Topologia:

Knowledge and understanding: students should demonstrate to have acquired a good understanding the basics of General Topology, homotopy and the fundamental group.

Applying knowledge and understanding: at the end of the course, students should be able to apply the basic techniques of topology for solving problems and exercises as well as in other areas. It is expected that students develop intuition about the most important topological spaces.

Making judgements: students should demonstrate self-evaluation skills on their understanding level of the topics of the course

Communication skills: students should be able to communicate, explain and present the notions and theorems learned in the course

Learning skill: students should be able to integrate the topics of topology that have been studied with the topics of other courses including those of MSc in Mathematics. It is expected that students develop a good level of independence in the study that is useful to understand the proposed textbooks and for making connections with the other branches of Mathematics.


Curve e Superfici:

KNOWLEDGE AND UNDERSTANDING: understanding the differential geometry
of curves and surfaces in the Euclidean 3-space.

APPLYING KNOWLEDGE AND UNDERSTANDING: carrying out exercises about regular curves and surfaces in the Euclidean 3-space; computing curvature, torsion and Frenet trihedron of a curve; computing curvatures and coefficients of fundamental forms of a surface.

LEARNING SKILLS: reading and understanding introductory texts concerning the differential geometry of curves and surfaces.

COMMUNICATION SKILLS: presenting in a correct and appropriate manner
definitions and theorems of the differential geometry of curves and
surfaces.

Calculus in one and more variables. Linear algebra, affine and Euclidean
geometry. Elementary notions of algebra.
The exams of Analysis 1 and 2, Algebra 1, Geometry 1 and 2 are propedeutical

Topologia:

General topology, covering spaces, homotopy, fundamental group.

Curve e Superfici:

Regular curves in R^3. Frenet formulas. Regular surfaces in R^3. First and second fundamental forms. Principal curvatures. Main curvature. Gaussian curvature. Theorema Egregium. Geodesics.

Topologia:

1) J. R. Munkres, Topology, Prentice Hall, 2000.

2) E. Sernesi, Geometria 2, Bollati Boringhieri, 2019.

3) I. M. Singer e J. A. Thorpe, Lecture Notes on Elementary Topology and Geometry, Springer-Verlag, 1967.

4) C. Kosniowski, A First Course in Algebraic Topology, Cambridge University Press, 2009.


Curve e superfici:

M.P. Do Carmo: Differential geometry of curves and surfaces, Prentice- Hall, 1976

M. Abate – F. Tovena: Curve e superfici, Springer Italia, 2006

Topologia:

General and algebraic topology: topologies and topological spaces, bases, closure, interior and frontier, limit points, subspaces, product topology, continuous maps and homeomorphisms, metric spaces, countability axioms, connected and path-connected spaces, compactness and limit point compactness, Aleksandrov compactification, compact metric spaces, separability axioms (Hausdorff spaces, regular spaces and normal spaces); homotopy and path homotopy, path and homotopy lifting properties, fundamental group, homotopic invariance and dependence on base point, fundamental group of the circle, of the spheres, of tori and projective spaces, retractions, non-retraction theorem and Brouwer fixed point theorem, Borsuk-Ulam theorem, homotopy equivalences and deformation retractions, homotopic maps and fundamental group. Free groups, group presentations and Van Kampen theorem. Computations of fundamental groups. Covering spaces and their classification, automorphisms of a covering, regular coverings and universal covering.




Curve e superfici:

Regular curves in R^3. Curvature and torsion. Frenet formulas.

Regular surfaces in R^3. Tangent plane. Differential of maps. Tangent and normal vector fields. First and second fundamental forms. Normal curvatures. Gaussian curvature. Mean curvature. Euler formula. Elliptic, hyperbolic, planar and parabolic points. Isometries. Christoffel symbols. Gauss' Theorema Egregium. Geodesics. Ruled surfaces. Minimal surfaces.

Topologia:

Lectures and exercises

Curve e Superfici:

Lectures. Significant exercises and examples will be explained during the lessons. Regularly we will assign to the students some exercises as
homework.

Information about the progress of the two modules and teaching materials will be posted on the website: http://moodle2.units.it

A tutoring activity will be organized.

Any changes to the methods described here, which become necessary to
ensure the application of the safety protocols related to the COVID19
emergency, will be communicated on the websites of the Department of
Mathematics and Geoscience - DMG and of the Study Program in
Mathematics.

Six possible dates. The final exam of each module consists of a written three hours exam, and of an oral exam, focused on theory and exercises.

The final score of the exam is the arithmetic mean of the grades obtained in the single modules.