Statement of Research Interests
نویسنده
چکیده
My primary area of research has been algebraic geometry and its interaction with the theory of linear systems of partial differential equations (PDEs). This interaction is provided through the study of D-modules; i.e., modules over the ring of algebraic differential operators. The computational aspects of the subject interest me most. The use of Gröbner bases yields relatively effective algorithms for many essential computations in the polynomial ring k[x1, . . . , xn] and its quotients. The theory is less clear in the non-commutative case, namely in the Weyl algebra D = k〈x1, . . . , xn, ∂ ∂x1 , . . . , ∂ ∂xn 〉, where xi ∂ ∂xj = ∂ ∂xj xi + δij . Many algorithms can be generalized to work in this case also, but there is a lot of room for improvement. Of particular interest is the relationship of D with the associated commutative ring k[x1, . . . , xn, y1, . . . , yn], obtained as the graded ring of D associated to particular weight filtration. This relationship gives rise to the notion of irregularity for a module D/I, which is closely related to the behavior of the solutions of the PDE corresponding to D/I. Understanding this behavior has been the main part of my thesis.
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