AN INTRODUCTION TO CONTEMPORARY MATHEMATICS

Foundations of calculus and algebra presented from the point of view of contemporary mathematics and its modern language.

PROGRAMME:

1. Sets, maps, equivalence relations, and the RSA algorithm.

2. Construction of numbers: natural, integer, rational, real,

3. p-adic, and complex.

4. From metric spaces to general topology.

5. The topology of metric spaces.

6. Basic notions of general topology.

7. Preorders and Alexandrov topology.

8. Examples and applications of preorders.

9. Limits of nets.

10. The concept of continuity.

11. Limits of maps at a point.

12. Limits of maps at infinity.

13. The topology of fields and other algebraic structures.

14. Supremum, infimum and completeness.

15. Limits and colimits in category theory.

The aim of this lecture course is to introduce undergraduate students into the world of contemporary mathematics. A training in an effective abstract thinking in solving concrete problems is proposed as a teaching method. This course is complementary to the standard calculus and algebra courses. It is focused on fundamentals in their broad mathematical context presented from the point of view and in the language of contemporary mathematics. Examples discussed during lectures will be supplemented by homework assignments.

The only prerequisites are the working knowledge of mathematics at an advanced pre-university level and a clear desire to understand and learn to handle not only standard, but also difficult topics. Criteria to pass the course will be adapted to the level of comprehension achieved by students. To pass the course, it is necessary to attend most of the lectures. The final grade will be based on the presentation of solutions of homework assignments.

Bibliography:

1. Wprowadzenie do topologii, Roman Duda, Biblioteka Matematyczna 61.

2. Categories for the Working Mathematician, Saunders Mac Lane, Graduate Texts in Mathematics 5.

3. General Topology, Chapters 5-10, Nicolas Bourbaki, Elements of Mathematics.