Instructor: Dr. Tamás KELETI

Text: M. Laczkovich: Conjecture and Proof and handouts

Prerequisites: Introductory math courses

Description: We try to follow the instruction given by the 100 years ago born Paul Erdôs: Conjecture and prove!
We shall also see some of his favorite problems and enjoy some proofs from - what he called - ``The Book''.
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), to give an introduction to some basic methods of proofs via the active problem solving of the students, and also to exhibit several unexpected mathematical phenomena. Among others we plan to find answers to the following questions:

• Can the real function f(x)=x be written as the sum of (finitely many) periodic functions?
• Is there a power of 2 whose first 2014 digits are all 7?
• Can you ``cut'' a ball into finitely many (not necessarily ``nice'') subsets and ``reassemble'' two(!) copies of the original ball?
• Can you tile a rectangle of size 1 x pi with squares?
• Can you construct two subsets of the plane so that their union is congruent (in the usual geometric sense) to both subsets?

### Topics:

Pigeonhole principle, counting arguments, invariants for proving impossibility, applications in combinatorics and number theory.

Irrational, algebraic and transcendental numbers, their relations to approximation by rationals and to cardinalities.

Vector spaces, Hamel bases, Cauchy's functional equation, applications.

Isometries, geometric and paradoxical decompositions: Bolyai--Gerwien theorem, Hilbert's third problem, Hausdorff paradox, Banach--Tarski paradox.