# 38 KAM Mathematical Colloquium

## Prof. VERA T. SOS

### Mathematical Institute, Hungarian Academy of Sciences, Budapest

## ORDER, DISORDER AND WHAT IS IN BETWEEN?

## (QUASI-PERIODIC STRUCTURES)

December 15, 2000 Lecture Room S6, Charles University, Malostranske nam. 25, Praha 1 13:00 (1 PM)

## Abstract

The study of sequences close to periodic goes back to Morse and Hedlund. The simplest examples are the $(\{n\alpha\})$ (and the $(\lfloor n\alpha\rfloor)$ (Sturmian) sequences. These sequences are fundamental in diophantine approximation, in ergodic theory etc..They have two -- seemingly contradictory -- features:

(a) these sequences have a very strict structure, are close to periodic in some well defined sense,

(b) they are randomlike in some sense and therefore can be used as quasi-random (deterministic) sequences.

In the last two decades quasi-periodic (almost periodic) structures, in higher dimension as well (e.g. the Penrose tilings in the plane) became important in many other fields, like in operation research, computer science, game theory, in physics, in the theory of quasi-crystals etc.. In this lecture we intend to indicate some of the large variety of links and connections, but we concentrate on results and problems with number-theoretical and combinatorial character.