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

(1)利用Fulton和MacPherson提出的双变理论，建立了原代数变异特征类的一般理论。这项研究似乎将我们引向迈克尔·格罗莫夫所说的“符号代数几何”。(2)在与国外合作者的联合研究中，j.p。Brasselet和J.Schurmann，我可以成功地“使用斋藤的混合Hodge模块构建Hirzebruch特征的特征类版本”，这是由首席研究员早些时候提出的。(3)在上述(2)得到的理论中，相对Grothendieck群是一个基本工具，事实证明它与动机整合理论密切相关。作为这一工作的双积，我们可以得到一种统一奇异变元特征类的三种典型理论的理论。(4)我们的原代数变体特征类的构造与数学物理中的顶点代数理论在某种意义上是相互联系的。我们想对这一点作进一步调查。(5)我们不能构造奇异变异的“atiya - singer指标定理”，但为了实现这一最终目标，我将在与j.s uramnn和Markus Pflaum的联合研究中继续利用相对Grothendieck群对特征类理论进行更多的研究。

(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.