A generalization of the Atiyah-Singer Index Theorem on Noncommutative manifolds

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The purpose of the present research are;1) to find an extension of the Atiyah-Singerindex theorem in the framework of Noncommutative Geometry;2) to apply such anoncommutative index theorem to Geometry and String theory. Achieving the presentresearch we obtained finally the following results.First we extended the classicalAtiyah-Patodi-Singer index theorem to foliated manifolds with boundary of higherdimensional leaves and obtained an index theorem involved with the Godbillon-Veyclass (a joint work with P. Piazza).Second, by exploiting the framework of Noncommutative Geometry, we extended the domain of the Godbillon-Vey class (differentiability of foliations), and related our cocycle to the area cocycle defined by T.Tsuboi.Third, we clarified relationship among the family index theorem of odd dimension, the Dixmier-Douady class (a characteristic class of gerbe) and the Godbillon-Vey class.