**Hillel Furstenberg** is known to his friends and colleagues as Harry. He was born into a Jewish family living in Germany shortly after Hitler had come to power and his Nazi party had passed anti-Semitic legislation. Problems for Jewish people became increasingly difficult over the first few years of Hillel's life and, in 1939, shortly before the start of World War II in the autumn of that year, the Furstenberg family emigrated to the United States. The family settled in New York but suffered financial hardships following the death of Hillel's father. However, he attended Talmudical Academy (now the Yeshiva University High School for Boys), located on the campus of Yeshiva University in northern Manhattan, and after graduating from the High School in 1951 he studied mathematics and divinity at Yeshiva College. Furstenberg recently recalled his time at university [1]:-

To me, as undoubtedly to many who attended Yeshiva College in the early1950s, the subject of mathematics was identified with one remarkable individual, Professor Jekuthiel Ginsburg. ... In the classroom, he communicated to his students the innate beauty of abstract mathematical ideas. ... It is hard to imagine a professional career that owes more to one individual and to one institution than my own career owes to Jekuthiel Ginsberg and Yeshiva University. Over and beyond the mathematics I learned, I experienced the love of mathematics blended with human-kindness, an experience I can only wish I could replicate for others.

At Yeshiva College, Furstenberg was fortunate to be able to attend lectures given by leading mathematicians [1]:-

... while still an undergraduate, I was exposed to a series of high-level lectures in advanced topics given by prominent professors who visited[Yeshiva College]from a number of institutions. These included Samuel Eilenberg and Ellis Kolchin from Columbia University, Jesse Douglas from City College, and Abe Gelbart who travelled from Syracuse University.

In 1955 Furstenberg graduated from Yeshiva College having been awarded both a B.A. and an M.Sc. He had already published a number of papers with *Note on one type of indeterminate form* (1953) and *On the infinitude of primes* (1955) both appearing in the *American Mathematical Monthly*. The paper on primes gives a topological proof that there are infinitely many primes. Also in 1955, the year he gained his first degree, he published *The inverse operation in groups* in the *Proceedings* of the American Mathematical Society. This is a lovely paper, giving results which could be incorporated into a group theory course. Bill Boone reviewed the paper:-

The author gives an elegant set of postulates for groups in terms of a single binary operation which occurs quite frequently in group theoretic analyses, ab^{-1}.

Let G be a system with an operation a*b such that

(1)a*b in G for any a, b in G,

(2) (a*c)*(b*c)=a*b for any a, b, c in G,

(3)a*G = G for any a in G.(

Then it follows that there is an e in G such that a*a = e for all a in G, that G is a group under the operation ab=a*e*b), and that a*b = ab^{-1}. If in addition(c*b)*(c*a)= a*b for all a, b, c in G, then G is abelian. In analogy with semi-groups, a "half-group" is a system G satisfying(1)and(2). (Not every half-group is a group.)A structure theorem for half-groups is demonstrated.

Furstenberg went to Princeton University to study for his doctorate, supervised by Salomon Bochner. At this time Bochner was interested in probability, having published his classic text *Harmonic Analysis and the Theory of Probability* in 1955, the year in which Furstenberg began research. After submitting his thesis *Prediction Theory* in 1958, Furstenberg was awarded his doctorate. This thesis was published as *Stationary processes and prediction theory* in 1960. P Masani writes in a review:-

In this work the limitations of the classical prediction theory of stochastic processes are first discussed. In the light of this discussion a new prediction theory for single time-sequences is formulated. The ideas uncovered in the course of this development are shown to have interesting ramifications outside prediction theory proper. ... the work stands as a first-rate and highly original dissertation on a very difficult subject.

After a year 1958-59 as an Instructor at Massachusetts Institute of Technology, Furstenberg worked at the Mathematics Department in the College of Science, Letters, and Arts of the University of Minnesota. In this Department he was a member of a strong group working on probability theory. In 1963 the two University of Minnesota Departments of Mathematics were merged into the School of Mathematics in the Institute of Technology and in the following year Furstenberg was appointed a full professor. In 1965, along with his wife Rochelle, he went to Israel when he was appointed as Professor of Mathematics at the Hebrew University of Jerusalem. Rochelle is a writer and magazine editor specialising in arts and contemporary culture. Harry and Rochelle Furstenberg have five children. Furstenberg remained at the Hebrew University until he retired in 2003. He has also taught at Bar Ilan University.

