DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCES
Master's Degree in Mathematics
Academic Year 2019/2020 - 1° Year - DIDACTIC Curriculum and THEORETIC Curriculum

ELEMENTS OF ADVANCED GEOMETRY

Teaching Staff

Learning Objectives

• MODULO 1
  

  The student will learn methods of set theory that will be applied to general topology.

• MODULO 2
  

  Differential Geometry studies geometrical objects like curves, surfaces and manifolds using the tools of mathematical analysis. The course aims to introduce the basics of the theory of differentiable manifolds both in its global and local aspects. We will also study Riemannian manifolds, focusing on curves and surfaces in Euclidean space.

Course Structure

• MODULO 1
  

  Lectures with slides. Discussion and correction of exercises.

• MODULO 2
  

  There will be lectures and exercise sets. Students may be invited to present their solution to some of the exercises at the blackboard.

Detailed Course Content

• MODULO 1
  

  Introduction to set theory.Ordinals and cardinals. Filters and Ultrafilters. The space of all ultrafilters on the integers. Complements of general topology.

• MODULO 2
  
  1. Survey of general topology and multilinear algebra.
  2. Topological and differential manifolds.
  3. The tangent space and vector fields.
  4. Tensors in tangent space, tensor fields.
  5. The external derivative.
  6. Curves in space, arclength, curvature, torsion.
  7. Parametrical surfaces, First and Second fundamental form, gaussian curvature and Teorema Egregium, geodetics.
  8. Linear connection and Riemannian varieties.

Textbook Information

• MODULO 1
  

  1. Notes of the lecturer.

  2.Topologia di M. Manetti.

• MODULO 2
  

  There won't be a textbook, but the following books will be useful:

  1. M. Abate, F. Tovena, “Geometria Differenziale”, Springer Italia, Milano, 2011.
  2. Loring W. Tu, “An introduction to Manifolds”, Springer-Verlag, New York, 2011.
  3. M. Abate, F. Tovena, “Curve e Superfici”, Springer Italia, Milano, 2006.
  4. W. Boothby, “An introduction to differentiable manifolds and Riemannian Geometry”, Academic Press, 1986.