## The history of the Institute

The swift developments and effectiveness that have characterized the teaching and the research of mathematics at the University of Szeged for ninety years have their origins in the spirit and achievements of the mathematical workshop of the Franz Joseph University, predecessor of the University of Szeged in Kolozsvár. Among other names, Gyula Vályi, Gyula Farkas, Lipót Klug, Lajos Schlesinger, Lipót Fejér, Frigyes Riesz, Alfréd Haar were professors at the University of Kolozsvár. Due to the whims of European history, the latter two became – as indicated in their memorial plaque in the National Memorial Hall in Szeged - the well-known founders of the Szeged mathematical school. At the end of their careers spent in Kolozsvár, Frigyes Riesz (at the age of 38) and Alfréd Haar (at the age of 33) were reckoned to be among the top young mathematicians in the world at that time.

At the end of the First World War, Transylvania was occupied by Romania. The Romanian authorities, without waiting for the conclusion of peace, demanded an oath of allegiance from the teaching staff of the University, and as this was refused, the University in Kolozsvár was closed down. The University, which was provisionally moved to Budapest in 1919, found a final home in Szeged in the academic year 1921-1922. Its mathematical departments, the Institute of Higher Mathematics, Institute of Elementary Mathematics and Institute of Mathematical Physics (collectively called the Mathematical Seminar at the time) and the Institute of Descriptive Geometry were set up in three rooms on the ground floor of the central building in Dugonics Square. Professors Riesz, Haar and Ortvay, as heads of the Mathematical Seminar received a common room; Haar, in addition was temporarily head of the Institute of Descriptive Geometry. One of the two other rooms served to accommodate the library and the assistant (i.e. Tibor Radó, the only teaching assistant), the other “reading room” was the room for journals. The Institute had a single room for educational purposes called Classroom No.1.

From 1924 on, Tibor Radó was assistant professor working alongside Riesz. At the same time, János Kudar became teaching assistant of the Institute of Descriptive Geometry, and he was also assigned to Ortvay in the same position. Kudar, who was an excellent theoretical physicist, later worked together with Schrödinger (he was the first to formulate the relativistic version of the Schrödinger equation). When Riesz and Haar considered it the right time to fill the post of professor, which had been vacant since the retirement of Lipót Klug, the previous head of the Institute of Descriptive Geometry, they suggested changing the name to the Institute of Geometry. This provoked disagreement at the Faculty. József Gelei, professor of zoology, explained the adherence to the old name with his opinion that this reflected the fundamental importance of the subject, and recommended Hildegárd Szmodits (assistant professor from the University of Technology, private lecturer in descriptive geometry) to fill the post of professor. Riesz and Haar supported the appointment of Béla Kerékjártó, who was already regarded as an internationally recognized topologist during his university studies, and had been a private lecturer at the University since 1922. In the end the Faculty decided to change the name of the department to the Institute of Geometry and Descriptive Geometry, but Szmodits was proposed for the position. In exchange for the compromise, Riesz and Haar got another classroom (Mathematics classroom No. 2) and two other rooms; one for the professor of the Institute, the other for his assistants. At the same time, to vent their dissatisfaction, they turned to the ministry with their differing opinion. As a result of this, Kerékjártó became public extraordinary professor of the University. Ortvay helped Kudar to get a three-year scholarship in Germany and István Lipka was appointed in his place. It should be added that Jenő Egerváry – who is known in the history of science as one of the creators of the “Hungarian method” used in operations research – was a private lecturer at the University in the twenties; however, he had his degree withdrawn for reasons not clear from our sources.

When Kudar reported that he had no intention of coming home after the end of his scholarship, Ortvay, at Kürschák’s recommendation, invited László Kalmár, research physicist of the VATEA Vacuum Tube Factory, to be his assistant. Kalmár eagerly accepted the offer, but he did not conceal the fact that he was interested first and foremost in mathematics. Ortvay viewed this as an advantage, since theoretical physics requires theoreticians who are well trained in mathematics as well. This is how Kalmár arrived at the Department of Mathematical Physics in 1927, where – besides performing his duties in theoretical physics – he also investigated mathematical topics with great vigour. After László Kalmár joined the staff, the new six-member team became the nucleus of the mathematics school in Szeged. This school functioned like a Swiss watch. Besides Riesz, Haar and Kerékjártó, the assistants Radó, Kalmár and Lipka not only produced work of a high standard, but thanks to their talents and knowledge they were also able to tackle the latest problems of the various new disciplines (sometimes frowned upon at that time).

The professional creed of the newly evolving mathematics school was declared by Riesz, the ‘master’. (Riesz was called the master by his fellow professors among themselves. For others, it was common and required to address him as “Professor”, which was something inherited from Kolozsvár. When a colleague from Budapest or a visitor coming to Szeged called Riesz ‘Honoured Sir’, he tactfully rejected the method of address and explained the usual custom in Szeged.)

This creed was as follows: “The duty of a professor is to transmit pure science just like an antenna, to a high standard. Whether it is understood or not, it is not up to him.” Riesz fulfilled this. His scientific clout was huge throughout the world, but during the decades he spent in Szeged, he only had three students: first Tibor Radó, then in the thirties Béla Szőkefalvi-Nagy and the American Edgar R. Lorch. This followed from the fact that assigning topics was not in Riesz’s nature. He used to say: if someone wants to take his doctorate under Riesz’s guidance, he should choose his own topic, outline his conceptions, because his mathematical talent can be judged from this. Nowadays, Riesz’s severely formulated creed seems to be archaic and we should rather adopt Kalmár’s words: “I think... the highest level of science is if I can explain things in such a way that everyone who is interested can understand it.”

## FRIGYES RIESZ (1880-1956)

Frigyes Riesz published his celebrated paper in 1907 where he showed that the space of Lebesgue square-integrable functions is isomorphic to the space of infinite sequences with a finite sum of squares. This recognition led to the evolution of the notion of a Hilbert space, which played a crucial role in theoretical physics in the 20th century. He introduced a general notion of topological space at the international mathematical conference in 1908 in Rome. At the same time, he established that Lebesgue integrable functions can be defined without the prior introduction of the notion of a Lesbesgue measure, as limit functions of almost everywhere convergent sequences of step functions.

During the time he spent in Szeged, he also worked on the theory of subharmonic functions. His fundamental theorem, which characterizes subharmonic functions as potentials of negative mass distributions, opened new perspectives for potential theory. It was here also that certain ideas came to him, which gave birth to investigations on semi-ordered linear spaces, later called Riesz spaces, first expounded by him at the international mathematical conference in Bologna in 1928, and later in an improved form in his inaugural talk at the Academy of Sciences. His book on functional analysis written jointly with Béla Szőkefalvi-Nagy, Leçons d’Analyse fonctionnelle, which appeared in French in 1952, was largely a result of their collaboration in Szeged. The book was later translated into English, German, Russian, Japanese, Chinese and finally into Hungarian [!] as well. It has served as a handbook on real analysis and functional analysis for several generations.

The first step of an academic degree at that time was a doctoral degree, followed by the degree of private lecturer (obtained by habilitation); the latter corresponding approximately to the candidate degree later on. In 1926, Radó became a private lecturer in Analysis and geometry; Kalmár in 1932 a lecturer in Arithmetic and analysis; Lipka in 1933 a lecturer in Algebra.

Paedagogium, the state-run Civil School Teachers College was transferred from Budapest to Szeged in 1928. As the mathematics teachers of this institute did not want to move to Szeged, in the first semester of the academic year 1928-30 Riesz assigned the duties of the lecturing of mathematics to Radó, Kalmár and Lipka. This teaching duty lasted one semester, then the college invited Gyula Szőkefalvi-Nagy, director of the Marianum commercial college of Kolozsvár, who had been a private lecturer at the University of Kolozsvár in Algebra and analysis since 1915, to fill the post of professor of mathematics. There was a good educational and academic connection between the University and the teacher training college (now the education faculty of the University). Up to 1949 this connection also used to be an institutional co-operation because of the following. Teacher training at that time was regulated and administrated by the National Institute of Teacher Training; trainee teachers had to take an examination in front of a teachers’ examining body. The academic freedom of university education then made it possible for the professor to lecture on whatever he liked; no one had any say in the matter. However, the National Institute of Teacher Training also required that education follow the curriculum approved by the Ministry of Public Education from the teacher training institutes, and this work was properly paid. This is why two different lectures in mathematics were given at the University. One type of the lectures helped one get a teacher qualification, and the other type were lectures that allowed one to obtain “a higher academic qualification”. University lectures could also be attended by students from the Civil School Teachers College.

In 1928 Ortvay left for the University of Budapest. At the same time, Radó applied for the position of professor of mathematics announced by the University of Debrecen. To assess the applications, an interacademic commission was set up, which unanimously recommended Radó. However, the position was given to someone else. Radó, exasperated, moved to the United States with his family, and worked as professor at the Ohio State University until his death. One of his remarkable results was a way of solving Plateau’s famous problem: to determine the minimal surface with a given boundary. He published a prestigious monograph on arc length and surface area when he was an American professor. He commenced his studies on this topic with the encouragement of Riesz. An additional benefit of his work was to process and make available the relevant results of Zoárd Geőcze, a high school teacher of significant talent and victim of the First World War.

The year 1929 brought a great change in the life of the mathematics departments. During his visit in Germany, Hungarian Minister of Public Education Klebelsberg sat beside the renowned mathematician Richard Courant at a dinner in Göttingen. The German academic not only captivated the Hungarian minister who was interested in and receptive to the sciences by telling him about the results and developments of mathematics in the 20th century, but he also explained in what way and to what degree the mathematicians from Szeged, namely Riesz, Haar, Kerékjártó, Radó, Kalmár and Lipka, had contributed to it all. Klebelsberg was not only delighted to hear this, but upon his return he saw to it that extra financial support for the mathematics departments in Szeged was provided to improve their circumstances. Thanks to a resolution passed by the university council, since 1929 the Mathematical Seminar and the Institute of Geometry and Descriptive Geometry have had the common name “Bolyai Institute”. The Institute found a new location on the second floor of the building at 2 Baross Gábor Street (today Egyetem Street, at that time also called the building in Szukováthy Square). The name Bolyai Institute was first used in academic correspondence. The Institute’s journal has had this name on its title page since the ninth volume. In 1932, the Institute of Mathematical Physics, led by Zoltán Bay since 1929, broke away from the Mathematical Seminar, hence from the Bolyai Institute as well, and became a separate department called the Institute of Theoretical Physics.

