Elementary set theory through the classical axiomatization by Zermelo and Fraenkel. Mathematical paradoxes. Ruler-and-compass constructions. Paper folding constructions.Diphantine equations. Elliptic curves. Hilbert’s tenth problem.
M. Barlotti “Complementi di Algebra” – appunti disponibili nella pagina e-learning dell’insegnamento.
M. Davis “Computability and Unsolvability” Dover, 1982
Y. Matyasievich “Hilbert's Thenth Problem” MIT Press, 1993
S. Wagon “The Banach-Tarski paradox” Cambridge University Press 1986
Learning Objectives
Knowledge acquired:
The axioms by Zermelo e Fraenkel. The Banach-Tarski paradox. A characterization of the real numbers which can be constructed with ruler and compass. Some paper folding constructions. Theory of diophantine equations. Hilbert’s tenth problem
Competence acquired:
The axiomatic construction of set theory. An interpretation of mathematical paradoxes. Ruler-and-compass constructions. Paper folding constructions. Types of algebraic equations of which one seeks integer or rational solutions.
Skills acquired at the end of the course:
Constructing the number sets by axioms. Splitting a sphere in five pieces and reassembling them to build two isometric copies of the same sphere. Constructing numbers with ruler and compass. Trisecting an angle by paper folding. Managing algebraic equations of which one seeks integer or rational solutions. Some of the uses of elliptic curves.
Prerequisites
Courses recommended: Algebra I and II
Teaching Methods
Total hours of the course: 250
Hours reserved to private study and other indivual formative activities: 170
Hours for lectures: 80
Further information
Attendance of lectures, practice and lab:
Not compulsory, but strongly recommended.
Teaching tools:
Books. Sheets of paper.
Office hours:
Marco Barlotti: Tuesday 3:30 – 5:30 P.M., Thursday 3:30 – 5:30 and by appointment.
Virgilio Pannone: Tuesday 8:00 – 8:45 a. m., Friday 8:00 – 8:45 a.m. and by appointment.
Contact:
Marco Barlotti
Dep. of math. and inf. “U. Dini”
via delle Pandette, 9 - 50127 Firenze
Phone: 055 4374669
Fax: 055 4374913
E-mail: marco.barlotti@dmd.unifi.it
Virgilio Pannone
Dep. of math. and inf. “U. Dini”
Viale Morgagni 67/A - 50134 Firenze
Tel: 055 4237125
E-mail: virgilio.pannone@unifi.it
Web: http://e-l.unifi.it/course/view.php?id=1569
Type of Assessment
Written and oral examination
Course program
Set theory through the classical axiomatization by Zermelo and Fraenkel. Construction of the sets N, Z, Q. Mathematical paradoxes. Characterization of the numbers which can be constructed with ruler-and-compass. Geometric constructions by paper folding.
Introduction to Diophantine equations. Pythagorean equations. The Chinese remainder theorem. Polinomial equations in one indeterminate and rational coefficients. Elliptic equations, Poincaré group, Mordell rank. Examples of elliptic curves with large rank. Introduction to Hilbert’s tenth problem. Recursive functions and Diophantine
functions. Non-existence of a finite algorithm to
decide the existence of integer solutions of poly-
nomial equations in several indeterminates.