Description: We try to follow the instruction given by Paul Erdős: Conjecture and prove! According to his spirit, the course intends to show the many and often surprising interrelations between the various branches of mathematics (algebra, analysis, combinatorics, geometry, number theory and set theory) and also to exhibit several unexpected mathematical phenomena via the active problem solving of the students.
Among others we plan to find answers to the following questions:
The main activity in our meetings is not lecturing. You will need to present your solutions to the assigned problems and to understand, check and discuss solutions of other students. To present your solutions you will need to have a drawing tablet or a normal tablet with a stylus.
A large part of the problems are pretty challenging, this is why strong problem solving skills are needed. The main goal of the course is not to improve but to use your already existing skills to reach deep, beautiful and surprising phenomena via problem solving.
Click here for the problem set of the first week to get an idea about the difficulty and the style of the problems.
Invariants for proving impossibility, applications in combinatorics and number theory
Irrational, algebraic and transcendental numbers, their relations to approximation by rationals
Vector spaces, Hamel bases, Cauchy's functional equation, applications
Countable sets, cardinalities, applications of Axiom of Choice and Zorn's Lemma
Isometries of the plane and the space
Isometries, geometric and paradoxical decompositions: Bolyai--Gerwien theorem, Hilbert's third problem, Banach--Tarski paradox
Finite and infinite games, winning strategies