Subject: GEOMETRY B (A.A. 2021/2022)
Unit Geometria B - mod I
Theoretical Studies (lesson)
Unit Geometria B - mod II
Theoretical Studies (lesson)
The aim of the Course is to complete some concepts of Linear Algebra that were given in previous Courses ("Algebra Lineare" and "Geometria"); to introduce the main notions of Projective Geometry and present a treatment of conics and quadric surfaces in projective, affine and Euclidean spaces; to introduce to the major notions of General Topology, with some hints to Algebraic Topology.
The basic notions of Linear Algebra (algebraic structures, matrix calculus, vector spaces, linear maps, eigenvalues and eigenvectors) and of Affine and Euclidean Geometry.
Classes of linear operators and matrices. Review of bilinear and quadratic forms. Congruence of symmetric matrices. Sylvester's theorem for real quadratic forms. (3CFU)
Projective spaces and subspaces. Projectivities and homographies. Homogeneous coordinates. Equations of a projectivity. Change of homogeneous coordinates. Cartesian representation of subspaces. Projective extension of an affine space. Proper and improper points. Cross ratios, harmonic quadruples. Fixed elements of a projectivity. Classification criteria for homographies; the characteristic of a plane homology. (3 CFU)
Quadrics of projective spaces. The rank of a quadric. The polar hyperplane of a point. Intersection of a quadric with a straight line. Double points. Tangent hyperplane. Projective classification. Parabolic, hyperbolic and elliptic points. Quadrics of affine spaces. Diametral hyperplanes and the center of a non-specialised quadric. Affine classification. Quadrics of Euclidean spaces. Principal hyperplanes. Axes. Canonical Forms. Focuses and directrices of a non-degenerate conic. Sheaves of conics. (3 CFU)
Elements of general topology. Internal, external and boundary points; adherent, isolated and accumulation points. Metric spaces. Continuous applications; homeomorphisms. Subspaces, products and quotients. Topological models of the real projective space. Separation axioms. Hausdorff spaces; limits of function; uniqueness of the limit. axioms of Countability (outline). Compactness. Theorems of Heine-Borel and Bolzano-Weiertrass. Local compactness. Alexandroff compactification. The n-sphere as the Alexandroff compactification of R^n. Connectedness and arc-connectedness. Jordan theorem. Homotopy of paths; fundamental group; simply connected spaces. Topological manifolds; hints on the classification of compact surfaces. (6 CFU)
The course consists of lectures and exercises in which concepts and theoretical methods are applied. On the MOODLE platform of this Course, students will find examples of exercises with full solutions, examination tests and exercises for self-assessment.
The final exam consists of a 2-hour written test followed by an oral interview. Admission to the oral interview is achieved with a score of 18/30 or more in the written test. Students who pass the written test but do not pass the oral interview must begin the proceess again, starting from the written test, at a later round of exams The written test requires the resolution of exercises of standard type, very similar to those presented during the lectures. The oral interview consists in a discussion of the main concepts of the course, definitions, main results with their proofs and, above all, a logical connection among the various notions. It will be possible to replace the written test with two partial tests given during the course: full details will be given "in progress".
Through lectures and individual study, knowledge of the main concepts presented during the course.
Through classroom exercises and individual work, ability to solve standard problems, similar to those presented in the classroom.
Attitude towards a methodological approach that leads to verify claims and methods through rigorous arguments.
Capabilities towards self-assessment of student skills.
Ability to deal with a dialectical discourse in a timely and consistent way, arguing with precision.
Acquisition of a permanent mathematical knowledge to be used at any time of the student's cultural journey.
Attitude towards a methodological approach, leading to an improvement in the method of study and to deeper learning capbilities.
C. GAGLIARDI, L. GRASSELLI, Algebra lineare e Geometria, Vol. 3, Società Editrice Esculapio, Bologna, 1993;
M.R.CASALI, C. GAGLIARDI, L. GRASSELLI, Geometria, Società Editrice Esculapio, Bologna, 2016;
E. SERNESI, Geometria I-II, Bollati Boringhieri, Torino, 1989-1994.
C. KOSNIOWSKI, Introduzione alla Topologia Algebrica, Casa Editrice Zanichelli, Bologna, 1988;