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Department of Mathematics and Applications
"Renato Caccioppoli"


Geometry 2

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Geometry 2

Credits

9

Prerequisites

Geometry 1.

Scientific-disciplinary sector (SSD)

MAT/03 Geometry.

Examination method

Written exam (exercises and numerical problems, possibly multiple-choice) and oral exam.

Learning
objectives

The aims of the course are:

  • to develop the basic concepts of projective geometry and general topology;
  • to acquire a rigorous mathematical language;
  • to acquire the ability to solve standard exercises;
  • to acquire the ability to place the notions learnt in an applied context.

Syllabus

General topology. Definition of metric space, definition of topological space, open sets, closed sets, neighbourhoods. Topologies induced by a metric. Bases of open sets and of neighbourhoods. Continuous functions, homeomorphisms. Subspaces. Product topology and quotient topology. Separation axioms. Connectedness, path-connectedness. Compactness. Countability axioms. Sequences, convergence.
Projective spaces. Definition, frames of reference, subspaces and their representation, Grassmann relation. Homographies. Projective extension of an affine space, proper and improper subspaces, homogeneous coordinates. Affinities. Symmetric bilinear forms and quadratic forms. Quadrics of a complex projective space. Study of the affine, projective and metric properties of a real quadric. Polarities. Affine, projective and topological classification of a real quadric.

Expected learning
outcomes

At the end of the course, students must demonstrate that they

  • know and understand the contents listed in the syllabus, and have a good ability to present, discuss and contextualise them, including in areas other than geometry.
  • are able to apply the knowledge acquired through mastery of proof techniques and the ability to discuss possible applications of a theorem;
  • are able to communicate ideas and solutions clearly, rigorously and effectively to both specialist and non-specialist audiences;
  • are able to identify the most appropriate methods to analyse and solve a problem related to the course topics and to interpret the results correctly.

Learning outcomes
to be assessed

Assessment will cover the knowledge and skills acquired on the topics developed during the course, mastery of the mathematical tools introduced, presentation skills and appropriate use of language, the ability to apply the knowledge acquired to the solution of simple problems, and the ability to place that knowledge in applied contexts; finally, the ability to support a discussion with examples and counterexamples will be assessed.