Projective Geometry
Budapest
Semesters in Mathematics
Lecture Notes by Balázs
Csikós
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CONTENTS
Part
1. Introduction.
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The invention of perspective drawing.
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The real projective space, points at infinity.
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Topological structure of the real projective straight
line and plane.
Part
2. Linear spaces and the
associated projective spaces
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Groups, rings, division rings and fields.
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Vector spaces and their subspaces.
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Basis, coordinates, dimension.
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The projective space associated to a linear space.
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Projective coordinate systems.
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The theorems of Desargues and Pappus.
Part
3. Examples
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Projective spaces over finite fields.
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Complex projective spaces and the Hopf fibration.
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Digression: Stereographic projection and inversion.
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The stereographic image of the Hopf fibration.
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Quaternions.
Part
4. The axiomatic treatment
of projective spaces
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The incidence axioms of an n-dimensional projective space.
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The duality principle, the dual space.
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Desargues' theorem and the incidence axioms.
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Collineations.
Part
5. Desarguesian projective
spaces
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Construction of the division ring F.
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Construction of a collineation between P and FP2.
Part
6. The Fundamental Theorem
of projective geometry
-
The Projective General Linear group.
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Collineations induced by automorphisms of F.
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The Fundamental Theorem of projective geometry.
Part
7. Cross-ratio preserving
transformations between lines
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Cross-ratio.
-
Characterizations of cross-ratio preserving transformations
between straight lines.
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Cross-ratio preserving transformations between coplanar
lines.
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Cross-ratio preserving transformations of a line, involutions.