9
None.
MAT/03 Geometry.
Oral exam.
The aim of the course is to provide an introduction to differential geometry, algebraic geometry and algebraic topology. The most important results in these three fields will be discussed, and the main techniques of proof and problem solving will be illustrated.
Differential geometry. Topological and differentiable manifolds. Tangent vectors. Differentiable maps: diffeomorphisms, coverings, immersions, submersions and embeddings. Bump functions and partitions of unity. Submanifolds. Vector fields. Integral curves and the flow of a vector field. Vector bundles, sections and bundle morphisms. Local frames. The cotangent bundle. Line integrals. Tensors and tensor calculus. Differential forms, orientability and integration on manifolds. Complexes of R-modules and their cohomology, first properties. De Rham cohomology. Poincaré lemma. Mayer- Vietoris sequence. Cohomology of spheres. Hairy ball theorem. Brouwer fixed-point theorem.
Algebraic geometry. Affine space and algebraic closed sets. Zariski topology. Noetherian rings and the basis theorem. Gauss’s lemma and unique factorisation domains. Nullstellensatz. Plane curves. Regular points and tangent line to a curve. Multiplicity of a curve at a point. Fractions and local rings. Asymptotic expression of multiplicity. Intersection multiplicity of two plane curves at a point. Curves in the projective plane. Bézout’s theorem.
Algebraic topology. Categories, functors and natural transformations. The homotopy category. Deformation retracts and contractible spaces. Free abelian groups. Review of affine spaces and convex cells. Singular chains and their homology. Homology and path-connectedness. Chain complexes. Connecting homomorphism. Fundamental theorem of homological algebra. Outline of: homotopy invariance of homology, homological invariance of homotopy, excision theorem, Mayer-Vietoris theorem.
On completion of the course, students must demonstrate that they
Knowledge and skills acquired on the course topics, the student’s presentation skills and command of language, the ability to apply the knowledge acquired to the solution of simple problems, the ability to support a discussion with examples and counterexamples, and command of the mathematical tools used in the course.