Kalmár became a joint assistant professor of Riesz and Haar in 1930. In 1931, modelled on the Eötvös József College in Budapest, the Eötvös Loránd College was founded. On the part of the Bolyai Institute, Kalmár gave special lessons in mathematics, usually on topics requested by the students.

In 1933, Alfréd Haar passed away at the age of 48. With his demise, the golden age of mathematics in Szeged marked by the names of the first great “triumvirate”, Riesz, Haar and Kerékjártó came to an end. Following Riesz’s suggestion, the Faculty asked Kalmár to undertake the duties of the Institute of Elementary Mathematics for the academic year 1932-33. Afterwards, for economic reasons, the ministry did not appoint a professor to Haar’s department, which practically meant the cessation of the institute. In 1933 Béla Szőkefalvi-Nagy became an unremunerated assistant of Riesz, and Lipka, in turn became assistant professor to Kerékjártó in 1935.

## ALFRÉD HAAR (1885-1933)

Alfréd Haar became a private lecturer of the University of Göttingen at the age of 24, working under David Hilbert. Earlier, in his doctoral thesis, he introduced the system of orthogonal functions bearing his name. His academic work reached its peak while he lived in Szeged. He generalized Du Bois-Reymond’s lemma, one of the fundamental tools in the calculus of variations, valid in one dimension, to several variables, called the Haar lemma. He summarized his results in the area of the calculus of variations, their applications and further problems in a series of lectures given in Hamburg in 1929.

His greatest result is the proof of the existence of the measure named after him, about which he spoke at his inaugural talk at the Academy of Sciences in 1931. One way to formulate this theorem is via the following: on any locally compact group having a countable neighbourhood basis, there is a nontrivial measure invariant to left multiplication by elements of the group. With this result, Haar made an essential contribution to the solution of David Hilbert’s famous fifth problem, posed in 1900, which was finally solved in 1952.

## BÉLA KERÉKJÁRTÓ (1898-1946)

After his habilitation as private lecturer in Szeged in the twenties, Béla Kerékjártó became a visiting professor in Göttingen, later for two years in Princeton, and won fame with his book of great influence, Vorlesungen über Topologie. Hermann Weyl, one of the great all-round mathematicians of the 20th century wrote of Kerékjártó’s book: ‘...While formerly strict proofs by intuition in topology were really laborious, and you could bet ten to one that you would not succeed, Kerékjártó has made this way smooth; he has brought thoughts and intuition into a close connection. From now on I will teach topology on the basis of his perception and conception...’

While in Szeged, he finished the first volume of his enormous monograph ‘The foundations of geometry’, which was later published in French as well. The author wrote the following words about the publishing of his work in Hungarian: ‘...I wish to further the aim of helping every secondary school teacher to get acquainted with the role of the theorems taught at secondary schools in geometry’s scientific system...’ In addition to his books, Kerékjártó’s name is preserved for posterity by his deep results. Besides his investigations into the topology of surfaces, he achieved outstanding results on Jordan curves: he showed that a homeomorphic mapping between two Jordan curves can be extended to the whole plane, and he proved the inverse of the classical Jordan Curve Theorem.

In 1938, Béla Kerékjártó left for Budapest. The management of his institute was taken over by Riesz in the following academic year, then Gyula Szőkefalvi-Nagy was appointed public ordinary professor and head of the Institute of Geometry and Descriptive Geometry in the summer of 1939. His son, Béla Szőkefalvi-Nagy, held the same post as his father at the teacher training college and later became a professor at the Bolyai Institute.

The significance of mathematics in Szeged within Hungarian mathematics is well illustrated by the fact that the Gyula Kőnig Prize, the most prestigious mathematics award in the period between the two wars (more precisely, the Gyula Kőnig Medal, which replaced the prize after the value of it went down due to inflation) was awarded to Gyula Szőkefalvi-Nagy, László Kalmár, István Lipka and at the end of the period to Béla Szőkefalvi-Nagy as well.

Information related to the teaching of mathematics at our University during these two decades reveals that Szeged was not only among the leading universities in mathematical research, but also in transmitting mathematics to the next generation. In today’s world such words about an institute consisting of only six teachers may appear somewhat exaggerated, but one should not judge by appearances. At that time even foreign universities with outstanding reputations and academic performance had a limited number of teachers with permanent tenures. The following lists some of the mathematics courses that were on offer at that time at the University of Szeged.

Frigyes Riesz: Differentiation and integration, Real analysis, Fourier series, Complex analysis, Function operations, Theory of Hilbert spaces, Integral equations, Differential equations

Alfréd Haar: Algebra and number theory, Analytic number theory, Group theory, Theory of continuous groups, Galois theory, Calculus of variations, Probability theory (Courses in algebra were announced by István Lipka after Haar’s demise)

Béla Kerékjártó: Non-Euclidean geometry, Complex geometry, Geometric group theory, Basics of geometry, Topology

Tibor Radó: Descriptive geometry, Arc length and surface area, Minimal surfaces

László Kalmár: Set theory, Introduction to mathematical axiomatics, Mathematical confirmability theory

Gyula Szőkefalvi-Nagy: Algebraic and geometric theory of curves, Topology of curves, Distribution of roots of algebraic equations, Elementary mathematics, Methodology of the teaching of mathematics, Theory of geometric constructions

Despite the above-mentioned conflicts and problems, mathematicians formed a respected unit of the Faculty and the University, insofar as in the first years after the University’s move to Szeged, Riesz, Haar and Ortvay followed each other as deans of the Faculty, and in the thirties this position was filled by Kerékjártó as well. In the academic year 1925-26 Frigyes Riesz was the rector of the University for one year (the period usual at that time). In addition to these local positions of prestige, the time for national appreciation arrived as well, though not so quickly. At the beginning of the twenties Alfréd Haar, followed by Béla Kerékjártó and Gyula Szőkefalvi-Nagy was elected corresponding member of the Hungarian Academy of Sciences. Frigyes Riesz, in turn, was elected full member after twenty years of corresponding membership.

The Franz Joseph University returned to Kolozsvár in 1940, but it ceased to function four years later in consequence of another turn of history. The government founded the Horthy Miklós Royal Hungarian University in Szeged in 1940. Among the professors of mathematics it was Gyula Szőkefalvi-Nagy – at that time dean of the Faculty of Sciences – who accepted the appointment to the University of Kolozsvár. László Rédei (who won the Gyula Kőnig Medal as a high school teacher in the same year) received the post at the Institute of Geometry as public extraordinary, then in the following year as public ordinary professor. The term “descriptive geometry” was omitted from the title of the Institute at the new university.

The ill effects of the war affected the mathematicians in Szeged as well. László Kalmár‘s degree of private lecturer was not acknowledged by the Horthy Miklós University. With the power of the racist decree of the Sztójay government, he was relieved of his position as assistant professor and special lecturer. Frigyes Riesz was protected by his Swedish safe conduct, but he was obliged to wear the yellow badge and he did so demonstrating that he was not ashamed of his family roots.

After the military front line had moved on, university and mathematical life resumed. Frigyes Riesz – one of the six professors of the University remaining in Szeged in spite of the official edict – was elected as Rector of the University, like twenty years before. He successfully fulfilled his aim in restarting the work of the University, a task that required much wisdom. Resuscitating mathematical life became the task of Kalmár, who was again appointed assistant professor. After the front line had moved on, several outstanding mathematicians from Budapest resided in Szeged, including Rózsa Péter, Paul Turán and his family, István Vincze, János Surányi as well as Alfréd Rényi with his future wife. Turán was employed as assistant lecturer by Riesz. Paula Soós and János Surányi recommenced the Mathematical Journal for Secondary Schools with the help of Kalmár. In the academic year 1944-45, when the university building in Ady Square was used as a military hospital, the Bolyai Institute was housed in the departments of physics and chemistry. István Lipka had to leave the University for political reasons in 1945. Earlier, he obtained valuable results in the theory of algebraic equations, and he became an acknowledged expert in technical mathematics in Budapest.

László Kalmár got his title of private lecturer and post back in 1945. Gyula Szőkefalvi-Nagy, who had to return from Kolozsvár, was appointed subordinate public ordinary professor and became head of department. Béla Szőkefalvi-Nagy in turn became titular public extraordinary professor. In 1946 Kalmár was appointed departmental teacher, and shortly after he became public extraordinary professor.

## LÁSZLÓ KALMÁR (1905-1976)

In addition to some results in analysis and number theory, László Kalmár published a pioneering paper on game theory (in today’s terminology, the theory of combinatorial games) during his early years as a teacher. Furthermore, as pointed out by Edmund Landau in the preface of his classic book about the foundations of analysis, he corrected a hidden error in the axiomatic basis of the natural numbers. He even had to fight for the justification of the two areas of mathematics where he was especially creative. One of these was mathematical logic, the other was computer science, which – as it later turned out – had a strong connection with mathematical logic, but in the 1930s, in one of the golden ages of mathematical logic, it didn’t yet exist.

When Kalmár’s first paper on mathematical logic was published, not even Riesz considered this discipline to be an area of mathematics (similarly, Kalmár’s first paper on computer science was not received with much enthusiasm). Kalmár gave a simpler proof for some key results in mathematical logic, which were treated as novelties at that time, and later became public property, and he had a deeper insight into their connections with areas of mathematics than anyone else. Due to this insight, when he first encountered computer programs, he became a committed supporter and researcher of computer science. In the 50s, informatics – called cybernetics at that time – was officially regarded as a pseudo-science, Kalmár’s seminars allowed some to become familiar with the latest results of this field. The Cybernetic Laboratory was founded, following his initiative in 1963, as a separate unit of the University, and he also became its first head. From 1965 on, the laboratory even possessed a computer. This institute and Kalmár’s department – which remained a part of the Bolyai Institute for a long time – were the predecessors and the basis of today’s Institute of Informatics. It was Kalmár who initiated the training of applied mathematicians and later programmer and software developer mathematicians at the University of Szeged. The world’s leading computer society, the IEEE Computer Society considers Kalmár to be one of the great pioneers of computer science. Kalmár was also a passionate teacher. His lecture notes on Analysis delivered over several decades were edited into a textbook called An Introduction to mathematical analysis by his students (Károly Tandori, Piroska Paár Csúri, Rozália Lévai Duró, József Németh and Antal Varga).

