AIMS OF THE PROPOSED COURSE: mastering basic topics in general and algebraic topology and in differential geometry
SYLLABUS:
1. General topology: The Tychonoff theorem, separation and countability axioms, The Urysohn lemma, The Tietze extension theorem, The Baire cathegory theorem
2. Fundamental group and coverings: homotopy, the fundamental group, covering spaces, the fundamental group of the circle, The Seifert vanKampen theorem
3. Simplicial complexes: geometry of simplicial complexes, barycentric subdivisions, the simplicial approximation theorem
4. Differentiable manifolds: definition and examples of manifolds, the tangent and the cotangent bundle, vector fields and differential forms, the algebra of differential forms, De Rham cohomology, inverse and implicit function theorems, submanifolds, integral curves, orientation, tensors, Riemannian structure
5. Homology: simplicial homology, De Rham's theorem
6. Riemannian geometry of surfaces: parallel transport, connections, structural equations, curvatures, geodesic coordinate systems, isometries, spaces of constant curvature
7. Surfaces in R3: interpretation of the Riemannian connection, curvature and parallel transport for surfaces embedded in R3. The second fundamental form.

 J. R. Munkres: Topology, 2nd edition
 I.M.Singer, J.A.Thorpe: Lecture Notes on Elementary Topology and Geometry
 V.Guillemin, A.Pollack: Differential Topology
 A.Hatcher: Algebraic Topology
 G.E.Bredon: Topology and Geometry
 W.S.Massey: A Basic Course in Algebraic Topology
 W. Fulton: Algebraic topology, a first course
 F.W.Warner: Foundations of Differential Manifolds and Lie Groups
 M.P.do Carmo: Differential geometry of curves and surfaces
 W.Kühnel: Differential Geometry, Curves  Surfaces  Manifolds
 J.M.Lee: Introduction to smooth manifolds
 J.M.Lee: Riemannian Manifolds, An Introduction to Curvature
