1. knowledge and understanding The student will become familiar with the basic notions and techniques of differential topology, specifically: the notions of smooth manifold, transversality, vector bundles and cohomology 2. applying knowledge and understanding The student will be able to use the theories and techniques taught in class for solving problems in differential topology and basic differential geometry 3. making judgements The student will be able to recognize the features of the considered problems and decide the appropriate techniques for approaching them 4. communication skills The student will be able to use the standard terminology and vocabulary from differential topology. 5. learning skills The student will be able to read advanced papers and textbooks on differential topology and integrate her/his knowledge.

Basic material of linear algebra, differential calculus, general topology.

Differentiable manifolds, tangent and cotangent spaces, vector bundles. Sard's Lemma and transversality. Vector fields. Differential forms and De Rham cohomology. Integration on manifolds and Stokes Theorem. Basics of Riemannian manifolds

- John M. Lee, "Introduction to Smooth Manifolds", Springer.
- M. W. Hirsch, "Differential topology", Springer.
- J. Milnor, "Topology from the differentiable viewpoint". Princeton
University Press, Princeton, NJ, 1997.
- Notes provided by the instructor

Differentiable manifolds and differentiable functions. Transversality and Sard's theorem. Tangent and cotangent spaces, vector bundles. Vector fields. Differential forms and tensor fields. Integration on manifolds and Stokes' theorem. De Rham cohomology.

Theoretical lessons and exercises sessions, including discussions on the exercises proposed in the lecture notes

The final exam is aimed at testing the knowledge of the subjects of the entire course program. It will be administered as an oral exam, touching on both theoretical aspects of the course, and the solution of exercises.