Illustrate, using a critical and unitary approach,ÿsome interesting and classical results and notions that are central for teaching mathematics in high school (focussing, principally, on arithmetics, geometry and algebra). The aim of the course is also to give a contribution to teachers training through the investigation on historical, didactic and cultural aspects of these topics.
teacher profile teaching materials
Based on the interests and requests of attending students, changes to parts of the program are possible.
Euclidean geometry: axioms, remarkable points in triangles, nine points circle, Morley's theorem, other theorems on triangles.
Affine geometry and barycentric coordinates, Ceva theorem, Menelaus theorem.
Projective geometry: axioms, the case of the plane over the finite field F2, Pappus and Desargues theorems, collineations and correlations.
Ordered geometry and the Sylvester problem on point collineation, generalizations.
Delaunay triangulations and Voronoi tassellation: properties and algorithms.
Ideals of polynomials, orderings of monomials and divisions between polynomials in several variables, Groebner bases.
Solving polynomial equations by elimination, by eigenvectors and eigenvalues, by resulting.
Polytopic geometry, mixed volume, Bernstein's theorem.
Materials, discussions, forum, videos on moodle https://matematicafisica.el.uniroma3.it
1) H.S.M. Coxeter Introduction to geometry, Wiley 1970;
2) D. Cox, J. Little, D. O’Shea Using Algebraic Geometry, GTM 185 Springer.
moreover
3) D. Cox, J. Little, D. O’Shea Ideals, Varieties, and Algorithms. An Introduction to Computational Algebraic Geometry and Commutative Algebra UTM Springer
4) M. Aigner, G. Ziegler, Proofs from THE BOOK, Springer, 1998;
5) S. Rebay, Tecniche di Generazione di Griglia per il Calcolo Scientifico-Triangolazione di Delaunay, slides Univ. Studi di Brescia;
6) B. Sturmfels, Polynomial equations and convex polytopes, American Mathematical Monthly 105 (1998) 907-922.
7) Shuhong Gao, Absolute Irreducibility of Polynomials via Newton Polytopes, J. of Algebra 237 (2001), 501-520.
Fruizione: 20410452 ME410 - MATEMATICHE ELEMENTARI DA UN PUNTO DI VISTA SUPERIORE in Matematica LM-40 SUPINO PAOLA
Programme
The program includes two intertwined courses: themes that have a didactic interest and more specifically computational applicative themes. Classical topics (Euclidean geometry, points and lines configurations ..) are chosen for their fallout in computer graphics, arguments of computational geometry are motivated by mathematical problems that have an elementary representation (Systems of polynomial equations in n unknowns ..).Based on the interests and requests of attending students, changes to parts of the program are possible.
Euclidean geometry: axioms, remarkable points in triangles, nine points circle, Morley's theorem, other theorems on triangles.
Affine geometry and barycentric coordinates, Ceva theorem, Menelaus theorem.
Projective geometry: axioms, the case of the plane over the finite field F2, Pappus and Desargues theorems, collineations and correlations.
Ordered geometry and the Sylvester problem on point collineation, generalizations.
Delaunay triangulations and Voronoi tassellation: properties and algorithms.
Ideals of polynomials, orderings of monomials and divisions between polynomials in several variables, Groebner bases.
Solving polynomial equations by elimination, by eigenvectors and eigenvalues, by resulting.
Polytopic geometry, mixed volume, Bernstein's theorem.
Materials, discussions, forum, videos on moodle https://matematicafisica.el.uniroma3.it
Core Documentation
1) H.S.M. Coxeter Introduction to geometry, Wiley 1970;
2) D. Cox, J. Little, D. O’Shea Using Algebraic Geometry, GTM 185 Springer.
moreover
3) D. Cox, J. Little, D. O’Shea Ideals, Varieties, and Algorithms. An Introduction to Computational Algebraic Geometry and Commutative Algebra UTM Springer
4) M. Aigner, G. Ziegler, Proofs from THE BOOK, Springer, 1998;
5) S. Rebay, Tecniche di Generazione di Griglia per il Calcolo Scientifico-Triangolazione di Delaunay, slides Univ. Studi di Brescia;
6) B. Sturmfels, Polynomial equations and convex polytopes, American Mathematical Monthly 105 (1998) 907-922.
7) Shuhong Gao, Absolute Irreducibility of Polynomials via Newton Polytopes, J. of Algebra 237 (2001), 501-520.
Type of delivery of the course
frontal lessons, plus debatesType of evaluation
oral presentation around a selected theme during the semester, plus a final oral examination