Description:

This challenging course delves into intriguing and profound areas of mathematics through the solution of difficult problems. Guided by the spirit of Paul Erdős, born a century ago, who famously advocated "Conjecture and prove!", the course aims to reveal the numerous and often surprising interrelations among various branches of mathematics—algebra, analysis, combinatorics, geometry, number theory, and set theory. Students engage in active problem-solving, uncovering unexpected mathematical phenomena. Some of the questions we seek to answer include:

• Can the real function f(x) = x be expressed as the sum of a finite number of periodic functions?
• Does there exist a power of 2 whose first 2014 digits are all 7?
• Is it possible to partition a ball into a finite number of subsets (not necessarily "nice") and then reassemble two copies of the original ball?
• Can a rectangle of size 1 x π be tiled with squares?
• Is it possible to construct two subsets of the plane such that their union is congruent (in the usual geometric sense) to both subsets?

Topics covered in the course:

• Countable and uncountable sets, using invariants for proving impossibilities.
• Rational, algebraic, and transcendental numbers, and their connections to rational approximations and cardinalities.
• Vector spaces, Hamel bases, and the application of Cauchy's functional equation.
• Geometric and paradoxical decompositions, including the Banach-Tarski paradox.
• Cardinalities and the application of Zorn's Lemma.
• Finite and infinite games.