2019/2020

2020/2021

DM270

DEPARTMENT OF MATHEMATICS "FELICE CASORATI"

MATHEMATICS

PERCORSO COMUNE

2°

1st semester (01/10/2020 - 20/01/2021)

9

72 lesson hours

Italian

ORAL TEST

BONSANTE FRANCESCO (titolare) - 6 ECTS
SLAVICH LEONE - 3 ECTS

The contents of the Algebra 1, Geometry 1 and 2, Linear Algebra courses and of the Analysis courses of the first two years of the Laurea in Mathematics curriculum

The course is an introduction to the basic concepts and methods of differential geometry and topology

Differentiable manifolds: tangent and cotangent spaces, vector fields and differential forms, vector fields and coordinates: the Frobenius theorem, Lie groups and Lie algebras.
Topics in differential topology: Sard’s lemma, Whitney immersion theorems. Differential forms, exterior differential, de Rham cohomology, cohomology of forms with compact support. Poincaré lemmas. Mayer Vietoris sequence. Poincaré duality. De Rham Theorem.
Riemannian geometry: riemannian manifolds and Levi-Civita connections, curvature, geodesics, completeness, the theorems of Hopf-Rinow and Whitehead; Jacobi fields.
Complex manifolds (if time allows): holomorphic functions of several complex variables and their basic properties, meromorphic functions, complex manifolds, Kähler manifolds

Lectures

Notes by Gian Pietro Pirola.

Frank Warner: "Foundations of differentiable manifolds and Lie groups". Graduate Texts in Mathematics, 94. Springer-Verlag, New York-Berlin.

Bott Tu Differential forms and Algebra Topology, Graduate Text in mathematics 82. Springer Verlag,

Milnor, Topology From differentiable viewpoint, The University Press of Virginia, Charlottesville.

Manfredo Perdigao Do Carmo: "Riemannian Geometry", Birkhaeuser.
Boothby, William M.: "An introduction to differentiable manifolds and Riemannian geometry". Pure and Applied Mathematics, No. 63. Academic Press, New York-London, 1975.

Th. Broecker and K. Jaenich: "Introduction to differential topology".

Milnor, J.: "Morse theory". Annals of Mathematics Studies, No. 51 Princeton University Press, Princeton, N.J. 1963.

D. Huybrechts: "Complex geometry. An introduction". Universitext. Springer-Verlag, Berlin, 2005.

Oral exam