## LÁSZLÓ RÉDEI (1900-1980)

László Rédei had already gained an international reputation during his twenty years as a secondary school teacher with his results concerning the invariants of class groups of quadratic number fields, which completed the classical investigations by Gauss. His abstract algebraic approach, which was already demonstrated by his results in number theory, was kindled by his collaboration with Tibor Szele – which was unhappily cut short by fate – and by encountering Kalmár’s abstract way of thinking. That is how Rédei became the author of the first comprehensive textbook in algebra, which went beyond the so-called modern algebraic approach initiated by Van der Waerden, and treated algebra as a general theory of sets endowed with operations (Algebra I; published in three languages). His further monographs: Theorie der endlich erzeugbaren kommutativen Halbgruppen, Lückenhafte Polynome über endlichen Körpern. The research on finite geometries, which is now intensively studied, is supported by the results described in the latter book. His book Begründung der euklidischen und nichteuklidischen Geometrien and his above-mentioned monographs were also published in English. His posthumous work Endliche p-Gruppen was edited by Péter Pál Pálfy. Rédei’s paper on two-step non-abelian finite simple groups, published in 1950, provided the first stimulus for the resumption of research on finite simple groups after several decades. As a result of this, at the end of the 70s, during Rédei’s lifetime, at the international congress of mathematicians in Helsinki a complete description of finite groups could be announced.

He played a decisive role in establishing the Hungarian school of algebra: most of the Hungarian algebraists have been his intellectual successors in some sense since the 50s. Another interesting thing is that in the world’s leading mathematical centres there is a wealth of legends and anecdotes that circulate about eccentric academics. In Szeged, many of them are associated with László Rédei. Let us mention one of his witty sayings: “To do mathematics, you only need a sphere and two hemispheres.”

## BÉLA SZŐKEFALVI-NAGY (1913-1998)

Béla Szőkefalvi-Nagy’s reputation was established by his monograph Spektraldarstellung linearer Transformationen des Hilbertschen Raumes (1942). He obtained several deep results in approximation theory, in the theory of orthogonal function systems, in Fourier analysis and geometry, but the most important field of his studies was the theory of linear operators of Hilbert and Banach spaces. He was described as the person “who knows the most in the world of what can be known about a linear operator”, a view which became widely accepted. His monograph Analyse harmonique des opérateurs de l'espace de Hilbert written together with Ciprian Foiaş, a Romanian mathematician who has lived in the United States in the last few decades contains several significant results of their joint research work. This book sprang from the consequences of a theorem by Béla Szőkefalvi-Nagy, which states that every contraction on Hilbert spaces has a unitary dilation, and one of its most important chapters gives a unitarily equivalent model of completely non-unitary contractions. From 1947 onwards, Béla Szőkefalvi-Nagy Béla was the editor of the Acta Scientiarum Mathematicarum over several decades, ensuring its high professional standard and developing its technical aspects. In 1975 he helped establish Analysis Mathematica, the joint journal of the Hungarian and the Soviet Academy of Sciences as the editor-in-chief. Szőkefalvi continued Riesz’s and Haar’s work in the development of the library, using his great international reputation and professional connections. His character, which reflected severity and discipline, commanded respect, but he was not unapproachable. He played an important role in preserving the original spirit of the Institute and conserving the primacy of science and education in the changing circumstances of society.

During the academic year 1945-46 Frigyes Riesz was invited and appointed to the Department of Mathematics No. 3 (formerly called “Suták’s department”) of the revived University of Budapest as head of department. An old desire of Riesz was realized with this appointment. His name had already come up in 1936, after the retirement of Suták. John von Neumann wrote the following to Ortvay about this possibility: “It would be a pity, if Szeged, which has so far been an important mathematical centre, continued to weaken…” It was Szeged’s luck that at that time the position of head of department was unfilled for financial reasons. Now Riesz received the appointment with great pleasure, but remained in Szeged until the end of the academic year to fulfil his duties as prorector (i.e. Rector in the previous year and hence vice-rector). With his departure, our city lost a great personality in Hungarian mathematics, who can only be compared to János Bolyai. During the quarter-century spent here he helped turn Szeged into one of the recognized mathematical research centres of the world. His successor was László Kalmár, first as public ordinary professor of higher mathematics, later as head of the Bolyai Institute. The latter post was filled by Gyula Szőkefalvi-Nagy for some time after Riesz’s departure. In 1948, the Institute of Descriptive Geometry was re-established, with Béla Szőkefalvi-Nagy as public ordinary professor. During the post-war years, Rédei’s resarch fellow, and later Kalmár’s assistant lecturer, was Tibor Szele, a renowned algebraist who left for Debrecen in 1948 and passed away before his time. János Aczél and István Fáry, who later continued their research career in America, also worked at the Institute. It is remarkable that the Bolyai János Mathematical Society, the national association of mathematicians, was founded in Szeged in 1947 and operated for two years with its centre in Szeged. Its first chairman was László Rédei.

Then in 1949 higher education underwent a significant reorganization. By one of the orders of the relevant decree the name of the Faculty of Mathematics and Sciences was changed to the Faculty of Sciences, but among other things, in principle it permitted the training of applied mathematicians – which in Szeged was initiated in 1957. The same decree dissolved the National Institute of Teacher Training. In 1950, following the suggestion of Kalmár, Rédei and Béla Szőkefalvi-Nagy, the three institutes that made up the Bolyai Institute were officially dissolved and operated until 1967 as an integrated institute. Its four professors each kept the title “head of department”.

Before 1950 the number of mathematics students had never exceeded twenty, but from 1950 onwards classes of about a hundred teacher trainees of mathematics appeared at the University. The Institute outgrew its once cosy place in Szukováthy Square (today’s Ady Square), which had been its home for almost a quarter of a century. In 1952 it moved to its present location, to the place of the former Piarist grammar school, to the first floor of the building No.1, Aradi vértanúk Square, acquiring spacious classrooms and schoolrooms on the second floor. The rooms were named after outstanding Hungarian mathematicians: besides the two Bolyais, Frigyes Riesz, Alfréd Haar and Béla Kerékjártó, rooms were named after two great personalities of the Kolozsvár school, namely Gyula Farkas and Gyula Vályi, and also Géza Grünwald, a student of the University of exceptional talent, who was sent to a labour camp and died there at a young age. From that time on, the building itself has also been called “the Bolyai building” by everyone at the University.

In 1953, after a long illness, giving his lectures until the last day of his life, Gyula Szőkefalvi-Nagy passed away. He was a genuine geometer and a conscientious teacher. In addition to his book Theory of geometrical constructions, he is noted for establishing the theory of curves of maximal index. He carried out research on the border of classical algebra and planar geometry as well. To supplement his educational work, Arthur Moór, an excellent researcher of differential geometry joined the Institute, but left in 1968 to work at the University of Sopron. Apart from him, first-rate geometers from Budapest worked in Szeged as well, namely Gyula Soós, and – for a long time – János Szenthe. The position of head of the Department of Geometry was only filled in 1975, when the Institute invited László Lovász to the department, a young mathematician of outstanding talent from Budapest, who had recently begun to make a name for himself in mathematics.

## LÁSZLÓ LOVÁSZ (1948-)

László Lovász became head of the Department of Geometry at the age of 27. His first talk in Szeged was worthy of his predecessor and of himself as well. The great Gauss raised a problem concerning a combinatorial property of planar projections of closed space curves in the 1820s. Gyula Szőkefalvi-Nagy was the first to make significant progress in 1927 on this issue, and in his first talk in Szeged Lovász provided a complete solution of the problem.

He attracted notice as a first-year university student by solving a famous algebraic problem of Jónsson and Tarski, and today, he is the Hungarian mathematician who is the most appreciated internationally. He is head of the Mathematical Institute of Eötvös Loránd University, considered by all to be a leading expert in combinatorics and complexity theory, which are of great importance in computer science and other disciplines, but he has obtained outstanding results in other areas of pure and applied mathematics. Among other things, he found important results concerning the information capacity of graphs as well as the algorithm of factorization of polynomials over rationals during his years spent in Szeged. He was elected a corresponding member of the Academy, and he received his first significant international awards, the Pólya Prize and Fulkerson Prize in the same period. Also, he wrote his book here called Combinatorial Problems and Excercises that had a great impact. Later, he became professor at Yale University and a senior researcher at Microsoft’s research centre. From among the most prestigious international prizes that can be awarded to a mathematician, he received the Wolf Prize in 1999 and the Kyoto Prize in 2010. He was elected president of the International Mathematical Union for four years in 2006.

Excerpts from an interview of Gyula Staar with László Lovász in 1979. (Taken from the volume Living Mathematics, published in 1990.)

“...The area that I am really interested in is called combinatorial optimization. In the case of combinatorial or discrete optimization, we have to choose the optimal from the elements of a given set, given a certain ‘goal function’. A typical problem, for example, is the following. There are companies producing materials and others processing them. Which company should supply which processing company? We would like to assign a store to each company. Of course, we want to choose the most economical solution (for example, we should not deliver to places too far away). So we have to devise a goal function that minimizes the overall cost of each assignment. Then we choose the most advantageous from all the possible assignments.

In traditional mathematics, people often said about such problems that there were finitely many assignments, all of which could be counted one at a time. The fact that this statement makes no sense in practice has contributed to the development of combinatorics to a great extent. Not even our fastest computers can examine all the possible assignments, let us say just those for fifty companies. Thus we have to find a method that chooses the best assignment based on the special nature of the problem.”

“No one can say that this area of mathematics has no practical benefits.”

“This is true, but this opinion may also be questionable. It is easy to say that something is an applied field of science, and then we expect everything done in this field to have direct practical benefits. Applicability motivates the general development of a field, but it does not mean that we only deal with topics that have profits measurable in financial terms. This is an incorrect attitude, because it hinders exactly the general development of applicable fields.“

“What is the reason for the recent rapid development of discrete mathematics?”

“Every discipline develops, if it can formulate problems that can be solved, but are not trivial. I think that nowadays the fact that the organization of the whole society is becoming evermore complex is the source of such problems, and these problems are too big to be solvable by hand. Combinatorial and graph theoretic problems have had no real foundations yet. It could be said that the problems arising from real-life situations were riddles solvable by examining them one at a time, or using special tricks; they did not require any mathematical theory. Theorems and theory become interesting in the case of problems with time-consuming calculations, which cannot be solved using the usual methods, but require some novel idea. Computers have extended the frontiers to some extent, so that today this theory is a fertile area for study. Solutions require powerful and effective mathematical methods.“

“...You seem to be only interested in difficult questions.”

