One of my few memories from Germany is Kristallnacht. I remember looking at the broken windows in our apartment. We lived right next to a shul and I remember standing there staring at the broken glass.

... soon became a student at Yeshiva Rabbi Moses Soloveichik on 185th Street, which he attended through the eighth grade. During those elementary school years, he had already shown early promise in mathematics, and his sister, three years his senior, became his tutor. "She was teaching me multiplication when my class was learning addition. I was always ahead of my class," he says. "When you're doing well in school, you tend to become interested in things. I'm not sure whether I had any kind of ambitions in mathematics at that time or had any plans about what I wanted to be."

When I was in high school, I really enjoyed the Euclidean geometry that was taught there. I guess I enjoyed the challenge of geometry exercises. You are able to do things your way. You do not have to follow definite rules, it is about your own thinking. If it is clear and logical, you get to the right answer. I enjoyed that. We learned about imaginary numbers when I was in high school. I thought I could make my name in mathematics if I proved that, when using imaginary numbers like √–1, it was going to lead to some contradiction in mathematics. I filled pages and pages of calculations and of course it didn't get anywhere, but it was a good experience just doing the calculations.

After having taught Mr Lichtenberg all he knows about Math, Harry is now venturing forth to teach Mr Greitzer the Fundamentals of Elementary Arithmetic.

To me, as undoubtedly to many who attended Yeshiva College in the early 1950s, 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.

... 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.

The Math Club, under the leadership of Harry Furstenberg, president, and Isaac Sadowsky, vice-president, endeavours to present advanced mathematical topics in simplified form. The topics prepared in lecture form and given entirely by student members of the club included: "Non-Euclidean Geometry", "Groups of Transformation of Geometry", "Topics in Topology", "Four Colour Theorem", and "Elemental Examples of Gilel's Theorem"[sic].

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.

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.

His roommate from Princeton University, where Furstenberg was then enrolled in the mathematics doctoral program, had seen a young woman on a subway train reading a philosophy book - not the most common sight in 1957. Rochelle Cohen had come from her hometown of Chicago to spend the year in New York City, where she was renting a room in Boro Park. A few months later, when Simchat Torah arrived, she made her way to a local shul to watch the men dance. As it happens, Furstenberg's roommate was celebrating Yom Tov at that same shul, and recognising the girl on the subway and thinking of his bachelor roommate, he approached Rochelle and said: "I know just the guy for you." True, a mathematician and a philosophy aficionada don't necessarily seem like a match made in Heaven. "But I often say that one of the reasons she married me," relates Furstenberg, "is because I convinced her that there's beauty in mathematics. Beauty comes from the hidden, not the revealed."

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!

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

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. In 1975 he 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.

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

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 revolutionised 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 generalisations thereof.

Dr Hillel Furstenberg is a Professor Emeritus of Mathematics at the Hebrew University of Jerusalem in the Einstein Institute of Mathematics. His research focuses on the interaction of stochastic phenomena and ergodic theory with other branches of mathematics; particularly group theory and number theory. He has studied the behaviour of random products of matrices in the context of random walks on Lie groups and the theory of harmonic functions. He has investigated the interaction of recurrence phenomena in ergodic theory with combinatorial number theory. Currently he is studying applications of ergodic theory to the geometry of fractals.

These lectures will focus on the role of ergodic theory in the geometry of fractals. We shall be looking at dynamical systems in which progression in time corresponds to progressively increasing magnification of fractals in Euclidean space. From this point of view the phenomenon of self-similarity for special fractals can be regarded as corresponding to that of periodicity of orbits in dynamical systems. The more general dynamical phenomena of almost periodicity and recurrence also have their counterparts in the geometry of fractals, and much of our discussion will be devoted to clarifying this. It will be convenient to deal with "fractal measures", i.e., measures supported on fractal sets, for which tools of ergodic theory will be available. These ideas will find application in questions involving Hausdorff dimension, but we will see that "ergodic fractal measures" are objects of independent interest.

... for pioneering the use of methods from probability and dynamics in group theory, number theory and combinatorics.

Hillel Furstenberg and Gregory Margulis invented random walk techniques to investigate mathematical objects such as groups and graphs, and in so doing introduced probabilistic methods to solve many open problems in group theory, number theory, combinatorics and graph theory. A random walk is a path consisting of a succession of random steps, and the study of random walks is a central branch of probability theory. "The works of Furstenberg and Margulis have demonstrated the effectiveness of crossing boundaries between separate mathematical disciplines and brought down the traditional wall between pure and applied mathematics," says Hans Munthe-Kaas, chair of the Abel committee. He continues: "Furstenberg and Margulis stunned the mathematical world by their ingenious use of probabilistic methods and random walks to solve deep problems in diverse areas of mathematics. This has opened up a wealth of new results, such as the existence of long arithmetic progressions of prime numbers, understanding the structure of lattices in Lie groups, and the construction of expander graphs with applications to communication technology and computer science, to mention a few."

Hillel Furstenberg was born in Berlin in 1935. His family was Jewish and they managed to flee from Nazi Germany to the U.S. in 1939. Sadly, his father did not survive the journey, and Furstenberg grew up with his mother and sister in an orthodox community in New York. When he published one of his early papers, a rumour circulated that he was not an individual but instead a pseudonym for a group of mathematicians. The paper contained ideas from so many different areas, surely it could not possibly be the work of one man? Following a career in mathematics at several universities in the U.S., he left the country in 1965 for the Hebrew University of Jerusalem, where he stayed until his retirement in 2003. Spending most of his career in Israel, he helped establish the country as a world centre for mathematics. Furstenberg has won the Israel Prize and the Wolf Prize.