Balogh was educated in Debrecen where he attended the Fazekas Mihaly Gymnasium. As did many Hungarian school children with mathematical talents at this time, he entered the mathematics competitions organised by KöMal, the Középiskolai Matematikai és Fizikai Lapok (High School Mathematics and Physics Journal). This Journal had begun posing competition problems for high school children in 1894 and, after a couple of short breaks due to World War I and World War II, had continued the competitions. Balogh entered these competitions and won awards. In the same year that the problem solving journal KöMal was launched, the Hungarian Mathematical and Physical Society decided to start up a mathematical competition for those graduating from secondary schools. It was called the Eötvös Competition, in honour of Lóránd Eötvös who was, in that year, made minister of education in the Hungarian government. The competition renamed the Kürschák Mathematical Competition after World War II, always consisted of three problems and those entering the competition had to answer the questions under examination conditions in an afternoon. The 1972 competition, the 72nd Kürschák Competition, posed the following three questions:
- A triangle has sides with lengths a, b, c. Prove that
a(b - c)2 + b(c - a)2 + c(a - b)2 + 4abc > a3 + b3 + c3.
- A class has n > 1 boys and n girls. For each arrangement X of the class in a line let f (X) be the number of ways of dividing the line into two non-empty segments, so that in each segment the number of boys and girls is equal. Let the number of arrangements with f (X) = 0 be A, and the number of arrangements with f (X) = 1 be B. Show that B = 2A.
- ABCD is a square side 10. There are four points P1, P2, P3, P4 inside the square. Show that we can always construct line segments parallel to the sides of the square of total length 25 or less, so that each Pi is linked by the segments to both of the sides AB and CD. Show that for some points Pi it is not possible with a total length less than 25.
In 1972 Balogh graduated from the Fazekas Mihaly Gymnasium and entered the Kossuth Lajos University of Debrecen to begin the five year course which would reach the level of a Master's Degree. In the year he entered university his father died from a circulatory disease at the age of 42. Balogh was aware that this disease was due to his father's genetic makeup and so he was likely to inherit similar health problems. While still an undergraduate, Balogh began to undertake research. He attended the Fourth Prague Topological Symposium in 1976 and delivered the lecture Relative compactness and recent common generalizations of metric and locally compact spaces. He presented a paper with this title to the conference proceeding and the 8-page paper was published in 1977. It became his first publication and is remarkable for introducing the concept of "relative compactness." This outstanding paper led to Balogh being awarded the Renyi Kato Memorial Prize by the Janos Bolyai Mathematical Society in 1977. The Society awards this Prize to "an outstanding young researcher in mathematics." In the same year he graduated from the Kossuth Lajos University of Debrecen.
With such an outstanding start to his research career, it was natural for Balogh to continue to undertake research at the Kossuth Lajos University of Debrecen. While undertaking research, he was employed as a Teaching Assistant and, later, as a Research Fellow. He attended the Fourth Colloquium in Budapest in 1978 and presented the paper On the heredity of being a paracompact M-space and its relation to the normal Moore space conjecture which was published in the Proceedings. He also attended the Meeting on General Topology at the University of Trieste in 1978 and presented the paper On recent common generalizations of metrizable and compact T2 spaces which was published in the Proceedings of that conference. His third publication in 1978 was Relative compactness and recent common generalizations of metric and locally compact spaces. Three of his papers were published in 1979, namely Metrization theorems concerning relative compactness; On the structure of spaces which are paracompact p-spaces hereditarily; and Relative countable compactness. As a result of these publications, the Janos Bolyai Mathematical Society presented Balogh with their Grunwald Geza Memorial Prize, which they award annually to an outstanding researcher under the age of 30. In 1980 Balogh was awarded a Candidate's Degree (equivalent to a Ph.D.) in Topology and Set-theory from he Kossuth Lajos University of Debrecen and a Ph.D. from the Hungarian Academy of Sciences.
While Balogh was an undergraduate at the Kossuth Lajos University of Debrecen he got to know Eva Balicza who was a student in the same class. They were married in 1976 when Balogh was still studying for his first degree. Their two daughters, Agnes (born 1978) and Judit (born 1979), were born before Balogh completed studying for his Candidate's Degree. The marriage, however, did not last and Balogh and his wife were divorced in 1981.