“I do not know whether this is true. Beyond doubt you are stimulated more by a problem that could not be solved by anyone for a long time. We can save ourselves some of the routine work by learning from their experiences. They have tried all the usual techniques and methods, and usually it becomes clear why they could not find the right solution.”

“...You became a head of department at the József Attila University of Szeged at the age of 27. Why did you come to Szeged?”

“The József Attila University had been searching for a head for the department of geometry for a long time. I was also asked by László Leindler, who is a member of the Hungarian Academy of Sciences. I had some moments of hesitation, but in the end I accepted the appointment. My wife’s family lives in Budapest and my family does as well, so it was not easy to move so far from them. I have to admit that I was a bit afraid at first. Now I am happy to live and work in Szeged.”

“Don’t you miss the atmosphere of the capital and the way it encourages creative thinking?”

“I think that in many ways Szeged is more suitable for encouraging creativity. Also, the lively atmosphere of our Institute encourages us to find new results...”

Lovász moved to Szeged, and in addition to the introduction of his own topics of research, as an all-round mathematician, he also supported research in classical geometry (for instance, he also contributed to Lajos Szilassi’s and Rozália Juhász’s surprising results in intuitive geometry). He led the department until 1982, when he became professor of the Eötvös Loránd University. After his departure, Péter Nagy was appointed head of the department. The wide spectrum of his research areas in differential geometry extended from Riemannian manifolds to the investigation of the connection of nets and webs with groups and loops. In 1995 he left for the Kossuth Lajos University in Debrecen. Following him, János Kincses, Péter Hajnal and Nándor Simányi each headed the department for a short time. Since 2000, Árpád Kurusa has been in charge of the department.

Péter Hajnal is an expert in graph theory, a research field that was initiated in Szeged by László Lovász; Kincses has carried out studies mainly in convex geometry. Simányi is a first-rate specialist in the theory of dynamical systems (e.g. abstract billiards), but in 2000 he moved to the United States, where he works as a professor. Kurusa himself does research work on Radon transformations.

In 1967 “The Bolyai Institute” became a common name of four departments. It was at this time that the Department of Analysis was founded under the direction of Béla Szőkefalvi-Nagy, the Department of Algebra and Number Theory under the direction of László Rédei, the Department of Foundations of Mathematics (from 1970 Department of Foundations of Mathematics and Computer Science, from the following year Department of Computer Science) under the direction of László Kalmár, as well as the Department of Geometry, with Béla Szőkefalvi-Nagy as mandatory head of department.

László Rédei moved to Budapest at the end of 1967 to take up his appointment as head of the Department of Algebra of the Institute of Mathematics of the Hungarian Academy of Sciences. After Rédei, the Department of Algebra and Number Theory was led by Béla Csákány from 1968 to 1972 and from 1974 to 1993, introducing the study of universal algebra, and establishing a renowned centre of this field at the department. The department was headed by Ferenc Gécseg from 1972 to 1974, and after 1993 by László Megyesi, Gábor Czédli and Mária B. Szendrei. Today its head is László Zádori.

Gécseg is discussed in detail in another chapter of this book. Megyesi, a fine researcher in semigroup theory, was a common student of László Rédei and Pollák, Rédei’s highly talented coworker from Szeged. Czédli, who is one of the leading authorities on lattice theory, was a student of András Huhn, a lattice theorist of outstanding talent, who died at an early age. Pollák was the mentor of Mária B. Szendrei as well, who is a leading authority on regular semigroups and their generalizations; moreover, she is the editor-in-chief of the international journal Periodica Mathematica Hungarica of Bolyai János Mathematical Society, edited in Szeged. Zádori is a member of the third generation of the workshop of universal algebra, one of the students of Ágnes Szendrei, a researcher belonging to the very best of her research field, who now works permanently in the United States.

The departmental system underwent another change in 1971. The Department of Applications of Analysis was founded, headed by Károly Tandori. The Department of Set Theory and Mathematical Logic separated from the Department of Computer Science, its first head being Géza Fodor.

László Kalmár retired in 1975. After his retirement, his department was headed by Ferenc Gécseg. It was he who initiated and established research on the algebraic theory of automata and the theory of formal languages. With the foundation of the Institute of Informatics in 1990, the department separated from the Bolyai Institute. (A detailed description of the work of Gécseg and his students can be found in the chapter about the Institute of Informatics of this volume.)

Géza Fodor died before his time in 1977. Two important theorems of combinatorial set theory bear his name. One of these led to the development of the theory of stationary sets, with the great forerunners Alexandrov and Urysohn. Fodor’s other theorem is one of the fundamental statements of the theory of set functions initiated by Paul Turán: the domain of a set function of low (infinite) order can be represented as the union of independent sets of at most the same order. Fodor’s department was taken over by László Leindler, and since 1990 it has been headed by Vilmos Totik.

## VILMOS TOTIK (1954-)

He was born in a distant corner of the country, in the village of Ásványráró in Little Rye Island (Szigetköz in Hungarian), but he graduated as a mathematician from the University of Szeged. After winning the prestigious annual Schweitzer Miklós Memorial Competition twice, as a student of professors Leindler, Tandori and Béla Szőkefalvi-Nagy, he became another leading researcher of the analysis school of Szeged established by Frigyes Riesz. (Here, we should mention the name of his best student, Péter Varjú, who also won the same competition twice like his master.) At the age of thirty-three, Vilmos Totik became a full professor. Since 1989, he has been a professor of the University of South Florida and also of the University of Szeged.

His fields of interest are illustrated by his monographs, written in part with coauthors (Moduli of Smoothness, General Orthogonal Polynomials, Weighted Approximation with Varying Weights, Logarithmic Potentials with External Fields, Metric properties of harmonic measures), which were published by leading academic publishers (Springer-Verlag and the Cambridge University Press).

Excerpts from an interview of Erzsébet Sulyok with Vilmos Totik. (From the volume called Gold-washing published in 1995.)

Professor Totik was elected member of the Hungarian Academy of Sciences at the age of 40, based on the unanimous recommendation of his colleagues. He is head of department. For one half of the school year he teaches at home, and for the other at the University of Tampa in Florida.

“Isn’t it difficult? Being in Florida from January to April each year?”

“It is interesting that you ask this. People are usually curious about the advantages.”

“First, I wanted to ask why you do not live abroad permanently instead of commuting. Florida may be a better place than Szeged. “

“Everything is relative. It has advantages and disadvantages... I am fond of teaching. I teach with pleasure not only my narrow research area, but also other branches of mathematics, because I think that the more areas you can comprehend, the better. Apart from analysis, I teach number theory, mathematical logic, combinatorics and set theory, both at home and abroad. It gives me the chance to relax. And, of course, it is a useful professional investment, since if I know a lot about other areas of mathematics and understand the connections among them, then I should be better able to understand my own special area. Another advantage of the position in the United States is professional connections. These are very important in mathematics, as it is in all disciplines – Florida also means this to me. But I still do not feel quite at home there. At first the whole family was with me, but as my children were growing up, it became increasingly difficult for them to accept the changes. Anyway, we decided at the beginning not to remain abroad permanently. To be honest, we feel more at home here.“

“Did you attend a grammar school that specialized in English?”

“No, and I had only a few lessons more in mathematics than students of most grammar schools. I only started to learn English during my university studies. I also grew interested in mathematics at a rather late stage. The first time I took part in a competition was in the eighth class.”

“But at university you won first prize in the Schweitzer competition twice, which is said to be an amazing achievement.”

“This mathematical competition is unique in the world, in the sense that there are no age categories, but students from each year, and even newly graduated students can participate. The problems, coming from different disciplines, are posted on the notice board, and students take them home and have to solve all of them in ten days. These are ten hard days, as not even a broad mathematical interest is sufficient; all of these problems are extremely hard. To be honest, even today what I am the proudest of is that I twice won the first prize in the Schweitzer competition.”

“Who do you think has the real mathematical talent? Someone who is familiar with every branch of mathematics, or someone who has the deep knowledge in just one particular discipline?”

“That is hard question to answer. It is not even certain that a student with good results in competitions will become a good mathematician. However, there is a strong correlation.”

“Can the career of a researcher of mathematics be planned?”

“I do not know whether it is possible to make such five-year plans in other disciplines, but it is absolutely impossible in mathematics. You can never be sure about being able to solve a problem before finishing it. Anyway, the main point of this discipline is explication -- explaining how something works.”

“Is it true that mathematicians can never really relax? Are they always thinking about a proof for a problem?”

“As for me, I have nothing in my mind, say, when I am hoeing in the garden. Maybe with the exception of not having any luck with the grass. The results each year are wonderful, I have green fingers – but the grass does not grow. Apart from this, I feel quite at home in the garden and I enjoy seeing how easily I can achieve tangible and even edible results.”

“At the age of forty, you had achieved everything that a major researcher can set as a goal – except the lawn in your garden. Don’t you think you could rest a bit now and take things easy?”

“The picture of a renowned scientist or academic sitting in his room and enjoying the respect of those surrounding him, as imagined by ordinary people, is just a myth. Anyway, I would not like to become dull too soon by doing nothing. I can see clearly the small part that I have created or am going to create in the future; I can only hope it is of value and will contribute to mathematics as a whole. And I hope that mathematics in its own way can contribute to the general welfare of society. I have just remembered a lovely quotation – this time about mathematicians: ”A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.”

From 1983, Béla Szőkefalvi-Nagy took part in the life of the Institute only as a professor emeritus. Although he was still present at the most important events of the Institute, his retiring from everyday university duties ended fifty years of the Institute, which got its magic and colour to a great extent from the presence of the “second triumvirate” (Kalmár, Rédei, Szőkefalvi-Nagy). The successors of Szőkefalvi-Nagy as heads of the department were László Leindler, László Hatvani and László Kérchy.

## KÁROLY TANDORI (1925-2005)

Károly Tandori began his career at the Bolyai Institute as a teaching assistant of László Kalmár in 1949, and he soon obtained significant results in the theory of orthogonal function systems. By proving the necessity of the sufficient conditions for the convergence of series with constant coefficients given by Rademacher, Menshov and Kaczmarz several decades earlier, he provided a complete answer to the question in the case of positive, monotone, non-increasing coefficient sequences, and he even found a complicated, but elegant, sufficient and necessary condition of convergence in the general case. He established a school on his research topic: at present several of his students work as acknowledged professors at Hungarian universities. His tranquil wisdom played a decisive role in fostering stability of the Institute over several decades.

