1. Algebraic properties of modules over the Weyl algebra D=Dn.

(i) Arithmetic filtration of the Weyl algebra. Filtrations of D-modules. Good filtrations.

(ii) Noetherian properties.

(iii) Associated varieties, Hilbert polynomials. Dimension and degree of a D-module.

(iv) Basic inequality.

(v) Holonomic D-modules and their properties.


2. Relation with analysis.

(i) D-modules and systems of differential equations. Solutions of D-modules. Left and right D-modules.

(ii) Digression on the theory of generalized functions and distributions.

(iii) Regularization of distributions x^\lambda and Q^\lambda using differential equations.


3. Standard functors for D-modules.

(i) Inverse image functor

(ii) Relation between left and right D-modules

(iii) Direct image functor

(iv) Fourier transform


4. Properties of functors on categories of D-modules.


5. Stability of the holonomic property.


6. Applications to analysis.

(i) Meromorphic continuation of P^\lambda. Notion of b-function of a polynomial P.

(ii) Regularization of different type of integrals.


5. Cohomological techniques in study of D-modules.


6. Geometric picture of the the algebra D of differential operators.

(i) Geometric filtration on the algebra D.

(ii) Associated varieties for D-modules. Noetherian properties.

(iii) Comparison of two approaches.

(iv) Equivalence of two approaches to holonomic modules.


7. D-modules on smooth affine algebraic varieties.

(i) Short digression into affine algebraic varieties. The sheaf O(X) and the category M(O(X)) of O-modules on X. Localization of O(X)-modules.

(ii) Recall of basic properties of smooth varieties.

(iii) Basic definition and the structure of the algebra D = D(X) of differential operators on a smooth affine algebraic variety X.

(iv) Category M(D) of D-modules on a smooth affine algebraic variety X. Localization of D-modules.

(v) Relation between left and right D-modules.


8. Basic functors between D-modules.

(i) Basic functors and their properties.

(ii) Kashiwara's lemma

(iii) Reduction to the case of affine space (Weyl algebra).


9. Study of D-modules using the geometric filtration.

(i) Associated graded module. Noetherian properties.

(ii) Singular support of a D-module. Relation with Kashiwara lemma.

(iii) Proof of the basic inequality.

10. If time permits: The Bernstein - Kashiwara theorem on the finiteness of the dimension of the space of solutions of a holonomic D-module.