After the award of his Candidate's Degree in 1980, Balogh remained at the Kossuth Lajos University of Debrecen. He had already been appointed as a Junior Research Fellow in August 1979 and during the next years he was promoted first to Research Fellow and then to Senior Research Fellow. He continued to hold this position until May 1984. He spent the summer of 1984 in Canada. He had been offered a visiting professor position at the University of Toronto for three months and spent June 1984 to August 1984 in Toronto. In September 1984 he returned to Debrecen and in November he married Agnes Polgar. Agnes was an undergraduate at the Kossuth Lajos University of Debrecen studying chemistry. Balogh, together with his wife Agnes, left Debrecen less than two months after their marriage and travelled to the United States where Balogh had been offered a Visiting Associate Professorship at Texas Tech University in Lubboch, Texas. This institution had been founded in 1923 as Texas Technological College and had formally become Texas Tech University in 1969. He had only been there for a few months when he had major heart problems :-
In June of 1985 Zoli was faced with his first life-threatening medical emergency when he underwent open heart surgery for a bypass operation. Zoli had been having some chest pains and only a few days earlier, he and a friend had noticed an unusual shortness of breath. While in the hospital, being prepared for surgery, Zoli had a heart attack and only the proximity of immediate medical help saved his life. He was 31 years old. In every other way, Zoli was a healthy young man - certainly, his weight was good and he was a non-smoker. Because his father died at an early age, Zoli knew that he had a genetic predisposition for circulatory disease. He began to watch his diet more carefully and for many years became an avid practitioner of various types of aerobic exercises, including walking, running, and bicycling.In July 1986, Balogh and his wife returned to Hungary and he was again employed at the Kossuth Lajos University of Debrecen. now as an Associate Professor. His wife Agnes had been near the end of her undergraduate chemistry course when she married and left for the United States so she now completed her degree. Their first son Adam was born in 1987. Balogh worked on his habilitation thesis and he submitted Set-theoretic investigations on the classes of compact and locally compact spaces (Hungarian) to the Hungarian Academy of Sciences in 1988. He returned to the United States in 1988 when he was appointed as a Distinguished Visiting Professor in the Department of Mathematics and Statistics of Miami University. He planned to spend two years in the United States before returning to Hungary and in 1989 his habilitation thesis was accepted by the Hungarian Academy of Sciences so everything was set for his return. First he had his second year as a visiting professor in the United States and he spent the first semester of this second year, September 1989 to January 1990, as a Visiting Associate Professor at the University of Wisconsin at Madison. For the second semester of 1989-90 he returned to his visiting position at the University of Miami.
At Miami University Balogh was working on the exceedingly beautiful Oxford campus in Oxford, Ohio. This city, a college town based largely on serving the university, is in the Miami Valley in south west Ohio. Balogh had fallen in love with the university and so when he was offered a tenure track position there he decided that he would not return to Hungary as he had originally planned but he would accept the permanent position at Miami University. He took up this position of Associate Professor in August 1990. Soon after, in January 1991, Zoli and Agnes Balogh's second son Daniel was born.
Dennis Burke and Gary Gruenhage give a detailed account of Balogh's mathematics in . Let us quote here their introduction:-
Zoli's research was in set-theoretic topology. ... Deep infinitary combinatorics lie at the heart of many problems in this field, and thus their solutions frequently make use of the tools of modern set theory, e.g. special axioms such as the continuum hypothesis or Martin's Axiom, or building models of set theory by Cohen's method of forcing. Statements shown to be true using special axioms or models are thereby proven consistent with the usual axioms of ZFC (the Zermelo-Fraenkel axioms plus the axiom of choice). Sometimes the negation of a consistent statement is also shown to be consistent, and hence the statement is independent of ZFC. ... A problem in set-theoretic topology is not considered "solved" until either its statement is proven independent, or a positive or negative answer in ZFC is found. Some of Zoli's best results were finding ZFC solutions to problems for which previously only a consistent answer was known. Zoli's research spans 25 years or so, and includes many significant contributions in diverse areas within set-theoretic topology. ... What makes Zoli's research especially stand out are a series of solutions to several long-standing problems in the field, which he obtained at an amazing pace starting in the mid-1980s, continuing essentially until his death.Balogh had always known that his health might let him down again and indeed it did in the summer of 1999. He suffered a massive stroke but was saved by some innovative new medical treatment. You can read an extract from  in which his collapse and subsequent treatment are described in detail at THIS LINK.
His recovery was remarkable, so much so that only a couple of weeks after his stroke he was on a flight to New York to attend the 1999 Summer Conference on Topology and its Applications. Balogh certainly did not ease up following this major stroke for he continued to undertake research with three papers appearing in print in 2001 and a further three in 2002. He continued to attend conferences and he was invited to speak in the set-theoretic topology section of the 'International Conference on Topology and its Applications' held at Shimane University in Matsue City, Japan from 24 to 28 June 2002. He was to lecture on Jig-sawed bases and metrization. He received another invitation to lecture in Tsukuba, Japan, before attended the conference in Matsue City. He died suddenly on the morning of the 19th June, the day he was due to fly to Japan. In fact six papers by Balogh appeared in print in 2003 and later, submitted for publication before his death.
Let us end with the tribute from the Editors of Publicationes Mathematicae Debrecen :-
He was a man of wide interests and of a broad intellectual horizon. He like to meet old and new friends and talk with them about mathematics. His early departure at the height of his creative power is a considerable loss for mathematics both in Hungary and in the USA. He was full of promising plans which will be realised no more. With him we lost an outstanding mathematician, a warm-hearted colleague, a good friend and a member of our Editorial Board.
Article by: J J O'Connor and E F Robertson