## LÁSZLÓ LEINDLER (1935-)

László Leindler excelled as a student with the synthesis of the theorems of Károly Tandori and the Russian professor Menshov concerning the convergence of orthogonal series. Later he became one of the leading researchers of this area. With the general proof of the equivalence of structural and coefficient conditions, he sharpened several classical theorems. The elaboration of the strong approximation of Fourier series and the generalization of the Hardy-Littlewood inequalities are also some of his results. His memorable advice to his students should be taken to heart by every young mathematician: “Do not respect prestige! Not even the greatest mathematical minds can think about everything. You have to set your mind on doing what they did not succeed in.”

## LÁSZLÓ HATVANI (1943-)

László Hatvani began investigating differential equations following the advice of Lajos Pintér, an iconic figure in the Bolyai Institute who helped students and looked for talented secondary school students. He established an internationally acclaimed school on the qualitative theory of differential equations. He founded the online journal called Electronic Journal of Qualitative Theory of Differential Equations, together with his American colleague Theodore A. Burton in 1998, with an international editorial board (the site of the journal is the Bolyai Institute, but it can be downloaded from the mirror sites of the European Mathematical Society as well). In 2002, he provided a complete solution to one of the central problems in the stability theory of functional differential equations.

László Kérchy is a student of Béla Szőkefalvi-Nagy who carries on the work of his master: he investigates the operators of Hilbert spaces, mainly contractions, proceeding in the direction laid out by Szőkefalvi-Nagy and another outstanding student of his called Ciprian Foiaş. Another former student of Szőkefalvi-Nagy, Endre Durszt is no longer living.

In 1995, Károly Tandori, a fine researcher and teacher of the period following the two triumvirates, became a professor emeritus. His successor at the Department of Applications of Analysis was his student, Ferenc Móricz until 2001, when the department split into two departments, namely the Department of Stochastics and the Department of Applied and Numerical Mathematics. The latter was headed by Ferenc Móricz until 2004, then succeeded by Tibor Krisztin.

The investigations of Ferenc Móricz spread over several areas of analysis: he achieved his most important result in approximation theory and Fourier analysis. At the moment he is the Hungarian editor-in-chief of Analysis Mathematica, joint journal of the Hungarian and the Russian Academy of Sciences, edited at the Bolyai Institute. As editor-in-chief of this journal, he is the successor of Béla Szőkefalvi-Nagy. Tibor Krisztin is already a student of László Hatvani’s school, who, for today, has established an internationally renowned centre for functional differential equations.

The Department of Stochastics was directed by its founder, Sándor Csörgő until his untimely demise in 2008. His successor is Gyula Pap, who moved from Debrecen to Szeged, and is a first-rate researcher and teacher of probability theory and mathematical statistics. Other professors at this department include András Krámli and Péter Major.

## SÁNDOR CSÖRGŐ (1947-2008)

Sándor Csörgő, the fifth student that came from the village of Egerfarmos in Heves County, made a distinguished career as a teacher and researcher. As a student of Professor Tandori, after his studies in Kiev, he received all the Hungarian awards a young mathematician can receive, and he established a stochastics workshop in Szeged, which has produced outstanding results right from the beginning. He was appointed full professor in 1987. He was a professor of Michigan University at Ann Arbor for a decade after 1989, and returning to the University of Szeged, he became one of the most prolific and most cited Hungarian mathematicians besides Paul Erdős and László Lovász.

He was a leading researcher in probability theory and mathematical statistics. He created the probability theory of empirical characteristic functions, established the approximation theory of empirical processes under random censorship, and investigating the almost 300-year old classical St. Petersburg paradox from a novel point of view, he was able to resolve it.

As a teacher, he had a charismatic personality. He launched the research career of several talented students. He received the main prize of the Foundation for Szeged in 2007. Owing to his demise, he could not receive the Széchenyi Prize awarded to him in the following year.

Excerpts from Zoltán Bátyi’s interview with Sándor Csörgő. (Published in the monthly ‘Szeged’ magazine in March 2007.)

“...When you asked me to take a seat in your room full of books, it was not only the portrait of an internationally acknowledged mathematician that is suggested in your notes, but also of a Hungarian who has frequently been subject to the afflictions of the 20th century.“

“...I chose this discipline because I longed for the ideal of purity. I had a strong desire to study in a discipline that was free of politics. I did not want to be a doctor, I had no idea of being an engineer, and I came to the conclusion that what I liked in chemistry was what could be described by mathematics. Therefore – perhaps also following the example of my brother (who was living in Canada at the time), who had already been working in the field of mathematics – I opted for mathematics. I did not dare to apply to Budapest, as I was afraid that if I got among students with such a head start, I would get lost. There remained Debrecen and Szeged to choose from, and I found it easy to decide between these two cities. Based on observations from my brother in Canada, the mathematics school of Szeged had such fame that even the lovely campus of Debrecen could not compete with it.

...In Szeged… attending the lectures of exceptional mathematicians, the desire of living my life as a mathematics researcher kept growing. In the fifth year, I became immersed in stochastics (which was not studied by any of the researchers at the Bolyai Institute at that time) to such an extent that Professor Tandori directed his attention to my work, and with his support Professor Szőkefalvi-Nagy offered me a position of research assistant.”

“...In 1972, you became an aspirant of Professor Skorohod in Kiev, one of the greatest figures in stochastic mathematics. …Years passed by, and the name of Professor Csörgő became well known beyond the borders as well. In 1989 – as he wittily put it – he fled from liberty to America.”

“I have never thought about living my life overseas. I feel at home in this country and in the city of Szeged. Egerfarmos is still important for me, but today the fact that I could put down roots on the banks of the Tisza and become acquainted with several leading academics living here, has similar importance. As I spend the greatest part of my time closed off in the world of mathematics, I can say that I contemplate this city from the window of the University. But Szeged is a developing, liveable, likeable town seen from here, from the Bolyai Institute as well.

...The Hungarian mathematical school might also become famous beyond the borders, because we laid special emphasis on managing talent over a century ago. I would like to continue this tradition as well. ...I know there are several things that can seduce the students from mathematics in our open world. At the same time, we must recognize that perhaps there has never been such a need for well-trained mathematicians as in the 21st century. In our world, let us just look around – the machines serving us, computer sciences, space research, the World Wide Web, which have completely changed our lives, along with the medicine of the future where people are treated according to their genetic profile, and a great part of the natural and social sciences could not subsist and develop without the results of mathematical research. In our everyday life revolving around banks, stock exchanges, assurance companies, there is also an increasing need for the services provided by mathematicians. How could we plan a better, more comfortable, more liveable future without the mathematical rationality of thought?”

“...What formula do you think is needed so that this future vision might some day come true?”

“...The institutions of everyday life – administrational or educational ones – should become more independent, based on sensible agreement. Only by doing this can we hope to create a country where most people can find a way that leads to recognition and success, where really true talent can get on and be successful, and where it is possible to make one’s living from work performed with talent and honesty. There is a great risk — without this we can lose our most talented young academics who decide to go abroad. Naturally young people wish to get away, to get to know the language and culture of another country, its work habits and the way people live there and have fun. But I would like to live in a country where they return with this knowledge, because in such a country they can really be at home, and feel at home.”

From its beginnings until 1957, there was only the traditional training of secondary school teachers of mathematics and physics in the Institute (some of the classes had one and a half majors with mathematics as the first major and physics as the second major or vice versa). As mentioned before, the previously small number of students began to increase after 1950; for instance, 120 students began their studies in 1951 with the above majors. Between 1957 and 1970 the Institute also trained teachers of mathematics and descriptive geometry. The training of teachers of mathematics and geography started in 1965.

In the eighties, future teachers had more freedom to choose their particular majors: teachers of mathematics and chemistry appeared, and teachers with two majors could take informatics as a third major. From the nineties there has been a possibility for the training of teachers of mathematics as the only major, and teachers of mathematics and informatics. At the same time, some students from the Faculty of Arts could also obtain the qualification of teacher of mathematics (teachers of foreign languages or philosophers).

The training of computer scientists (under various names) has been continuously one of the main tasks of the Institute since 1957; and since 1990 under the direction of the Institute of Informatics (also called the Kalmár László Institute) established in that year. The training of teachers of mathematics - applied mathematicians was started by László Kalmár in 1957 with an emphasis on computer science – with three students in the first year. The independent training of mathematicians started in 1963 (until that time mathematics students were chosen from among the teacher trainees who successfully completed the first two years). Almost every student who graduated as a mathematician found a job as a computer scientist. To satisfy the rapidly growing need for specialists, the three-year training of programmer mathematicians leading to a college degree was started in 1972. At the same time the number of students of mathematics increased as well, e.g. 44 mathematicians (title of degree: “software developer mathematician”) and 38 programmer mathematicians graduated in 1975. These two majors were combined as the two-level major (training of 3+2 years) programmer-software developer. At the same time the ‘modelling mathematician’ major was launched with a small number of students; this helped foster the training of future researchers. As the number of graduate students looking for employment related to economics kept increasing, it was reasonable to start the training of economist programmer-software developer mathematician (from 1988, in common with the Budapest University of Economic Sciences).

Although the strictness of examinations has scarcely changed during the past decades, and quite a lot of the students beginning one of the majors in mathematics change their profession during their training – just as in the previous decades – the number of specialists graduating in the Institute has significantly increased. This tendency is shown by the following data: while 36 teachers and 8 mathematicians received a diploma in 1965, the number of graduated teachers was 39 in 1980, and 44 software developer mathematicians and 30 programmer mathematicians also finished their studies at the Institute (the latter students with the support of the consolidated Laboratory of Cybernetics). In 1995, besides 57 teachers of mathematics and 9 mathematicians (not specialized in informatics), 70 software developer mathematicians and 50 programmer mathematicians graduated at our University, the latter students already as a result of the education in co-ordination of the Institute of Informatics, and 34 of them within the framework of the above-mentioned training of economist mathematicians.

The possibilities of meeting foreign colleagues reappeared after the war in the sixties. Compared to the 21st century there were also few opportunities for a personal exchange of ideas between the two wars; this is why e.g. the photo taken during the visit of the two American researchers, George D. Birkhoff and Oliver D. Kellogg in 1928 is still one of the most precious relics of the Institute.

