2007 Fiscal Year Final Research Report Summary
Boundary value problems and Index Theorem for D Modules
| Project/Area Number |
16540150
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Basic analysis
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| Research Institution | Osaka University |
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Principal Investigator |
UCHIDA Motoo Osaka University, Graduate School of Science Depantment of Mathematics, Associate Professor (10221805)
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| Co-Investigator(Kenkyū-buntansha) |
NISHITANI Tatsuo Osaka University, Graduate School of Science, Dept. Math., Professor (80127117)
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| Project Period (FY) |
2004 – 2007
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| Keywords | D Modules / boundary value problem / microlocal analysis |
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| Research Abstract |
We have found an idea to formulate elliptic boundary value problems for systems of differential equations in terms of D-Modules and to construct their characteristic cycles (or microlocal Euler classes). Let us consider a system of differential equations M on a manifold M with boundary N. Let M_<tan> denote its pull back to the boundary. Given a system of differential equations N on the boundary and a D_N linear morphism α: N → M_<tan>. By definition, this boundary value problem is said to be elliptic if a induces an isomorphism from ε_N [○!×] N to a coherent quotient module M^+(tan)of ε_N [○!×] M^<tan> defined microlocally from the boundary. (ε_N is the sheaf of rings of microdifferential operators on the boundary.) It is still difficult to construct a characteristic cycle in this naive setting, and we want to translate this setting of BVP as a module (or an object of a derived category of modules) over some ring. If we introduce the ring BD as BD = D_M [○!+] D(N,M) [○!+]D_N, we can get an object B (M, N) of the derived category D^b(D_M [○!×] B), with B the ring of upper half triangle matrices of degree 2. One can possibly define a characteristic cycle (or a microlocal Euler class) associated to the pair of a D_M [○!×]B-module and a B-module (Z_M, Z_N) by the diagonal argument under the condition of ellipticity of α. We expect that one can prove (by chasing diagrams) an index theorem for boundary value problems in terms of characteristic classes defined here, since their construction is almost totally functorial.
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