**Preface**

This brochure gives a concise presentation of the past and the present of the Alfréd Rényi Institute of Mathematics, birthplace of many important scientific discoveries and venue of numerous high-level meetings of mathematics. Besides an overview of the activities and the history of the institute, you will find here information on our current members as well as some former members and regular visitors whom we are proud of.

A first version of this text was compiled in 2000, on the occasion of the 50^{th} anniversary of the Institute. The present revised and updated version contains a lot of additional information and we hope that it gives a faithful picture of the Institute at the dawn of the third millenium.

A brief chronology of the Institute

**1950** Founding of the Institute, under the name Institute for Applied Mathematics of the Hungarian Academy of Sciences. Alfréd Rényi, aged 29 at the time, is appointed as the first director.

**1955 **The name of the Institute is changed: it becomes the Mathematical Institute of the Hungarian Academy of Sciences.

**1958 **The institute moves to its current premises.

**1970 **Death of Alfréd Rényi. László Fejes Tóth is the new director.

**1983 **László Fejes Tóth is followed by András Hajnal in the director’s chair.

**1984 **Paul Erdős, member of the Institute, receives the Wolf Prize in Mathematics.

**1992** Domokos Szász is appointed as the new director of the Institute.

**1996 **Gyula Katona becomes director of the Institute. Members of the institute take an active part in the organization of the Second European Congress of Mathematics, held in Budapest.

**1999 **The Institute takes up the name of its founder Alfréd Rényi.

**2000 **The Institute receives the grant “Centre of Excellence” from the European Union.

**2001 **A new extension of the building of the institute is inaugurated, giving rise to a great improvement in working conditions. A joint graduate program is launched, in cooperation with Central European University.

For a more detailed history, see the corresponding chapter in the second half of this brochure.

Alfréd Rényi (1921-1970)

B

Alfréd Rényi

orn and educated in Budapest, he became a student of *Lipót Fejér*, and wrote his dissertation in 1945 under the supervision of *Frigyes Riesz*. In 1946 he went to Leningrad (now Saint Petersburg) where *Yu. V. Linnik* and *I. M. Vinogradov* were his advisors. His solution of the so-called quasi-Goldbach conjecture in his 1947 thesis attracted considerable attention. Partially due to this result, he became a corresponding member of the Hungarian Academy of Sciences in 1949. He received an associate professorship at the Loránd Eötvös University of Budapest in 1947, and a full professorship in Debrecen in 1949. In 1950 he was appointed as director of the Applied Mathematical Institute of the Hungarian Academy of Sciences. From 1952 on he was also chair of the department of probability Theory at Loránd Eötvös University. He kept both positions until his death in 1970. He took part in public life with amazing energy: he was a member of the editorial board of several journals and secretary general of the János Bolyai Mathematical Society. As director, he provided shelter at the Institute for various prominent scientists who were persecuted after the 1956 revolution.

I

Alfréd Rényi

n his scientific work he made notable contributions to several branches of mathematics. Some of his most important results concern probabilistic methods in number theory (including a breakthrough in Goldbach problems), the development of the theory of random graphs (jointly with *Paul Erdős*), and the introduction of the so-called Rényi-entropy. The famous Hungarian school in statistics grew out from his work. His textbook on probability theory, first published in German and then translated into several languages, has been in widespread use. Aside of his theoretical work, he published regularly on popular mathematics, and encouraged applications of mathematics. He was chair or co-chair of numerous mathematical conferences, and was visiting professor at various universities around the world. Through his students, his spirit is still alive today.

For his contribution to the axiomatization and application of probability theory he received the Kossuth Prize (the highest scientific prize awarded by the government) twice: in 1949 and in 1954. In 1956 he became a full member of the Hungarian Academy of Sciences. Two years after his untimely death in 1970, the Institute founded a prize in his memory; since 1999 it also bears his name.

Research at the Institute

Currently, researchers at the institute work in 10 research divisions, and some of them also in smaller research groups focused on more special subjects. The 10 current research divisions are:

Algebra

Algebraic Geometry and Differential Topology

Algebraic Logic

Analysis

Combinatorics and Discrete Mathematics

Convex and Discrete Geometry

Information Theory

Number theory

Probability Theory and Mathematical Statistics

Set Theory and Set-theoretic Topology

In addition, research groups on cryptology, database theory, mathematical immunology and statistical physics are also active at the Institute.