It is a fact that apart from the big names, Kalmár, Lipka and Radó participated in the above-mentioned international conference in Bologna, and in the thirties Béla Szőkefalvi-Nagy and László Rédei from among the young researchers could travel abroad for a longer study-tour. This, however, was the exceptional rather than the rule. The Second World War and the following two decades did not improve international relationships much either. Shorter or longer study trips and the attendance of foreign congresses and conferences became regular only after 1970. All of the present-day professors of the Institute have worked – some of them still do – at foreign universities as visiting professors or researchers and the same holds for a great portion of the teachers in other positions. The visits were reciprocated by distinguished foreign researchers, whose fact is testified by the entries in the visitors’ book kept in the library of the Institute over four decades. International mathematical conferences are regularly organized in different fields of mathematics by the departments of the Institute, sometimes together with the Bolyai János Mathematical Society, usually once a year. The topics of these conferences were universal algebra (1971, '75, '79, '83, '89), semigroup theory (1972, ’76, ’81, ’87, 2000), the theory of algebraic automata (1973, ‘77), lattice theory (1974, ‘80), algebraic methods in graph theory (1978), qualitative theory of differential equations (1979, ’84, ’88, ’92, ’96, ’99, 2003, 2007), matroid theory (1982), ordered sets and lattices (1985), geometry of webs (1987), intuitive geometry (1991), functional analysis (1993), lattices and universal algebra (1993, ’96, ’98, 2002, 2005), a conference in memory of Béla Szőkefalvi-Nagy (operator theory) (1999), a conference in memory of László Rédei (algebra) (2000), and algorithmic complexity and universal algebra (2007).

From the middle of the eighties, academic research was financed by an application system. The members of the Institute took part as supervisors or participants in calls for applications announced by the Hungarian Scientific Research Fund (OTKA) and other Hungarian or international institutions. This fact has crucially contributed to the Institute being properly equipped with electronic computers, the satisfactory development of the library and the chance of participating in major international academic programmes and gaining experience in acknowledged foreign research centres. To illustrate this point, we will list the applications won in 2010 or active in this year:

The European Research Council awarded a grant of 400,000 euros for five years for the project Potential Theory (principal investigator: Vilmos Totik); 800,000 euros for five years as well for the Epidelay Project (principal investigator: Gergely Röst; the topic is optimization of defence against epidemics using the theory of functional differential equations). The Social Renewal Operational Programme (TÁMOP; one of the programmes of the New Hungary Development Plan) has supported the two-year-long project Sensor network based data collection and processing of information started in 2009 with 341 million forints. The manager of this project is Miklós Maróti, and 16 members of the Bolyai Institute have helped to realize it. OTKA and the National Office for Research and Technology support research in the fields of algebra, differential equations, functional analysis and combinatorics with 48 million forints (principal investigators: Mária B. Szendrei, Gábor Czédli, Miklós Maróti, Tibor Krisztin, László Kérchy and Péter Hajnal).

We ought to mention the high-quality professional work in support of the preparation of our teacher trainees for their careers and finding young talent. Lajos Pintér mentioned above was assisted by Jenő Berkes, later by Rozália Lévai Duró in the preparation of teacher trainees, and at present most of this work is done by József Kosztolányi. The academic research performed by students is coordinated by the Academic Students' Association of the Institute, led by Ferenc Fodor. Our students have achieved excellent results in international mathematical competitions; their participation and preparation is organized and managed by Gergely Röst. The former centralized system of postgraduate training (“aspirantura”) was changed to the current PhD programme in the nineties; 109 young postgraduate students have participated in this training so far at the Bolyai Institute. The list of the eligible research topics reflects the great variety of today’s research carried out at the Institute, and shows convincingly that – like in the period between the two world wars – several fields of mathematics can be studied at the highest level in Szeged. PhD students also attend the academic seminars at the departments, which have been run for several decades and follow the traditions of the ancient functional analysis seminars. Nowadays the majority of the topics of these seminars come from the fields of algebra, geometry and the qualitative theory of differential equations. There is also an all-institute seminar with mathematical topics of general interest.

The doctoral school of mathematics is currently led by Gábor Czédli, the successor of László Hatvani, head of the former integrated doctoral school of mathematics and computer sciences. We shall list the current supervisors and the research topics of the PhD students to give the reader an idea of our current areas of research.

1. Doctoral programme in algebra:

Mária B. Szendrei: Regular semigroups and their generalizations. Finite inverse semigroups.

Gábor Czédli: Congruence lattices and Maltsev conditions. Related lattices and closure operators.

Béla Csákány: Discrete mathematical games and algebraic structures.

Miklós Maróti: Varieties and quasivarieties. Residuated lattices.

Ágnes Szendrei: Structure of the clone lattice. Examination of finite algebras with clone theoritical methods.

László Zádori: Clones, relations. Structure theory of finite algebras.

2. Doctoral programme in analysis:

László Kérchy: Operators of Hilbert spaces.

László Leindler: Orthogonal series, inequalities, function classes.

Ferenc Móricz: Harmonic means of multiple series. Behaviour of sums of absolutely convergent Fourier series.

József Németh: Characterization of function classes by the order of Fourier coefficients.

László Stachó: Automorphism groups of Banach manifolds and manifolds modelled in topological vector spaces and their resulting algebraic and geometric structures.

Vilmos Totik: Orthogonal polynomials, polynomial inequalities and potential theory.

3. Doctoral programme in dynamical systems:

Tibor Csendes: Automatic simplification of nonlinear optimization problems with symbolic methods.

László Hatvani: Non-autonomous second order differential equations. Problems from stability theory with applications.

János Karsai: Computer-aided examination of models in life sciences.

Tibor Krisztin: Stability theory of functional differential equations. Examination of attractors of nonlinear dynamical systems.

Béla Gábor Pusztai: Integrable systems.

Gergely Röst: Nonlinear dynamics in mathematical epidemiology.

4. Doctoral programme in geometry, combinatorics and theoretical computer sciences:

János Barát: Graph theory.

János Csirik: Learning methods in information retrieval.

József Dombi: Continuous logics.

Zoltán Ésik: Algebra and logic in computer sciences.

Ferenc Fodor: Discrete and analytical convex geometry.

Gábor Gévay: Polytopal and non-polytopal cellulated spheres. Construction and examination of perfect polytopes.

Péter Hajnal: Extremal problems of geometric and combinatorial structures. Combinatorial complexity theory.

Csanád Imreh: Competitive analysis of online algorithms.

János Kincses: Combinatorial geometry of convex sets. Integral geometry of convex sets.

György Kiss: Semiovals of finite planes.

Miklós Maróti: Algebra and algorithmic problems.

Gábor Péter Nagy: Geometric algebra.

András Pluhár: Extremal and algorithmic problems in graph theory. Combinatorial games.

György Turán: Theory of machine learning.

5. Doctoral programme in stochastics:

Krisztina Boda: The problem of multiple comparisons in biostatistics. Biostatistical systems in support of medical decisions.

András Krámli: Probabilistic methods in the examination of large graphs. Statistical behaviour of hyperbolic deterministic dynamical systems.

Péter Major: Nonlinear functionals of probability distributions. Almost sure limit distribution theorems.

Gyula Pap: Branching processes.

László Viharos: Asymptotic distribution of norms. Tail behaviour of distributions.

6. Doctoral programme in mathematics – didactics:

János Karsai: Computer-aided problem solving: experimentation, the role of mathematical programming languages. Possibilities in the development and application of interactive textbooks, problem books.

Lajos Klukovits: Antique and early medieval roots of secondary school mathematics. Generating of heuristic conjectures by mechanical principles and their rigorous proofs. The book Liber quadratorum by Leonardo of Pisa.

József Kosztolányi: Possibilities of improving problem solving skills at different levels of mathematical education.

Lajos Pintér: Evolution of the notions used in analysis, with particular attention to Euler and his followers.

István Szalay: Comparison of solutions of elementary and secondary school problems based on the knowledge of pupils and teachers. Conversion of mathematical knowledge and notions to a level suitable to the age characteristics of pupils of classes 1-6, based on the comparative analysis of the schoolbooks. Framing of the evidence level of mathematical knowledge during the general mathematical training of elementary school teachers.

Lajos Szilassi: Critical points of computer-aided teaching of mathematics. Actual problems of teaching geometry.

Apart from the publication of the results of academic work in international journals and talks given at international conferences, several monographs have been produced at the institute. Gábor Szász, who worked in the Institute for several years, published his book here first in Hungarian, later in German with the title Einführung in die Verbandstheorie. The book later appeared in English as well. Also Ferenc Gécseg published his three monographs (see the chapter about informatics) while he was a member of the Institute. Among the works of the present-day professors of the Institute – besides the papers and books of Vilmos Totik listed elsewhere – we should mention the monographs entitled Strong Approximation by Fourier Series by László Leindler and Clones in Universal Algebra by Ágnes Szendrei.

Apart from these works, several university textbooks and handbooks have been written in the Institute over the past few decades with the Institute’s publishing house called Polygon playing an increasing role. This was established in 1991, and publishes books and textbooks besides the Polygon journal, which contains mainly mathematical and methodological articles for teachers and university students. Antal Varga had a crucial role in the establishment of the publishing house; the first managing editor was Lajos Pintér, who was followed by János Kincses in 1995. Books by several teachers of the Institute (Gábor Czédli, Béla Csákány, Sándor Csörgő, Péter Hajnal, László Hatvani, József Kosztolányi, György Kiss, Géza Makay, József Németh, Lajos Pintér, László Imre Szabó, Ágnes Szendrei, Tamás Szőnyi, Antal Varga) have been published in the Polygon Library Series, along with some classic works of earlier authors (e.g. János Bolyai, Paul Erdős, János Surányi, Barna Szénássy, Béla Szőkefalvi-Nagy) being republished. The correspondence of László Kalmár with Hungarian mathematicians living in or outside Hungary also appeared in the Polygon Library Series in two volumes (under the title Kalmárium, edited by Péter Gábor Szabó). Almost fifty books written by the members of the Institute have been published in the Polygon Textbook Series (Mónika Bagota, Mária B. Szendrei, Béla Csákány, Tibor Csendes, Ferenc Gécseg, Péter Hajnal, László Hatvani, Balázs Imreh, Márta Kalmárné Németh, Tamás Kámán, Eszter Katonáné Horváth, László Kérchy, Lajos Klukovits, Zoltán Kovács, Tibor Krisztin, István Krisztin Németh, Árpád Kurusa, László Leindler, Géza Makay, László Megyesi, Ferenc Móricz, Judit Nagy-György, József Németh, Zoltán Németh, Éva Osztényiné Krauczi, László Stachó, László Szabó, László Imre Szabó, Péter Gábor Szabó, Tamás Szabó, Ágnes Szendrei, József Terjéki, Vilmos Totik, Endre Vármonostory, László Viharos). Several researchers of the Institute (Gábor Czédli, Endre Durszt, László Hatvani, László Kérchy, László Leindler, József Németh, Lajos Pintér) have published textbooks at the University’s publishing house (JATEPress).