Many important results due to Furstenberg are presented in his classic monograph *Recurrence in ergodic theory and combinatorial number theory* (1981). Here are extracts from a review by Michael Keane:-

This very readable book discusses some recent applications, due principally to the author, of dynamical systems and ergodic theory to combinatorics and number theory. It is divided into three parts. In Part I, entitled "Recurrence and uniform recurrence in compact spaces", the author gives an introduction to recurrence in topological dynamical systems, and then proves the multiple Birkhoff recurrence theorem ... From this theorem a multidimensional version of van der Waerden's theorem on arithmetic progressions is deduced, and applications to Diophantine inequalities are given. Part II carries the title "Recurrence in measure preserving systems". After a short introduction to the relevant part of measure-theoretic ergodic theory, this section is devoted to a proof of the multiple recurrence theorem ... From this result the author deduces a multidimensional version of Szemerédi's theorem on the existence of arbitrarily long arithmetic progressions in sequences of integers with positive density. Part III, called "Dynamics and large sets of integers", investigates the connections between recurrence in topological dynamics and combinatorial results concerning finite partitions of the integers(e.g., Hindman's theorem, Rado's theorem). Here the notion of proximality plays a central role. In reading this book, the reviewer found that the first part tickled his imagination and made him want to continue, the second part provided a good deal of work and tested his technical ability, while the last part led him to imagine the future possibilities for research. An excellent work!

Let us look at some of the awards that Furstenberg has received so that we can mention his greatest mathematical achievements. The Israel Prize, an award made by the State of Israel that is regarded as the state's highest honour, was presented to Furstenberg in 1993. In the same year Furstenberg received the Harvey Prize, awarded annually by the Technion in Haifa, Israel, for:-

... ground-breaking work in ergodic theory and probability, Lie groups and topological dynamics.

In 2004 he received the EMET Prize, an annual award given for excellence in academic and professional achievements that have far reaching influence and significant contribution to society. The prizes are sponsored by the Foundation for the Advancement of Science, Art and Culture in Israel acting for the Prime Minister of Israel. Here is an extract from the citation for the Prize [3]:-

Professor Furstenberg's immigration to Israel had great influence on the field of mathematics in the country, and helped transformed Israel into an important international centre in ergodic theory in particular and in mathematics in general. In Jerusalem, which was the centre of his academic activities, he continued producing a long series of monumental mathematical works. In1975he inaugurated, along with Professor Benjamin Weiss, an ergodic theory research year in Jerusalem. That year is still remembered as the year that entirely changed the face of research in the field. Through the years, he has been a guest lecturer at many universities around the world, including Stanford, Yale, and others. Professor Furstenberg has taught and guided many students studying towards advanced degrees in his field of research and in other, wider fields - thus ushering a new generation of mathematicians who today serve as professors at institutions of higher learning in Israel and abroad.

The most prestigious award given to Furstenberg has been the 2007 Wolf Prize [4] (see also [2]):-

... for his profound contributions to ergodic theory, probability, topological dynamics, analysis on symmetric spaces and homogenous flows.

The citation goes into more details of Furstenberg's contributions to these areas which led to the award [4] (see also [2]):-

Professor Harry Furstenberg is one of the great masters of probability theory, ergodic theory and topological dynamics. Among his contributions: the application of ergodic theoretic ideas to number theory and combinatorics and the application of probabilistic ideas to the theory of Lie groups and their discrete subgroups. In probability theory he was a pioneer in studying products of random matrices and showing how their limiting behaviour was intimately tied to deep structure theorems in Lie groups. This result has had a major influence on all subsequent work in this area - which has emerged as a major branch not only in probability, but also in statistical physics and other fields. In topological dynamics, Furstenberg's proof of the structure theorem for minimal distal flows, introduced radically new techniques and revolutionized the field. His theorem that the horocycle flow on surfaces of constant negative curvature is uniquely ergodic, has become a major part of the dynamical theory of Lie group actions. In his study of stochastic processes on homogenous spaces, he introduced stationary methods whose study led him to define what is now called the Furstenberg Boundary of a group. His analysis of the asymptotic behaviour of random walks on groups, has had a lasting influence on subsequent work in this area, including the study of lattices in Lie groups and co-cycles of group actions. In ergodic theory, Furstenberg developed the fundamental concept of dynamical embedding. This led him to spectacular applications in combinatorics, including a new proof of the Szemerédi Theorem on arithmetical progressions and far-reaching generalizations thereof.

In addition to these honours, Furstenberg has been elected to the Israel Academy of Sciences and the United States National Academy of Sciences. In 2003, on the occasion of Furstenberg's retirement, the Israel Science Foundation organised a research workshop *Conference on Probability in Mathematics* in his honour. Finally we note that Furstenberg was a plenary speaker at the British Mathematical Colloquium at Bristol in 1984 when he gave the lecture *Ergodic theory and Diophantine problems*.

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