**Kazimierz Urbanik** was born in Krzemieniec in Eastern Poland. He was brought up there and entered the famed educational establishment of the city, the Lyceum. However World War II was to cause a major disruption throughout his school years.

On 1 September 1939, after signing a non-aggression pact with the Russians, the German armies invaded Poland. On 17 September the Russians invaded and by 28 September Hitler and Stalin had agreed to partition Poland. Krzemieniec, by this agreement, came under Soviet control. A couple of months later it became part of Soviet Ukraine. However, in June 1941 the Germans attacked the Soviet Union and Krzemieniec eventually came under German control. By 1944 the Soviet armies had driven the Germans out and the Yalta Conference in February 1945 saw the district become part of the Ukraine (which it still is). However the Yalta agreement saw the town of Brzeg, 50 km south-east of Wroclaw in Lower Silesia, being returned to Poland. The Urbanik family set up home in Brzeg.

Completing his secondary education in Brzeg in 1948 by passing the matura, Urbanik entered the University of Wroclaw. There he studied a wide range of scientific topics, but he concentrated on mathematics and physics for his final degree which he obtained in 1952. Two of his lecturers at Wroclaw had a major influence on the direction of his studies, namely Hugo Steinhaus and Edward Marczewski. These two mathematicians had played a large part in the development of the Polish University of Wroclaw which had been set up in August 1945, built on the previous University and Technical University of Wroclaw. Lectures had begun at the new Polish University of Wroclaw on 15 November 1945. Marczewski, who had been held as a prisoner by the Germans in Wroclaw during their wartime occupation, had been appointed as a professor and Steinhaus had been appointed as Dean of the Faculty of Mathematics, Physics and Chemistry.

After graduating Urbanik was employed to teach at the university while he studied for his doctorate with Marczewski as his thesis advisor. His first work was on topology and he published *Sur les espaces complets séparables de dimension 0* Ⓣ in 1953 which was a joint paper with B Knaster. Gordon Whyburn writes:-

An orderly presentation is given of results, mostly new, concerning the topological structure and classification of ... the0-dimensional separable complete spaces.

Urbanik then began to mix an interest in topology with measure theory and probability and his 1954 papers show this mix: *Sur un problème de J F Pàl sur les courbes continues* Ⓣ; *Limit properties of homogeneous Markov processes with a denumerable set of states*; *Sur la structure non-topologique du corps des opérateurs* Ⓣ; and* Quelques théorèmes sur les measures* Ⓣ. Let us give Paul Halmos's description of this last mentioned paper:-

This paper is motivated by the measure-theoretic formulation of the Steinhaus cake problem. The problem is to divide a cake among n people, with possible different standards of value for the various parts of the cake, so that the value of each person's share is, in his own estimation, at least1/n times the value of the whole cake. In the author's mathematization of the problem it is assumed that the n normalized measures that enter are non-atomic, have the same null-sets, and are not all identical. The main theorem asserts that the cake can be divided so that the value of each person's share is, in his own estimation, exactly p times the value of the whole cake, where p is strictly greater than1/n, and, moreover, a best possible p can be attained. The number of distinct partitions of this kind, with the same optimal p, is either1or else greater than or equal to the power of the continuum.

During the years 1955 and 1956, sixteen of his papers appeared in print, some written in English, some in Polish, some in French, some in German and some in Russian. He was awarded his doctorate in 1956 for his work on a stochastic model of a cascade. He followed a suggestion of Marczewski and gave a model for a temporally homogeneous cascade process where individual lines of descent can be distinguished. Among the results he proved were:

(1) Every cascade process defined in extensive form determines a temporally homogeneous Markov process whose probabilities satisfy the usual relations for cascade processes; and

(2) Every Markov process of the type mentioned can be generated by an appropriate extensive cascade process.

Urbanik remained at the University of Wroclaw and habilitated there in 1957 becoming a dozent. His work on generalized stochastic processes is of great importance and he published many papers of the topic such as the two 65 page papers *Generalized stochastic processes* and *Local characteristics of generalized stochastic processes* both published in 1958. The depth and quantity of his research meant that he was soon promoted to professor, which happened in 1960 following his visit at Tulane University in New Orleans in the United States, in the academic year 1959-60.

The article [3] divides Urbanik's research into five different major areas: topology, measure theory and analysis; probability theory; stochastic processes; information theory and theoretical physics; and general algebras. In the light of the comments we have already made about Urbanik's research, we need only explain where his interest in the last two of these five areas arose. Although information theory and theoretical physics might sound like a topic unrelated to his other interests, in fact most of the work was related to statistical physics and was closely linked with his important results on generalized stochastic processes. His work on general algebras was, however, essentially unrelated to the other topics he studied. His interest here came through his mentor Edward Marczewski who, in 1958, introduced the concept of independence in universal algebras. Urbanik published a number of highly significant papers on this topic such as *Representation theorem for Marczewski's algebras* (1959), *A representation theorem for Marczewski's algebra* (1960), (with E Marczewski) *Abstract algebras in which all elements are independent* (1960), and *Linear independence in abstract algebras* (1966). We have spoken of the five areas in which he undertook research, but we should also mention the large number of research papers he wrote; the article [3] list 181 such papers.

In 1965 Urbanik was elected to the Polish Academy of Sciences, later serving two terms as Vice President. He was Director of the Institute of Mathematics of the Wroclaw University from 1967 to 1978 and again from 1981 to 1996. From 1975 to 1981 he served as Rector of the University. He was a fine teacher [2]:-

As a teacher Urbanik developed a large and faithful following. His delivery was crisp and velvety, and we were all mesmerized by his lectures in which deep theories unfolded effortlessly in front of our eyes without any help from notes or textbooks. He had developed original approaches to almost every subject he lectured on and we regret that most of his course offerings were never converted into published textbooks.

The authors of [3] describe his character:-

Urbanik's fairness, warmth, generosity and devotion to students were legendary and they reciprocated in kind. He loved doing and teaching mathematics and, despite his long and incapacitating illness about which he never complained, continued working with the students, publishing and fulfilling his editorial duties almost to the last days of his life.

Urbanik married Stefania; they had a son Witold and a daughter Jadwiga. He retired in 2000 but continued to undertake research.

**Article by:** *J J O'Connor* and *E F Robertson*