Quite a number of mathematicians trained at the Institute have made a successful career in other Hungarian or foreign institutes since the war. Let us mention some of the most outstanding (besides those already mentioned): András Ádám and András Hajnal (students of László Kalmár), Ádám Korányi, István Kovács and Lajos Pukánszky (students of Béla Szőkefalvi-Nagy), István Peák, Ottó Steinfeld and János Szendrei (students of László Rédei), Attila Máté (student of Géza Fodor), Lajos Horváth (student of Sándor Csörgő), István Szalay (student of László Leindler) and Zoltán Szabó (geometer, describing himself as a student of the Institute’s library); Tibor Bakos (later editor in chief of the Mathematical Journal for Secondary Schools), Imre Rábai (later a renowned teacher of the Fazekas Grammar School in Budapest) and Lajos Stachó. However, mathematicians from other institutes have joined the work of the Institute for longer or shorter period. Besides those mentioned, these include Károly Böröczky, László Czách, András Frank, Géza Freud, György Kiss, László Márki, Tamás Matolcsi, Pál Révész, Gyula Strommer, Árpád Szabó, Antal Szederkényi, László Székely, Tamás Szőnyi, András Szűcs, Lajos Tamássy, Endre Vármonostory. Those deserving a special mention are the teachers of the Teacher Training College (today the Faculty of Education at the University) as well as the teachers of Ságvári Endre Grammar School and other secondary schools, who have aided the work of the Institute for several years like József Csúri, Éva Hegyi, Ottokár Honti, István Kállai, József Miskolczi, Tibor Szerényi, Balázs Tóth and Ottó Varga.

The leaders of the former Mathematical Seminar and Bolyai Institute had different titles (managing director, director, chairman of the Institute council, head of the Institute). Their names are listed in the chronology at the end of the chapter, together with the heads of the institutes forming the Bolyai Institute and the heads of the departments. Their work has always been aided by one or two assistants. In the year of the completion of this volume, the head of the Institute is László Kérchy and his assistants are József Kosztolányi and Gábor Péter Nagy. A great strength of the Bolyai Institute is that its parts have worked with a common budget, common administration and common library since the beginning. As a result of development over nine decades, the library of the Bolyai Institute has one of the best mathematical collections in the country – possibly the best.

### The library of the Bolyai Institute and the Acta Scientiarum Mathematicarum

Frigyes Riesz and Alfréd Haar set about building a high quality mathematical library in Szeged intended for research and education, based on the contents of the University of Kolozsvár. Riesz brought with him his catalogue of the library and the list of the borrowed books and borrowers. The borrowers were asked politely to return the books to the Library of the Mathematical Seminar of the University of Szeged. The budget of the mathematics departments available for the purchase of books and journals was rather modest at that time. As a supplement, the Ministry of Public Education allowed the Institute to buy the valuable mathematics library of Mihály Demeczky, a professor of the University of Budapest, who passed away in 1920; and Gyula Farkas donated his book collection to the Mathematical Seminar. The legacy of Ágoston Scholtz (predecessor of Lipót Fejér at the University of Budapest) also came into the possession of the Mathematical Seminar. The situation is well illustrated by the following true story, which sounds like an anecdote. Professor Haar, who was on good terms with Minister Klebelsberg, described how dreadfully the Institute was furnished with books and journals. Klebelsberg discovered that the industrialist and Member of Parliament Pál Hámori Bíró owed 25,000 pengős to the state for his “de Hámor” title of nobility, and Klebelsberg arranged that Bíró should transfer this sum to the Mathematical Seminar of the University of Szeged for the augmentation of the library. Out of this sum, 43 books and 297 volumes of 12 journals were purchased.

By the end of the twenties the mathematical library became a university research library of a satisfactory standard: its catalogue lists the contents of almost 3000 volumes. The real development, however, would be the collection of journals of the library. Riesz and Haar came up with the idea of establishing a world-class mathematical journal. The costs for this were covered by a local association. The wealthy citizens and the big names of the city created the Society of Friends of Franz Joseph University. This society, led by Dezső Várnai, director of the Szeged Press and Publishing House became the publisher of the journal. Initially, the title of the journal founded in 1922 was Acta Litterarum ac Scientiarum Regiae Universitatis Hungaricae Francisco-Josephinae. Sectio Scientiarum Mathematicarum. Today, the title is Acta Scientiarum Mathematicarum, or Acta Sci. Math. (Szeged) for short by the top review journal (Mathematical Reviews). Riesz and Haar appealed to leading mathematicians of the day, including several of their acquaintances, and asked them for papers. Some names on this extraordinary list of authors in the first years were John von Neumann, Norbert Wiener, George D. Birkhoff, Henri Cartan, Antoni Zygmund, Lipót Fejér, György Pólya and Gábor Szegő. The editors of the journal themselves published their papers that they considered were the best, and asked their colleagues, Tibor Radó, Gyula Szőkefalvi-Nagy and others to do likewise. This way, the journal soon acquired an international ranking. Not long after, several universities and bigger libraries considered it important to order the Acta from Szeged or acquire it by other means. Riesz and Haar appealed to editors, publishers of renowned mathematical journals, and offered the Acta in exchange. In several cases they agreed to barter; hence even Russian and Japanese journals, which were viewed as rarities at that time, but which were of a high professional standard, appeared in the library. These exchanges were also facilitated by Hungarian mathematicians working abroad. In the thirties, the Library of the Mathematical Seminar had the most extensive collection of mathematical journals in the country, both in terms of variety and quality.

The Acta has also served to increase of the Szeged library, as several publishers sent their publications to the editors for refereeing, and after being refereed in the Acta, the books came into the library’s possession. In essence, this symbiosis of the Institute, the journal and the library still works this way. In 2012 the 78th volume of the Acta appeared, making it by far the oldest journal published at the University of Szeged. For further details, please see the homepage

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Another source for the development of the library was ‘Swedish money’: the brother of Frigyes Riesz, Marcell Riesz, who was also a famous mathematician living in Sweden asked for and received donations for the mathematicians of Szeged from the Swedish mathematicians. Furthermore, the Rockefeller Foundation assisted in the development of the library. Albert Szent-Györgyi (several years before receiving the Nobel Prize) negotiated with the Rockefeller Foundation about support for scientific and mathematical research in Szeged. In 1931, an agreement was reached about a one-off payment of 119,000 dollars for basic equipment and on the decision of an annual allowance provided that the Ministry of Public Education also contributed its share. According to the separate inventory books, this source alone contributed to the increase of the library’s contents by 806 volumes of journals and 395 books between 1931 and 1938. This is how the library obtained, for example, “Journal de Mathémathiques Pures et Appliquées” back to its first volume (1836); the acquisition of this journal has been continuous since then.

During the war the most important concern was of course not acquisition, but preservation. In 1944 the Minister of Public Education ordered that valuable equipment, instruments, books and documents of the University be sent to different places for safekeeping. Many valuable items of the University were sent to the western part of Hungary following this decree. Frigyes Riesz, as director of the Institute did not allow contents of the mathematics library to be broken up. Thus the library witnessed the passing of the military front line in its original place in the building at Egyetem Street. However, a couple of days later this building was turned into a military hospital. At that time Professor Riesz, together with László Kalmár, who had returned from forced-labour service and some students remaining in Szeged helped move the library’s collection in carts to the Institute of Theoretical Physics at Rerrich Béla Square. The library moved back to its original place in Egyetem Street in March 1946. However, the contents were reduced from shunting them back and forth. In the process, several books got damaged or even lost.

When the Bolyai Institute moved to the renovated building of the Piarist grammar school in Aradi vértanúk Square in December 1952, the library was moved to the former headmaster’s office and staff room on the first floor. The library, which had grown quite a lot in the preceding years, had to overcome another serious problem: the floor of the library rooms could not take the excessive load. The problem was solved by adding an iron trestle and a gallery in one room (completed in 1970), and also in the other room in 1979. The next problem – still existing today – is that these two rooms are insufficient for the fast growing contents. Therefore shelves were put up in some of the lecturer’s offices for the journals. In 1985, a locker system was designed which increased the capacity of the library by several hundred metres. The Study and Information Centre in Ady Square completed in 2004 also took over a part of the contents – mainly the journals of general scientific interest.

Another great problem after the war was that of the systematic ordering of the contents. This is why the director of the Institute asked the dean to create a new librarian position as, until then, the duties of the librarian were performed by an unpaid assistant, alongside his other tasks. The request was finally granted in 1949 by the Faculty. An inventory of the contents was drawn up, and the old book numbering system was deleted. (The library still has these records up to the present day.) Based on this inventory, the library had 3470 books, 417 offprints and 5872 volumes of various journals altogether on the 28th October in 1949. Afterwards, the development of the library was continued following the old traditions. In the next ten years the contents doubled, and this was followed by further continuous developments. In consequence, the number of books increased to 18,935 by 2009. The exchanges initiated by the founders are still active sources for the growth of the collections of journals. These sources did not cease even in the years of political isolation: just due to the exchange of journals between 1950 and 1965, the library received about 2000 volumes of journals. The current number of journals received is 263; 141 of which came into the possession of the library by exchange, 88 by purchase, and the others by donation. Today the Bolyai Institute has one of the best mathematics libraries in the country. Exploiting the developments of modern electronics, interested specialists can browse its contents from anywhere in the world.

We should also mention the joint award by the Bolyai Institute and the Acta Scientiarum Mathematicarum called the Béla Szőkefalvi-Nagy Medal, which was initiated by Erzsébet Szőkefalvi-Nagy in 1999, dedicated to the memory of her father. The medal is awarded once a year by the council of the Bolyai Institute based upon the recommendation of the editorial board of the Acta, with the stipulation that the person must be ‘an excellent mathematician who has published important and deep results in the Acta’. The mathematicians who have received the award in chronological order are Ciprian Foiaş, Károly Tandori, László Leindler, György Grätzer, Ferenc Móricz, Tsuyoshi Ando, Béla Csákány, Hari Bercovici, Tamás E. Schmidt, Heinz Langer and Pierre A. Grillet. Apart from the Acta, three further international mathematical journals and one in Hungarian are edited in the Bolyai Institute. We have already mentioned them elsewhere in this chapter.

