|Politecnico di Torino|
|Academic Year 2017/18|
Introduction to Algebra and Geometry
1st degree and Bachelor-level of the Bologna process in Mathematics For Engineering - Torino
The main goal of the course is to integrate basic notions acquired by the student in their first years, especially in the courses Analisi Matematica and Geometria. Elements of algebra and topology will be introduced, right after an introduction regarding the principles of mathematics.
Expected learning outcomes
- Basic language and main tools and techniques of the branches of mathematics described above.
- Ability to apply the concepts you learned in order to formalize simple algebraic and topological problems.
- Ability to apply the concepts learned in order to solve simple algebraic and topological problems
- Knowledge and understanding of the basic language, notations, and main results that allow one to be able to read and understand algebra and topology textbooks.
Prerequisites / Assumed knowledge
A working knowledge of the mathematical tools presented in the courses Analisi Matematica I and Algebra Lineare e Geometria is required.
ALGEBRA (5 cfu)
Correspondences and functions.
Equivalence relations and partial orders.
Axiom of choice and Zorn's lemma.
Countability of a set: some notions on cardinality.
Induction principle, weak and strong version.
Integer numbers, congruence classes, prime numbers, factorial, euclidean algorithm.
Groups, subgroups, normal subgroups: examples (finite abelian groups, matrix groups, dihedral group, symmetric group).Group homomorphisms.
Rings, subrings, ideals: examples (integer numbers, ring of polynomials, ring of square matrices). Ring homomorphisms.
Fields and division rings: examples (Q, R, C, H, O, field of rational functions).
Finite and infinite fields, algebraic and transcendental extensions.
Some ideas of cryptography.
TOPOLOGY (5 cfu)
Metric spaces and their properties. Continuous functions and isometries.
Topological spaces. Neighborhoods and closed sets. Hausdorff spaces. Induced topology. Topology associated to a metric function.
Topology bases and neighborhood bases. Interior of a set, closure, derived set and boundary set. Dense sets. Limits and closure in metric spaces. Continuous function, open and closed functions, homeomorphisms. Separation properties.
Connectedness, connected components. Product of connected spaces. Path-connection.
Compactness. Product of compact spaces. Heine-Borel theorem and sequential compactness.
Cauchy sequences in metric spaces, complete metric spaces and completeness of R and R^n. Quotient topology. Projective spaces. Topological manifolds and surfaces.
Lectures and exercise sessions will be carried out by the teacher.
Texts, readings, handouts and other learning resources
A. Conte, L. Picco Botta, D. Romagnoli, Algebra, Levrotto e Bella.
M. Artin, Algebra, Bollati Boringhieri.
P. Shick Topology, Wiley.
C.Kosniowski Introduzione alla topologia algebrica, Zanichelli.
V. Checcucci - A.Tognoli - E. Vesentini, Lezioni di topologia generale, Feltrinelli.
S. Lipschutz, Topologia: teoria e problemi di... Collana Schaum 39. ETAS
B. Mendelson Introduction to topology, Dover.
T. Gamelin, Introduction to topology, Dover
Assessment and grading criteria
The exam will check your knowledge of the course material, and your ability to apply theories and methods to solve problems and prove simple statements.
The exam will consist in a compulsory written test, followed by a compulsory oral test.
During the written test you cannot bring any kind of books/notes/lecture notes/computation or communication tools. The written test has two 60 minutes sections, an algebra and a topology one. For each section you will need to solve some exercises (usually 2 or 3), each structured in several parts. You pass the written test if you obtain a score of at least 15/30 on each of the two sections.
You have to take the oral test in the same exam session as the written test.
The oral test has two sections as well, an algebra and a topology one. The oral test is intended to test your knowledge of the theory discussed in the course, and it includes the correction of the written part.
For each section of the oral test you obtain a score in the range between -15/30 and 15/30.
For both the algebra and the topology section you obtain a score calculated as the algebraic sum of the scores in the corresponding written and oral test.
You pass the exam if you get a grade that is less or equal 18/30 in both the sections. The final grade is the rounded up average of the grades in the two sections of the exam.
Programma definitivo per l'A.A.2017/18