Modern Geometry

Course code

MLM6224.DT

old course code

MLM6224

Course title in Estonian

Kaasaegne geomeetria

Course title in English

Modern Geometry

ECTS credits

4.0

approximate amount of contact lessons

52

Teaching semester

autumn

Assessment form

Examination

lecturer of 2019/2020 Autumn semester

Mart Abel (eesti keel) tavaline kursus

lecturer of 2019/2020 Spring semester

lecturer not assigned

Course aims

To introduce notions and theorems from modern geometry and their applications. To deepen the knowledge of students about the geometry. To demonstrate the connenctions between geometry and other branches of mathematics.

Brief description of the course

Axiomatic structure of mathematics. Euclidean geometry based on the axiomatics of Hilbert. Absolute geometry. Affine geometry. Projective geometry: perspective, projective plane, projective space, Theorem of Desargues, ratio of division of the segment, cross ratio, homogenuous linear coordinates and duality.

Lobachevsky geometry: angle of parallelism, direction of parallelism, equidistant, horocycle, horosphere.

Spherical geometry, problems of navigation, maps, applications of stereographic projection, elliptic geometry, spherical trigonometry, hyperbolic trigonometry.

Geometry based on the axiomatics of Weyl: euclidean space, pseudo-euclidean space, hyperbolic space.

Erlanger Programme of Klein.

Topology and its connections with geometry.

Lobachevsky geometry: angle of parallelism, direction of parallelism, equidistant, horocycle, horosphere.

Spherical geometry, problems of navigation, maps, applications of stereographic projection, elliptic geometry, spherical trigonometry, hyperbolic trigonometry.

Geometry based on the axiomatics of Weyl: euclidean space, pseudo-euclidean space, hyperbolic space.

Erlanger Programme of Klein.

Topology and its connections with geometry.

Independent work

Work with the lecture materials and literature.

Learning outcomes in the course

Knows the basic notions and main results of euclidean geometry; knows the basic notions and main results of projective geometry; knows the basic notions and main results of hyperbolic geometry; knows the basic notions and main results of spherical geometry; knows different axiomatic approaches to the geometry; is able to prove some theorems about the subject.

Assessment methods

Written test. The mark is evaluated by the score of maximum of 100 credits. These 100 credits are given as maximum 50 credits for the test during the semester and as maximum 50 credits for the test after the end of the lectures. The mark is positive if the student gathers at least 51 credits during the tests.

Teacher

prof Mart Abel

Study literature

K. Ariva, Lobatševski geomeetria, Valgus 1992;

M. Väljas, Analüütiline geomeetria, TTÜ kirjastus, 2012;

R. Kolde, M. Väljas, Teisenduste rühmad geomeetrias Valgus 1991.

M. Väljas, Analüütiline geomeetria, TTÜ kirjastus, 2012;

R. Kolde, M. Väljas, Teisenduste rühmad geomeetrias Valgus 1991.

Replacement literature

G. A. Jennings. Modern geometry with applications, Springer, 1994

David A Thomas. Modern geometry, Grooks/Cole 2002.

H. S. M. Coxeter. Non-euclidean geometry, The Mathematical Association of America, 2002.

H. S. M. Coxeter. Introduction to Geometry, Wiley, 1989.

David A Thomas. Modern geometry, Grooks/Cole 2002.

H. S. M. Coxeter. Non-euclidean geometry, The Mathematical Association of America, 2002.

H. S. M. Coxeter. Introduction to Geometry, Wiley, 1989.