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ADVANCED GEOMETRY A

Degree course: 
Corso di Second cycle degree in MATHEMATICS
Academic year when starting the degree: 
2019/2020
Year: 
1
Academic year in which the course will be held: 
2019/2020
Course type: 
Compulsory subjects, characteristic of the class
Credits: 
8
Period: 
First Semester
Standard lectures hours: 
64
Detail of lecture’s hours: 
Lesson (64 hours)
Requirements: 

Linear algebra, general topology, advanced calculus as in the first two years of the undergraduate curriculum.

Final Examination: 
Orale

Oral examination

Assessment: 
Voto Finale

Students will acquire knowledge on some of the most important homotopic invariants of smooth varieties in arbitrary dimension and will acquire some general methods to calculate them in practice. They will also be introduced to the theory of vector or principal bundles and their topological invariants

Coomology of de Rham. Poincaré duality. Vector bundles and Thom isomorphism. Fasci. Cech cohomology and comparison with cohomology of de Rham. Characteristic classes of vector bundles.

De Rham complex on R^n. Differenzial forms with Compact support. The Mayer-Vietoris sequence. Differential manifolds, orientation and integration. Stokes' theorem. Poincaré lemmas.Finite dimensionality of de Rham cohomology. Poincaré duality in the orientable case. Kunneth's formula and Leray-Hirsch theorem. Poincaré dual of a closed sub-variety. Vector bundles and their cohomology. Thom isomorphism, Thom class and Poincaré duality. Cech-de Rham complex. Sheaves, Cech cohomology, Cech-deRham isomorphism. Sphere bundles. Euler characteristic and Hopf index theorem. Monodromy. Chern classes of a complex bundle. Splitting principle. Pontrjagin classes of real bundles.

R. Bott, L. Tu,
Differential Forms in Algebraic Topology,
3rd edition,
Springer 2010

Class lectures. Homework.

Professors

RE RICCARDO

Borrowers