Affine and projective geometry

Completed course Algebra 1.

Affine Geometry: affine spaces, affine transformations, the fundamental theorem of affine geometry.
Projective Geometry: projective spaces, embedding of affine spaces into projective spaces, collineations and projectivities, the fundamental theorem of projective geometry, projective coordinates, cross-ratio, harmonic ratio, perspectivities.
Conics in projective plane: poles and polars, cross-ration on a conic, Pascal's Theorem, classification of conics.
Additional topics: classification of isometries in the Euclidean plane, Leonardo's Theorem, frieze groups and wallpaper groups, finite groups of isometries in Euclidean 3-space.

1. M. Berger: Geometry. 1, Berlin : Springer, cop. 1987.
2. M. Berger: Geometry. 2, Berlin : Springer, cop. 1987.
3. T. Košir, B. Magajna: Transformacije v geometriji, Ljubljana : Društvo matematikov, fizikov in astronomov Slovenije, 1997.
4. E. G. Rees: Notes on geometry, Berlin : Springer, 1983.
5. R. A. Rosenbaum: Introduction to projective geometry and modern algebra, Reading : Addison-Wesley, cop. 1963.
6. I. Vidav: Afina in projektivna geometrija, Ljubljana : Društvo matematikov, fizikov in astronomov SRS, 1981.

The main objective is to introduce affine and projective geometry using the tools from algebra and linear algebra. The student develops geometric intution.

Knowledge and understanding: The understanding of the fundamental notions of affine and projective geometry. The ability to apply the knowledge obtained in algebra and mathemetical analysis courses in geometry.
Application: The application of geometric techniques in other subjects and in practice.
Reflection: The ability to connect different approaches: analytical, algebraic and geometric.
Transferable skills: The ability to apply theoretical knowledge in practice.

Lectures, exercises, consultations

2 midterm exams instead of written exam, written exam
Oral exam
grading: 5 (fail), 6-10 (pass) (according to the Statute of UL)

Tomaž Košir:
BUCKLEY, Anita, KOŠIR, Tomaž. Plane curves as Pfaffians. Annali della Scuola normale superiore di Pisa, Classe di scienze, ISSN 0391-173X, 2011, vol. 10, iss. 2, str. 363-388. [COBISS-SI-ID 15928409]
BINDING, Paul, KOŠIR, Tomaž. Root vectors for geometrically simple two-parameter eigenvalues. Transactions of the American Mathematical Society, ISSN 0002-9947, 2004, vol. 356, no. 5, str. 1705-1726. [COBISS-SI-ID 13013081]
KOŠIR, Tomaž. Root vectors for geometrically simple multiparameter eigenvalues. Integral equations and operator theory, ISSN 0378-620X, 2004, vol. 48, no. 3, str. 365-396. [COBISS-SI-ID 12895321]
Aleš Vavpetič:
CENCELJ, Matija, DYDAK, Jerzy, VAVPETIČ, Aleš, VIRK, Žiga. A combinatorial approach to coarse geometry. Topology and its Applications, ISSN 0166-8641. [Print ed.], 2012, vol. 159, iss. 3, str. 646-658. [COBISS-SI-ID 16094809]
CENCELJ, Matija, DYDAK, Jerzy, MITRA, Atish, VAVPETIČ, Aleš. Hurewicz-Serre theorem in extension theory. Fundamenta mathematicae, ISSN 0016-2736, 2008, vol. 198, no. 2, str. 113-123. [COBISS-SI-ID 14551385]
VAVPETIČ, Aleš, VIRUEL, Antonio. Symplectic groups are N-determined 2-compact groups. Fundamenta mathematicae, ISSN 0016-2736, 2006, vol. 192, no. 2, str. 121-139. [COBISS-SI-ID 14185305]
VAVPETIČ, Aleš. Afina in projektivna geometrija. Ljubljana: samozal. A. Vavpetič, 2011. VI, 114 str., ilustr. [COBISS-SI-ID 15994969]