A general study on topology, analysis and geometry of singular spaces

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(1)We developed a general theory of characteristic classes of proalgebiiac varieties, using the bivariant theory introduced by Fulton and MacPherson. This research seems to lead one to what Michael Gromov calles "Symbolic Algebraic Geometry".(2)In a joint research with the foreign collaborators, J.-P.Brasselet and J.Schurmann, I could succeed in "constructing a characteristic class version of Hirzebruch characteristics using Saito's mixed Hodge modules", which was proposed by the head investigator earlier.(3)In a theory obtained in the above (2), the relative Grothendieck group is a fundamental tool and it turned out that it is heavily related to the theory of motivic integrations. And as a biproduct of this work, we could obtain a kind of theory of unifying the three typical theories of characteristic classes of singular varieties.(4)Our construction of characteristic classes of proalgebraic varieties and the theory of vertex algebra in mathematical physics are in a sense related to each other. We want to make a further investigation on this point.(5)We could not construct a "Atiyah-Singer index theorem" for singular varieties, but to accomplish this final goal, in a oint research with J.Schuramnn and Markus Pflaum I will continue doing more research on the theory of characteristic classes using the relative Grothendieck groups.