Let us now briefly review the history and activities of the various research groups.

The school in *combinatorics* is the first to be mentioned. As the result of the work of* Paul Erdős, Tibor Gallai, András Hajnal, Alfréd Rényi, Pál Turán, Vera T. Sós* and their students, the Institute (in close collaboration with Eötvös University) has become one of the world centers in combinatorics. Here is the birthplace of the theory of random graphs, of the so-called probabilistic method in combinatorics, of basic results about extremal graphs and extremal set systems, or the regularity lemma describing the structure of huge graphs. Our researchers have achieved breakthrough results concerning discrepancy, combinatorial methods in group theory, and in combinatorial applications of entropy and graph entropy. Among the combinatorial applications of theoretical computer science, important results have been achieved in search theory, in the complexity of combinatorial algorithms, in cryptology and in the combinatorial theory of databases (in cooperation with the Computer and Automation Research Institute of the Hungarian Academy of Sciences).

From the very beginning, *number theory* has also been one of the most ardently cultivated areas of research at the Institute. A significant analytic and additive number theoretic school was formed under the leadership of* Pál Turán*, in close collaboration with researchers from Loránd Eötvös University. Basic results about the Riemann hypothesis, the distribution of prime numbers, multiplicative functions, dense sequences and probabilistic constructions of number theory have been achieved by *Gábor Halász*, *András Sárközy*, *Endre Szemerédi*, *Imre Ruzsa *and their collaborators.

The

*analysis* group was founded by

*Béla Szőkefalvi-Nagy*, the famous functional analyst. Following the footsteps of

*Lipót Fejér,* *Gábor Szegő, P**ál Erdős, Pál Turán* and

*Géza Freud,* research in approximation theory and the theory of orthogonal polynomials has been the main focus of analysts at the Institute. They initiated lacunary interpolation within interpolation theory, an area still active today. More recently, rational approximation and weighted approximation have also been the object of thorough study.

R

Handwritten notes by Paul Erdős

esearch in *set theory* and in *mathematical logic* has also been successful and highly acclaimed. Outstanding results have been achieved in combinatorial set theory, culminating in a monograph about the theory of partitions co-authored by *András Hajnal*, former director of the Institute and later director of DIMACS in New Jersey, *Paul Erdős*, and *Richard Rado*. Presently, set-theoretic topology and mathematical logic also have active groups at the Institute. During the last decades, Budapest has become an important center for *algebraic logic* as well. This school continues related work by Alfred Tarski exploring a flourishing connection between cylindric algebras and logic.

*Discrete geometry* has become an independent discipline in mathematics as a consequence of the work of *László Fejes Tóth*, another former director of the Institute. Hungarian researchers have played an essential role in this process. The discrete geometry group of the Institute, together with researchers from Loránd Eötvös University, achieved significant results in the field.

Being the main area of research cultivated by

*Alfréd Rényi*, the founder of the Institute,

*probability theory* and

*mathematical statistics* have always been present, with particular emphasis on applications. It was also

*Rényi* who started to work on

*information theory* in his last years. Later the Information Theory Division became famous under the leadership of

* Imre Csiszár*. One of the most significant results here was the elaboration of multiuser information theory. The group’s work opened up new directions in the application of information theory, notably in statistics (maximal entropy), in probability theory (measure concentration, theory of large discrepancies), and in combinatorics (graph entropy, graph conductivity).

The *Algebra Division* was formed about forty years ago by *László Rédei*. Its research profile initially focused on group and semigroup theory, lattices and universal algebra. This spectrum has gradually expanded with the introduction of category theory, linear algebra and ring theory. Presently a group working on asymptotic properties of groups is also very active at the Institute.

*Biometrics,* the branch of statistics connected with medicine, has always played an important role in the life of the Institute. Outstanding advances connected with genetics and recognition of cancer cells have resulted from statistical investigations conducted by *Gábor Tusnády*.

