# Dončo Dimovski

### Born: 1 October 1954 in Bitola, Yugoslavia, now Northern Macedonia

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**Dončo Dimovski**was born in Bitola, the second largest of the cities of Macedonia situated about 15 km north of the border with Greece. We note that when Dimovski was born, Macedonia was officially part of the People's Federal Republic of Yugoslavia and was one of the six republics of the Yugoslav federation. In 1963 it was renamed the Socialist Republic of Macedonia, part of what was then named the Socialist Federal Republic of Yugoslavia. Macedonia declared independence in September 1991, calling itself the Republic of Macedonia. Since June 2018 it has been agreed that the country will be called Northern Macedonia to distinguish it from the Macedonian region of Greece.

In was in Bitola that Dimovski completed both his primary and secondary education, graduating from the high school in 1973. At the beginning of the academic year 1973-74 he enrolled in the Faculty of Natural Sciences and Mathematics in the University of Skopje. This University had been founded in 1946 but it was almost completely destroyed in an earthquake in 1963 and so the university that Dimovski entered was a large modern one completely rebuilt after the earthquake. His outstanding abilities in mathematics had been demonstrated during his school years in Bitola, so it was natural for him to enrol in the Mathematical Group of the Faculty of Natural Sciences and Mathematics.

Dimovski graduated in June 1976, having competed the course in less than three years, continuing to demonstrate his outstanding abilities with an average grade of 9.48. His undergraduate project had been supervised by Georgj Čupona who was interested in algebra, particularly in general algebraic systems and group theory and its generalisations. At the time that he was advising Dimovski, Čupona was working on representations of universal algebras and on quasigroups. Dimovski's project,

*Additive semigroups of integers*(Macedonian), was an excellent piece of work and was published in 1977 by the Macedonian Academy of Sciences and Arts. In this work Dimovski studied decompositions of the semigroup of positive integers under addition. He continued his studies at the University of Skopje, undertaking research for his Master's Degree advised by Smilka Zdravkovska.

Smilka Zdravkovska was born in 1947 in Skopje. She won many mathematics prizes in the competitions she entered while at school. She studied at the University of Moscow from 1964 to 1969 and, returning to Skopje, she taught at the Mathematical Institute of the University while studying for her Master's degree. She went to England in January 1971 and, advised by Frank Adams, wrote the Ph.D. thesis

*Topological Objects in Homotopy Theor*y, being awarded the degree in 1973. In January 1974 she returned to the University of Skopje and, two years later, she began supervising Dimovski's Master's Degree. Dimovski worked as an associate at the Institute of Mathematics in Skopje in 1976 and 1977. From 1977 to 1980 he worked as an assistant at the Faculty of Mathematics. He submitted his thesis

*P-*1

*localization-related CW-complexes*and, in 1979, he completed his master's degree in Mathematics with the defence of his thesis. We note that in September 1979 his advisor Smilka Zdravkovska went to the United States where she later married Allen Shields.

In September 1980 Dimovski enrolled in doctoral studies at the State University of New York at Binghamton, USA. His thesis advisor was Ross Geoghegan who had obtained his Ph.D. at Cornell University in 1970 with the thesis

*Topological and Simplicial Properties of Function Spaces, and a Stability Theorem for Spaces of Homeomorphisms and Embeddings*. Geoghegan was born in Dublin, Ireland, in 1943 and had studied at University College, Dublin, before undertaking research for his Ph.D. at Cornell. He was appointed to the State University of New York at Binghamton in 1972. Dimovski submitted his Ph.D. thesis

*Non-simply connected Casson Handles*and defended this doctoral dissertation in May 1983. His dissertation was awarded a prize as the best in the year at the State University of New York at Binghamton. His 160-page thesis was published in English in 1984 with a Macedonian preface and summary. In fact, before submitting his thesis, Dimovski had published another paper, this one in English with a Macedonian summary, in 1982 entitled

*Some existence conditions for vector valued groups*. This paper gave partial answers to questions asked by Georgj Čupona concerning the existence of finite (

*n*,

*m*)-groups and about the embeddability of commutative, cancellative (

*n*,

*m*)-semigroups in commutative (

*n*,

*m*)-groups, when

*m*is not a divisor of

*n*.

Returning to Macedonia, Dimovski was appointed as an assistant professor in the Faculty of Mathematics and Natural Sciences of the University of Skopje in 1984. He attended the fourth Yugoslav algebraic conference "Algebra and logic" held in Zagreb 7-9 June 1984 and published the paper

*On a class of vector-valued groups*in the Proceedings of the conference, the paper being written jointly with Georgj Čupona. He published

*Examples of vector valued groups*in the 1985 volume of the Macedonian Academy of Sciences and Arts which only appeared in print in 1987. He also attended the conference "Algebra and Logic", held in Cetinje, Montenegro, 12-14 June 1986, publishing two papers in the Proceeding of the Conference,

*Free vector-valued semigroups*and

*On*(3, 2)

*-groups*. The first of these papers gave a combinatorial description of free vector-valued semigroups answering a question raised by Georgj Čupona in his 1983 paper in which he defined this class of semigroups. The second of these two papers is summarised by Dimovski as follows:-

Dimovski was promoted to associate professor in 1989 and then, in 1994, to full professor. In 1990 he was a visiting professor at Cornell University in the United States from January to June, then moved to Carleton University in Ottawa, Canada, where he spent July, August and September. In addition to these 1990 research visits, he has made several others, visiting: the University of Michigan, Ann Arbor, USA; the University of Tennessee, Knoxville, USA; the Research Institute for Mathematical Sciences, Berkeley, USA; the Stefan Banach Mathematical Centre, Warsaw, Poland; Binghamton University, USA; the Mathematical Research Centre in Oberwolfach, Germany; the Mathematical Institute, Ljubljana, Slovenia; the University of Sofia, Bulgaria; the University of Warsaw, Poland; the University of Heidelberg, Germany; and Bangor University, Wales. In his career of 40 years of teaching he lectured to graduate and postgraduate students at universities in Bitola, Štip and Priaština.The goal of this paper is to put together some known facts about(3,2)-groups. Some equivalent definitions for(3,2)-groups are given. It is mentioned that finite(3,2)-groups do not exist. An elementary proof that finite(3,2)-groups with less than12elements do not exist is given. At the end, it is shown that(3,2)-groups do exist, by giving a combinatorial description of a free(3,2)-group without generators. Such a group is countably infinite.

His scientific work includes: geometric and combinatorial topology (especially 4-manifolds, knots and clusters); single parameter theory of fixed points; homogeneous compacts; dynamical systems; and algebra, topology and combinatorial vector value structures with applications to physics, statistics and computer science.

For his outstanding contributions, Dimovski was awarded the "Goce Delčev" state prize for 1989, for the research results he had obtained in the fields of value vector algebra, topology and combinatorial structures. The Goce Delčev Award Board President said:-

Dimovski is a Member of the Union of Mathematicians of Macedonia and its Management Board, the American Mathematical Society, and the Balkan Mathematical Union and its Executive Board. On 27 May 2009 he was elected a full member of the Macedonian Academy of Sciences and Arts. In 2014 he was named as one of the seven best scientists of the "Saints Cyril and Methodius" University of Skopje for his scientific achievements in 2013.... the highest state award in Macedonia for outstanding scholarly contributions is a modest, but significant way of celebrating, promoting, and raising the awareness of the significance of academic work.

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

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