Mathematics

GUIDO PEZZINI
VITO FELICE ZENOBI

General objectives: to acquire basic knowledge in general topology, with an introduction to
algebraic topology and differential geometry.

Specific objectives:

Knowledge and understanding: At the end of the course the student will have acquired the concepts and the results
basic general topology, with various possible approaches to the notions of topological space,
continuous application, homeomorphism; then constructions of topologies on subspaces, products and
quotients, topological properties of separation, numerability, compactness, and connection
connection for arches. The student will also have acquired the notion of fundamental group and the its use together with the relevant calculation techniques, and the fundamental elements of the theory of topological coatings. Finally, the student will have acquired the basics of geometry differential of curves and surfaces in three-dimensional Euclidean space.

Apply knowledge and understanding: At the end of the course the student will be able to solve
simple topology problems, even with the use of elementary algebraic topology. He will also know use the notion of curvature in the contexts of the differential geometry of the curves and of the surfaces.

Critical and judgmental skills: The student will have the basis for analyzing the similarities and relations between
the topics covered and the fundamental notions of the theory of continuity and differentiability,
also with tools that have historically led to the solution of classical problems.

Communication skills: Ability to expose the contents in the oral part of the verification and in the any theoretical questions present in the written test.

Learning ability: The acquired knowledge will allow a study, individual or given in a subsequent three-year or master's degree course, related to more advanced aspects of geometry.

Knowledge of basic general topology (definitions, connected spaces, compactness, countability axioms, quotient topology, etc.), basic algebraic topology (fundamental group and covering spaces), and differential geometry (manifolds, curves on the plane and in space, surfaces in space with fundamental forms and curvatures). Being able to analyze concrete examples of topological spaces, and whether they satisfy the properties seen in the course. Being able to prove statements or formulas in general topology, simple or similar to those seen in the course. Being able to compute fundamental groups of concrete examples of topological spaces, or prove properties of these groups based on topological properties of the spaces. Being able to compute curvatures and other differential properties of curves and surfaces, and deduce topological properties from differential properties.

Topics of the courses Geometria I, Analisi I and Algebra I. Familiarity with baic linear algebra as introduced in the course Linear Algebra (vector spaces, linear maps, etc.).

First part: general topology

Topological spaces and continuous maps.
Subspaces, products, quotients.
Topological properties: separation, countability properties, compactness, connectedness, path connectedness.
Metric spaces.
Topological manifolds.

Second part: introduction to algebraic topology

Homotopy of maps and path homotopy.
Fundamental group, homotopy invariance.
Theorem of Van Kampen, applications.
Covering spaces and fundamental groups.
Monodromy.
Universal cover.

Third part: introduction to differential geometry of curves and surfaces.

Differentiable curves in two and three dimensional euclidian space.
Tangent line, curvature, torsion, theorem of rigidness.
Differentiable surfaces.
Tangent plane, first fundamental form.
Second fundamental form.
Gaussian curvature and Theorema Egregium.

M. Manetti, Topologia, Springer
E. Sernesi, Geometria II, Boringhieri
A. Hatcher, Algebraic topology, Cambridge University Press
M. Abate e F. Tovena, Curve e Superfici, Springer

M. Manetti, Topologia, Springer
E. Sernesi, Geometria II, Boringhieri
A. Hatcher, Algebraic topology, Cambridge University Press
M. Abate e F. Tovena, Curve e Superfici, Springer

Lectures (60%), recitation classes (40%).

Lectures (60%), recitation classes (40%).

The exam will have a written part (consisting of exercises similar to those seen during the semester) and an oral part (on the most relevant results seen in the course).
The minimum required note is 18/30.

Study properties of topological spaces explicitly defined (e.g. subspaces or quotients of known spaces).

Computation of fundamental groups of given topological spaces.

Computation of curvature and torsion of curves, and of fundamental forms of surfaces.

• Topologia generale
  • Books: M. Manetti, Topologia, Springer


• Topologia algebrica
  • Books: M. Manetti, Topologia, Springer


• Curve e superfici
  • Books: E. Sernesi, Geometria II, Boringhieri