The founder of the Hungarian school in *statistical physics* is *Domokos Szász*. The Institute has been one of the centers of this field. The main areas investigated are the dynamic theory of Brownian motion, infinite differential equations, and the mathematical foundations of the Boltzmann hypothesis. In the latter area of investigation, work of researchers of the Institute has produced breakthrough results. The 1979 Kőszeg conference in this area was of essential importance; here researchers from the Soviet Union and from Western countries worked together on the topic for the first time.

The group in *algebraic geometry and differential topology* is the youngest of the Institute. It came into existence in 1998 when young researchers having obtained their PhD from prestigious foreign universities launched a weekly seminar. Members of the group are working on a wide range of topics in the forefront of current research but previously not cultivated in Hungary, including classification of algebraic varieties, singularity theory, the theory of motives or low-dimensional topology.

Contributions to applications of mathematics

The Institute was founded as a research centre for applied mathematics, and applications determined its main profile during the 1950’s. This included close collaboration with industrial partners. Over the decades, the Institute’s main focus shifted towards theoretical research, especially after the foundation of the Computer and Automation Research Institute of the Academy which took over several fields of research in applied mathematics from the Institute.

Nevertheless, applications still play an important part in the Institute’s life. Results of our colleagues working in information theory and database theory are implemented in cryptography and computer science. Within statistics, pre-estimation of population size and other problems related to living organisms are among the most cultivated areas of application at the Institute. In the 1980’s the Institute’s help was essential for the revival of a profession forgotten in Hungary for over forty years: that of insurance mathematics. The Institute also took an active part in preparing the reform of the Hungarian pension system.

In collaboration with the Hungarian Oncology Institute, our researchers have developed a computer program for recognizing cancerous cells. Another joint project, developed together with ASK Ltd., was a device for cutting curtains, presented at the 2000 World Exhibition in Hannover. Among ongoing collaborations, the most important one is a joint project in cryptography together with Hewlett Packard Hungary, aiming at the elaboration of “digital watermarking”.

With the evolution of industrial applications in Hungary, it is expected that there will be a growing demand for theoretical mathematics and its applications. The Institute is ready for the challenge.

Scientific meetings at the Institute

Most of our research divisions run a *weekly seminar* for specialists in the field during the academic year. These have met at the same hour for decades now and form part of the Institute’s tradition; for instance, everyone knows that Thursday afternoon is for combinatorics, or Monday morning is for algebra.

About once a month, always on a Monday afternoon,

*a colloquium lecture* is organized for all members of the Institute and for interested visitors. The aim of these lectures is to present significant results of researchers of the Institute to a wider audience, and also to give an opportunity for prominent researchers from Hungary and abroad to speak about less specialized topics.

Every two years, the Institute organizes jointly with the János Bolyai Mathematical Society the *Pál Turán Memorial Lectures*. This is usually a series of three lectures given by a famous expert active in one of the research areas of the late Pál Turán: number theory, analysis or combinatorics. Recent Turán lecturers include *Hugh Montgomery*, *Peter Lax* and *Peter Sarnak*.

Since the 1960’s, the Institute has been hosting and organizing *international conferences *ranging from 2-day workshops devoted to a specialized topic to large conferences with hundreds of participants. During recent years, the largest meetings organized by members of the Institute were the 2^{nd} European Congress of Mathematics in 1996, whose organizing committee was chaired by the director of the Institute, the 1999 conference on ‘Paul Erdős and His Mathematics’, which had 450 participants from all over the world, and the 2001 Euroconference on ‘Finite and Infinite Combinatorics’. In September 2001, the Institute hosted a 3-week summer school and workshop in algebraic geometry devoted to ‘Higher Dimensional Varieties and Rational Points’ which received an important European subvention, and in 2003 the Von Neumann Centennial Conference on linear operators and the foundation of quantum mechanics, co-organized by the Bolyai Society and the American Mathematical Society. In June 2004, a 3-week workshop in low-dimensional topology will take place at the Institute, organized jointly with the Clay Mathematical Institute.

Apart from these large events, the Institute regularly hosts a number of international conferences and workshops for smaller groups of specialists, which since 1998 are grouped in a *Turán Workshop Series* in mathematics.

The webpage http://www.renyi.hu/old-conferences.html contains a list of meetings organized at the Institute since 1999.