Affine and projective geometry, Autumn 2011

Instructor: Vladimir Dotsenko

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This module is intended to introduce students to basic concepts of geometry and use those concepts to demonstrate how geometry and algebra can interact in a mutually beneficial way. The course will be accompanied with exercise classes; the work in class is not assessed but is extremely important as it prepares the students to approaching questions from home assignments and the final exam paper.

Syllabus

  • Geometric objects in two and three dimensions. Affine and projective plane: axiomatic approach.
  • Higher dimensions. Coordinate vector space and linear transformations.
  • Affine spaces and transformations. Convexity in affine spaces.
  • Projective spaces and transformations.
  • Coordinate systems. Change of coordinates. Proving theorems by change of coordinates. Example of a projective invariant: cross-ratio.
  • Affine and projective classification of plane quadrics. Euclidean classification. Geometric definitions of ellipse, hyperbola and parabola.
  • Complex numbers and quaternions, and their geometric interpretation.

Materials

... are available via the "Moodle" platform to which all students have access. Be warned that any handouts you might get hold of via "Moodle" or otherwise are incomplete, and by no means can they serve as a substitute for attending classes.

Assessment

There will be several home assignments of varying lengths, depending on the topics covered. However, the primary method of assessment is via a written examination. The final mark is computed as MAX(F,0.8F+0.2C), where F is the percentage earned in the final exam, C is the percentage earned by means of continuous assessment (home assignments). Thus, if a student does well in the final exam paper, their possible poor performance during the semester would not affect the final mark. However, since the final exam paper questions are modelled on questions from home assignments, students are strongly recommended to attempt as many problems from home assignments as they can.

Literature

No specific textbooks are recommended; some handouts and xerox copies of relevant literature will be disseminated in class.