. In the first phase of this project we have applied Hopf algebras and their cohomology to develop new theories for characterization of foliations. Some of these theories have no counterpart in the theory of foliations. For instance the twisting via Stable-Anti-Yetter-Drinfeld modules is merely due to Hopf cyclic cohomology. Our recent investigations on this subject make Hopf cyclic theory a new home for characteristic classes of foliations. Our goal is to continue reformulating and solving foliations problems with the help of Hopf cyclic theory. We propose to apply Hopf cyclic cohomology in the study of characteristic classes of foliations from the Noncommutative Geometry point of view. This is another piece of our long term research on the subject by which we plan to reformulate foliations and their invariants. Our ultimate goal is to prove realizablility and recognizability conjectures. Our research so far has established the fact that Hopf algebras are suitable sources of symmetry and infinitesimals for noncommutative spaces. One of the noncommutative spaces on which one can exercise all aspects of Noncommutative Geometry is the space of leaves of a foliation, on which Differential Geometry is powerless. In another parallel but not far project we plan to derive quantum invariant of knots from the category of Yetter-Drinfled modules over quantum groups. We would like to extend our recent results on the enveloping algebras to quantum algebras. Noncommutative geometry developed out of coalescence of operator algebras and differential geometry. This is in contrast to Differential Geometry, where spaces are sets of points and functions on them are secondary objects. This exchange of roles is well understood by the Gelfand-Naimark theorem which establishes the equivalence of commutative C*-algebras and locally compact Hausdorff spaces. However, as a crucial axiom we do not deprive noncommutative algebras of being coordinates algebras of a "spaces". In this project we apply and also develop a wide variety of concepts and objects of Hopf cyclic cohomology including: Hopf algebras of transverse geometries, stable-anti-Yetter-Drinfled modules, cup products, twisted cyclic cocycles, and local index formula.For example consider shuffling and cutting of a deck of cards. Then there is a difference between cutting then shuffling and shuffling and then cutting. Assume that there is a complicated puzzle. Via classical geometry you cannot see the picture. NCG provides you with the notebook of the locations of the pixels. In this example the scattered puzzle together with the notebook resembles an example of noncommutative spaces; while the solved puzzle represents a classical space. To see the meaning of symmetry you may draw a square in a sheet. Then the moves of the sheet with no effect on the square are given by the rotation around the center of the square by 90 degree and the reflection with respect to a line that divides the square into two equal rectangles. The group of symmetries of square comprises the repetition and combination of theses two simple moves.