Instructor: Dr. Róbert FREUD

Text: M. Laczkovich: Conjecture and Proof and handouts

Prerequisites: Introductory math courses

Description: We try to follow the instructions of Paul Erdôs: Conjecture and prove! We shall see also 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:


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 and fields, field extensions, geometric constructions (doubling the cube, squaring the circle, trisecting the angle, constructing regular polygons), Hamel bases, Cauchy's functional equation, finite fields.

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

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