Topological spaces, open and closed sets, basis, closure, interior and boundary.
Subspace topology, Hausdorff spaces. Continuous maps: properties and constructions; Pasting Lemma. Homeomorphisms. Product topology.
Connected, path-connected and locally connected spaces. Lindelof and compact spaces, locally compact spaces, one-point compactification and Tychonoff's theorem.
Paracompactness and partitions of unity (if time permits).
Countability and separation axioms, Urysohn embedding lemma and metrization theorem for second countable spaces. Urysohn's lemma, Tietze extension theorem and applications. Complete metric spaces. Baire category theorem and applications.
Quotient topology: Quotient of a space by a subspace. Group action, orbit spaces under a group action. Examples of topological manifolds.
Topological groups. Examples from subgroups of GLn(R) and GLn(C).
Homotopy of maps. Homotopy of paths. Fundamental group.
Review of fundamental groups, necessary introduction to free product of groups, Van Kampen's theorem. Covering spaces, lifting properties, universal cover, classification of covering spaces, Deck transformations, properly discontinuous action, covering manifolds, examples.
Categories and functors. Simplicial homology. Singular homology groups, axiomatic properties, Mayer-Vietoris sequence, homology with coefficients, statement of universal coefficient theorem for homology, simple computation of homology groups.
CW-complexes and cellular homology, simplicial complex and simplicial homology as a special case of cellular homology, relationship between fundamental group and first homology group. Computations for projective spaces, surfaces of genus g.
Polynomial rings, Hilbert basis theorem, Noether normalization lemma, Hilbert Nullstellensatz, affine and projective spaces, affine schemes, elementary dimension theory, smoothness, curves, divisors on curves, Bezout's theorem, abelian differentials, Riemann-Roch theorem for curves.
Smooth manifolds: Manifolds in Rn, submanifolds, manifolds with boundary. Smooth maps between manifolds. Regular values. Examples of manifolds: curves and surfaces in R2 and R3, level surfaces in R1, inverse image of regular values.
Tangent spaces, derivatives of smooth maps, smooth vector fields, existence of integral curves of a vector field near a point.
Geometry of curves and surfaces: Parametrized curves in R3, length, integral formula for smooth curves, regular curves, parametrization by arc length. Osculating plane of a space curve, Frenet frame, Frenet formula, curvature, invariance under isometry and reparametrization.
Surfaces in R3: Existence of a normal vector of a connected surface. Gauss map. The notion of a geodesic on a surface. The existence and uniqueness of a geodesic on a surface through a given point with a given velocity vector thereof.
Gauss curvature and mean curvature. Gauss-Bonnet theorem (statement only).
Differential forms and orientation: Differential forms, orientation of manifolds, integration of forms, Stokes' theorem (proof to be given if time permits). Proof of Gauss-Bonnet theorem (if time permits).