### A1 Course. Differential Cohomology I

#### Cohomological Algebra

- Modules and algebras.
- Complexes, differential algebras, differential ideals.
- Differential modules and their morphisms.
- Cohomologies, homotopies.
- Basic techniques in homological algebra.
- Spectral sequences and their morphisms.
- Spectral sequence associated with a filtered complex.

#### de Rham Cohomology

- de Rham complex.
- Compact support and relative de Rham complexes.
- Differential homotopy.
- Suspension theorem.
- Cohomological surgery.
- de Rham cohomologies of "simple" manifolds.
- Cohomological Newton-Leibnitz formula.
- Cohomological theory of integration for orientable manifolds.
- Degree of a proper map of manifolds.
- CW-complexes and group-valued differential cohomology.
- Cohomology of compact surfaces.

#### Algebraic theory of Linear Connections

- Modules and vector bundles.
- Der-operators in a module.
- Linear connections in a module.
- Parallel transport.
- Curvature of a connection.
- Simple operations with connections.
- Bianchi identities.

#### Cohomological Bundles

- Fiber bundles and their morphisms.
- Vertical morphisms and vertical vector fields.
- Vertical forms and the vertical de Rham complex.
- Cohomological bundle associated with a fiber bundle.
- A natural flat connection in the cohomological bundle.

#### Differential Leray-Serre Spectral Sequence

- Filtered complex associated with a fiber bundle.
- Differential Leray-Serre spectral sequence (DLSS).
- The term E
_{0} of the DLSS.
- The term E
_{1} of the DLSS.
- The term E
_{2} of the DLSS.
- Leray-Serre theorem and Künneth formula

#### Applications of the Leray-Serre Theory

- Suspension theorem.
- Thom isomorphism.
- Fixed point theorem.
- Intersection index.
- Cohomological theory of integration for non-orientable manifolds.
- Computation of cohomologies via Leray-Serre theorem.