Syllabus
1. The origin of the concept of space in Renaissance painting and its transfer into projective geometry
2. Origins and some interesting results of non-Euclidean geometry
3. The origin of the concept a model, Beltrami-Klein's model and Klein's classification geometries
4. The birth of algebra and the lengthy journey to standardized algebraic symbolism
5. Solving algebraic equations of the 3rd and 4th degree by Cardano, and the notion of a resolvent
6. Origins of complex numbers and the fundamental theorem of algebra
7. The proof of insolvability of the problem of trisecting an angle, duplicating the cube and constructing of a regular heptagon
8. The birth of the concept of a group and the proof insolvability of the general equation of the fifth degree.
9. The emergence of the concept of integral and the discovery of its relation to differentiation.
10. Infinite series as approximations tools and as means to define new functions
11. Euler and infinite series in the complex domain
12. The concept of fractals and the notion of a fractional dimension
Annotation
The goal of the course is on the example of three disciplines -- geometry, algera and mathematical analysis -- to illustrate the changes in the understanding of the fundamental notions of mathematics during the 16th, 17th and 18the centuries. That enables us to understand the motives of the creation of modern abstract mathematics based on the abstract notions of space, group and continuity.