Next, some words about the staff that manage the library are in order. In the twenties Tibor Radó and László Kalmár performed these duties, then, after Radó had left, László Kalmár and István Lipka took over. The library was managed by István Sándor Gál for a time after the war; later he became a mathematics professor in the United States. Mrs Ferenc Ádám became the librarian in 1951. She was succeeded by Erzsébet Árvay Kalmár. When Erzsébet Árvay Kalmár became head of the university dormitories, she was replaced by Erzsébet Kristó Horváth (‘Auntie Lizzy’), a teacher of mathematics and physics, who ensured the continuous operation of the library over several decades, even after her retirement. From September 1987 she had a new colleague, Piroska Fekete Varga, Piroska, also a teacher of mathematics and physics. Thanks to the excellent organising ability and wonderful memory of Erzsébet Kristó Horváth, the visitors of the library always got helpful advice from her. Piroska Fekete Varga introduced a new computer-based catalogue, which was a novelty at that time and which – continuously developed and updated – is still used by the readers.

The Institute has two computer rooms that are separate from the departments and bear the names of László Rédei and Béla Szőkefalvi-Nagy.

The Hungarian Academy of Sciences has supported mathematical research work in Szeged since 1956. It was then that the Department of Functional Analysis of the Mathematical Research Institute of the Hungarian Academy of Sciences was founded at the Bolyai Institute under the leadership of Béla Szőkefalvi-Nagy. The following year, in turn, the Group of Mathematical Logic and its Applications of the Mathematical Research Institute of the Hungarian Academy of Sciences was founded under the leadership of László Kalmár. At that time, under the political regime, these units not only helped extend research possibilities, but also provided the opportunity for professional development for talented persons who could not get a job at the University because of their ”inappropriate family descent”. These two units were turned into the Research Group of Analysis and the Mathematical Logic and Automata Theory Research Group of the Hungarian Academy of Science, which have functioned since 1967. The Institute is supported by the Hungarian Academy of Sciences even today e.g. the Analysis and Stochastics Research Group led by Vilmos Totik.

Mathematicians have also participated in the management of the University and the Faculty of Sciences since 1945. László Kalmár, Géza Fodor, Béla Csákány and János Csirik were rectors of the University, Géza Fodor, Béla Csákány, László Leindler and János Csirik (professor of the Institute of Informatics of the Faculty of Sciences, who began his career in the Bolyai Institute) were vice-rectors. László Rédei, Béla Szőkefalvi-Nagy (twice), László Leindler, Károly Tandori (elected for two subsequent periods), Ferenc Gécseg and László Hatvani were deans of the Faculty of Sciences. The former Gyula Kőnig Medal was replaced by several professional awards; since 1948 the Kossuth Prize, later replaced by the State Prize of the Hungarian People’s Republic, and afterwards by the Széchenyi Prize have become the most valuable and prestigious Hungarian awards. Riesz was even awarded a Kossuth Prize twice in Budapest at the end of his life. This prize or one of its later equivalents was awarded to Béla Szőkefalvi-Nagy three times, to László Rédei, László Kalmár, László Lovász and Károly Tandori twice, and once to Sándor Csörgő, László Hatvani and Vilmos Totik. In addition, Béla Szőkefalvi-Nagy received the greatest award of the Hungarian Academy of Sciences, the Gold Medal of the Hungarian Academy of Sciences.

Professors of the Institute in 2010 (Members of the Hungarian Academy of Sciences being marked by an asterisk) are: Mária B. Szendrei, Gábor Czédli, László Hatvani*, László Kérchy, András Krámli, Tibor Krisztin, Péter Major*, Gyula Pap, Ágnes Szendrei and Vilmos Totik*. Professors emeriti are: Béla Csákány, László Leindler* and Ferenc Móricz. Titular professors are: László Megyesi, Lajos Pintér and János Szendrei. Associate professors are: Ferenc Fodor, Gábor Gévay, Péter Hajnal, Jenő Hegedűs, János Kincses, Lajos Klukovits, József Kosztolányi, Árpád Kurusa, Géza Makay, Miklós Maróti, Gábor Péter Nagy, József Németh, Zoltán Németh, Tibor Ódor, László Stachó, László Szabó, László Imre Szabó and Tamás Zoltán Szabó, Titular associate professor: László Gehér. Assistant professors are: Mária Bartha, Miklós Dormán, Vanda Fülöp, Miklós Hartmann, Kamilla Kátai-Urbán, Eszter Katonáné Horváth, Zoltán Kovács, József Kozma, Miklós Maróti, Béla Nagy, Judit Nagy-György, Béla Gábor Pusztai, Gergely Röst, Róbert Vajda, Mónika van Leeuwen-Polner and Tamás Waldhauser. Teaching assistants are: László Csizmadia, Attila Máder and Gábor Szűcs. Assistant research fellows are: Attila Dénes, Péter Kevei, Viola Mészáros, Anna Pósfai, Zoltán Sáfár, Tamás Varga, Péter Varjú, Gabriella Ágnes Vas and Antal Veres. When we look back on the successes of the mathematics school of Szeged and ask why, the answer may be that the general aim has always been to educate the members of the school – both researchers and students – and to strike a balance between teaching and research; to find a balance between being too strict and too easy going; and between self-ambition and altruism.

Mathematics Institute*

Bolyai Institute**

* Name in use since 1984.

** Name in use since 1929.

The departments of mathematical research and education had the following two units up to 1940:

Mathematical Seminar (1921—1940)

Managing director: Frigyes Riesz 1921—1940

Departments of the Mathematical Seminar:

Institute of Higher Mathematics

Director:

Frigyes Riesz 1921—1929

Institute of Elementary Mathematics

Director:

Alfréd Haar 1921—1933

(The post was left vacant after Haar’s death.)

Institute of Mathematical Physics (until 1932)

Director:

Rudolf Ortvay 1921—1928

Pál Fröhlich 1928—1930

Zoltán Bay 1930—1936

(In 1932 it separated from the Mathematical Seminar and got the name Institute of Theoretical Physics.)

Institute of Descriptive Geometry (1921—1925)

Institute of Geometry and Descriptive Geometry (from 1925)

Director:

Alfréd Haar 1921—1925

Béla Kerékjártó 1925—1938

Frigyes Riesz 1938—1939

Gyula Szőkefalvi-Nagy 1939—1940

Bolyai Institute (from 1940)

Director:

Frigyes Riesz 1940—1946

Gyula Szőkefalvi-Nagy 1946—1947

László Kalmár 1947—1958

Béla Szőkefalvi-Nagy 1958—1959

László Rédei 1959—1960

László Kalmár 1960—1961

Béla Szőkefalvi-Nagy 1961—1962

Károly Tandori 1962—1963

László Rédei 1963—1964

Chairman of the Institute Council:

László Rédei 1964—1967

Béla Szőkefalvi-Nagy 1967—1969

Károly Tandori 1969—1977

Béla Szőkefalvi-Nagy 1977—1980

László Leindler 1980—1984

Head of the institute:

Károly Tandori 1984—1990

Ferenc Móricz 1990—1993

Vilmos Totik 1993—1996

László Hatvani 1996—1999

Gábor Czédli 1999—2005

László Kérchy 2005— 2011

Tibor Krisztin 2011- date

Departments of the Bolyai Institute:

Institute of Mathematics (1940—1950)

Director:

Frigyes Riesz 1929—1946

László Kalmár 1946—1967

Institute of Geometry (1940—1950)

Director:

László Rédei 1940—1950

Institute of Descriptive Geometry (1948—1950)

Director:

Béla Szőkefalvi-Nagy 1948—1950

The above-mentioned units of the Institute were dissolved in 1950. Their heads – as well as Gyula Szőkefalvi-Nagy – officially kept the title ”head of department”, but there were no departments in the Bolyai Institute until 1967.

Department of Analysis (from 1967)

Head of department:

Béla Szőkefalvi-Nagy 1967—1983

László Leindler 1983—1999

László Hatvani 1999—2006

László Kérchy 2006—date

Department of Applications of Analysis (1972—2001)

Department of Applied and Numerical Mathematics (from 2001)

Head of department:

Károly Tandori 1972—1995

Ferenc Móricz 1995—2004

Tibor Krisztin 2004—date

Department of Stochastics (from 2001)

Head of department:

Sándor Csörgő 2001—2008

Tibor Krisztin 2008—2009

Gyula Pap 2009—date

Departmental Analysis Research Group of the Hungarian Academy of Sciences (1967-1975)

Head:

Béla Szőkefalvi-Nagy 1967—1975

Analysis Research Group of the Hungarian Academy of Sciences (2003—2007)

Analysis and Stochastics Research Group of the Hungarian Academy of Sciences (from 2007)

Head:

Totik Vilmos 2003—date

Department of Algebra and Number Theory (from 1967)

Head of department:

László Rédei 1967-date

Béla Csákány 1968—1972

Ferenc Gécseg 1972—1974

Béla Csákány 1974—1993

László Megyesi 1993—2004

Gábor Czédli 2004—2007

Mária Bálintné Szendrei 2007—2010

László Zádori 2010—2011

Mária Bálintné Szendrei 2011-date

Department of Geometry (from 1967)

Head of department:

Béla Szőkefalvi-Nagy 1967—1975

László Lovász 1975—1982

Péter Nagy 1982—1994

János Kincses 1994—1995

Péter Hajnal 1995—1997

Nándor Simányi 1997—2000

Árpád Kurusa 2000—date

Department of Foundations of Mathematics and Computer Science (1967—1971)

Department of Computer Science (1971—1990)

Head of department:

László Kalmár 1967—1975

Ferenc Gécseg 1975—1990

(They separated from the Bolyai Institute in 1990 and became part of the Institute of Informatics.)

Department of Set Theory and Mathematical Logic (from 1971)

Head of department:

Géza Fodor 1971—1977

László Leindler 1977—1989

Vilmos Totik 1989—date

Mathematical Logic and Automata Theory Research Group of the Hungarian Academy of Science (1967—1980)

Automata Theory Research Group of the Hungarian Academy of Science (1980—1990)

Head:

László Kalmár 1967—1975

Ferenc Gécseg 1975—1990

(Since 1990, it has been part of the Institute of Informatics.)