Zdzisław Pogoda

Conversation with Andrzej Turowicz

Tyniec is a small village close to Krak6w, the ancient capital of Poland. On the bank of the river Wisia (Vistula), on a huge rock, a Benedictine Abbey was founded in the 11th century. The monastery from those days was rebuilt in the 15th and 17th centuries and nowadays is a great attraction for tourists. Several dozen friars and priests live in the monastery. One of them is a particularly interesting person, a Polish mathematician, a priest, and a monk, Professor Andrzej Turowicz. The 83-year-old priest leaves the monastery two or three times a year on special occasions. Any meeting he attends attracts many mathematicians who want to hear his stories. It is said that Professor Turowicz knew all the famous Polish mathematicians from the pre-war period. His excellent stories about mathematics and mathematicians of this time, coloured by interes t ing anecdotes , are l is tened to wi th bated breath. Andrzej Turowicz studied mathematics at the Jag ie l lonian Un ive r s i t y in Krak6w. He w o r k e d in Krak6w for some years and then in 1937 he moved to Lw6w (Lvov), where he started lecturing at the Technical University. In 1941 he came back to Krak6w and stopped working on mathematics. Immediately after the war he entered the Benedictine order and became a priest. In 1946 he started teaching mathematics at the Krzysztof Ciesielski (left), Andrzej Turowicz (center), and Zdzis~aw Pogoda (left) during the interview.

On Ordering the Natural Numbers or The Sharkovski Theorem

Note the rather transparent method in this madness. First come the odd numbers, be ginning with 3, arranged in increasing order. This sequence is repeated with each odd integer multiplied by 2. The initial sequence is again repeated with each odd integer multiplied by 22, and so on. The terminal sequence consists of the nonnegative powers of 2 arranged in decreasing order (note that 1=2?). We welcome the usual skeptical question of what this elaborate ordering is for. The integers have been arranged in a logical way, but it is not difficult to invent many other orderings. However, this particular ordering is an indispensable component of one of the most original and surprising theorems of the second half of the 20th century! What adds spice to the story is that the theorem was published in the 1960s and was virtually unnoticed for a number of years. Today this theorem is one of the classical results in the theory of dynamical systems and is known to many mathematicians specializing in other areas. Before stating the theorem we will talk some more about continuous functions and discrete dynamical systems. Suppose we are given a set and a mapping of this set into itself. These are the ingredients of a discrete dynamical system determined by the iteration of our mapping. From the viewpoint of their motions, the points of the set are divided into two classes. One class consists of the so-called periodic points, and the other of the nonperiodic points. A point is periodic if it returns to its initial position after a definite number of iterations and it is nonperiodic otherwise. It is not difficult to guess that the periodic points must be very important in a variety of investigations. Indeed, if a dynamical system describes a certain phenomenon or process, then it is important to know that a certain concrete situation will repeat itself. Note that the motion of a periodic point is completely determined as soon as we know its first few steps. Specifically, if a periodic point x returns to its initial position after n steps, then after n + \ steps it will be in the same position as after the first step. Formally, a point x is said to be periodic with respect to a function / if for some n the image of x under the n-fold composition of / is x; n is said to be a period of x.

Cauchy boundary andb-incompleteness of space-time

It is shown that if a space-time (M, g) is time-orientable and its Levi-Civita connection [in the bundle of orthonormal frames over (M, g)] is reducible to anO(3) structure, one can naturally select a nonvanishing timelike vector fieldξ and a Riemann metricg+ onM. The Cauchy boundary of the Riemann space (M, g+) consists of “endpoints” ofb-incomplete curves in (M, g); we call it theCauchy singular boundary. We use the space-time of a cosmic string with a conic singularity to test our method. The Cauchy singular boundary of this space-time is explicitly constructed. It turns out to consist of what should be expected.

Some Remarks on Popularizing Mathematics or a Magic Room

In this chapter, we first discuss activities for raising the public awareness of mathematics in Poland. Then, we write about the personal experiences of the authors concerning these activities. In the last and main part of the chapter we describe how the authors present mathematical terms and ideas to the general public, with a few examples.

Admissible operations and product preserving functors

In the paper lifts of tensor fields of type (1, k) and (0, k) and linear connections to a product preserving functor F are studied. A concept of “admissible” operations which send a finite sequence of tensor fields into a tensor field is introduced and a general procedure to prove formulas for lifts are proposed.