Hillel Furstenberg

Quick Info

29 September 1935
Berlin, Germany

Hillel Furstenberg is a German-born American mathematician working in combinatorics, probability theory, ergodic theory and topological dynamics. He has been awarded major mathematical prizes including the Wolf Prize and the Abel Prize.


Hillel Furstenberg is known to his friends and colleagues as Harry. He was the son of Solly Furstenberg (1900-1940) and Berta Gryzb (1908-1987). Harry had an older sister, Ruth Charlotte Furstenberg, born 27 May 1932. He was born into a Jewish family living in Germany shortly after Hitler had come to power and the Nazi party had passed anti-Semitic legislation. Problems for Jewish people became increasingly difficult over the first few years of Hillel's life and, because his grandparents were from Poland and Russia, the family were ordered to leave Germany in November 1938. They were sponsored by Berta's sister Anna Haller, and planned to go to England, spend a year in London, then emigrate to the United States. Before they left, on the night of 9-10 November 1938, Jewish businesses were smashed up, synagogues were destroyed on what has become known as Kristallnacht. Although Harry was only three years old, the memory of that night stayed with him [6]:-
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.
It must have been a traumatic experience for the young boy. At the time of the 1939 census, shortly before the start of World War II in the autumn of that year, the family was living at 2 Everswell Avenue, Downham, Norfolk, England. Solly's occupation is given as Furniture Dealer and Polisher with his wife Berta records her occupation as Unpaid Domestic Duties. Berta's brother, Morris Giles, had a poultry farm in New Jersey, United States, and the family planned to go there. Solly, however, had a medical condition which he believed might prevent him entering the United States so he underwent surgery in London. The operation was not successful and Solly died in February 1940. He was buried in the East Ham Jewish Cemetery in Newham, London.

On 18 October 1940 Berta, Ruth and Harry Furstenberg sailed from Glasgow, Scotland, to New York on the Cameronia. Although many ships were crossing the Atlantic in convoy because of fear from German submarine attacks, the Cameronia was unescorted. It was, in fact one of the Cameronia's last passenger carrying journeys across the Atlantic since she became a troop carrying ship in December 1940. The Furstenberg family arrived in New York on 28 October. On entering the United States, Harry's record shows: Age, 5; Height, 4 ft; Hair, light brown; Eye colour, blue; Complexion, medium; Citizenship intention, yes. The family went, as intended, to live with on Morris Giles's poultry farm. Harry's mother worked on the farm while Harry began his education in an elementary school in East Brunswick. They did not live there for very long, however, and they moved to Washington Heights, the attraction there being a large German Jewish community, but they found a house in a multi-ethnic area near Yeshiva University. Harry [6]:-
... 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."
The Furstenberg family suffered financial hardships; only having an income from Freda working as a sewing machine operator. To help support the family, Freda decided that Harry should learn a trade and, after taking a test, he was admitted to Brooklyn Technical High School with the aim of studying electrical engineering. The local Rabbi, however, persuaded Harry's mother that he should train to become a Rabbi, so after only one day at the Technical High School he entered the Talmudical Academy (now the Yeshiva University High School for Boys), located on the campus of Yeshiva University in northern Manhattan. He said [10]:-
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.
The 1950 US Census records Berta, Ruth and Harry Furstenberg living in New York City. Berta is a machine operator in an underwear factory, Ruth is a clerk in an insurance agency while Harry is at school. In his final year at Talmudical Academy, Harry Furstenberg was vice-president of the Science Club and consulting physicist to the Chemistry Club. He won an honourable mention in the Tenth Annual Westinghouse Science Talent Search Contest. He poked fun at his mathematics teachers in the 1951 yearbook [32]:-
After having taught Mr Lichtenberg all he knows about Math, Harry is now venturing forth to teach Mr Greitzer the Fundamentals of Elementary Arithmetic.
Joseph Lichtenberg, B.A., Columbia University, 1920; M.A., 1921, taught mathematics at Talmudical Academy while Samuel L Greitzer, M.A., Columbia University, 1936, taught both mathematics and physics.

After Furstenberg graduated from the Academy in 1951 he studied mathematics and divinity at Yeshiva College. Furstenberg recalled his time at university [12]:-
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 Ginsburg 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.
You can read more of Furstenberg's memories of Jekuthiel Ginsburg at THIS LINK.

At Yeshiva College, Furstenberg was fortunate to be able to attend lectures given by leading mathematicians [12]:-
... 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.
Furstenberg was president of the Math Club at Yeshiva College in 1955 [24]:-
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].
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 [4]:-
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, ab1ab^{-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 ab=ab1a^{*}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 the review [23]:-
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.
It was while he was at Princeton University that he met the girl he was to marry. The story is related in [6]:-
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."
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 [22]:-
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 [17]:-
... 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 [27]:-
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.
One of the highly prestigious awards given to Furstenberg has been the 2007 Wolf Prize [20] (see also [13]):-
... 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 [20] (see also [13]):-
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.
In addition to these honours, Furstenberg has been elected to the Israel Academy of Sciences (1974), the United States National Academy of Sciences (1989), and the American Academy of Arts and Sciences (1995). The Academy recorded his membership as follows [18]:-
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.
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. He delivered the Paul Turán Memorial Lectures in 2006. The topics of the three lectures were [24]:-
Lecture 1. Number Theory, Combinatorics and Recurrence in Dynamical Systems; the Correspondence Principle.
Lecture 2. Ergodicity, Mixing, Conventional and non-Conventional Ergodic Theorems.
Lecture 3. The Long Term Memory of Dynamical Systems and the Strange Role of Nilpotent Groups and Nilflows.
In 2008 he delivered the Twenty-Eighth Annual Bowen Lectures at Berkeley. He gave the following Abstract [1]:-
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.
We also 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.

The Abel Prize is recognised as the highest possible award to a mathematician. It was presented jointly to Hillel Furstenberg and Gergory Margulis in 2020 [33]:-
... for pioneering the use of methods from probability and dynamics in group theory, number theory and combinatorics.
The Press Release states [34]:-
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.
There is more about this Abel prize at THIS LINK.

As a final comment, let us note that Ruth Furstenberg, Harry's older sister, married the engineer Herbert Berger (born in New York City on 3 October 1931). They emigrated to Israel and Ruth died there on 8 August 2001.

References (show)

  1. 2008 Bowen Lectures, University of California, Berkeley.
  2. A biography of Hillel Furstenberg, International Mathematical Union. https://www.mathunion.org/fileadmin/IMU/Prizes/Abel/2020/biography_english_HF.pdf
  3. A biography of Hillel Furstenberg, Abel Prize, Norwegian Academy of Science and Letters (2020).
  4. W W Boone, Review: The inverse operation in groups, by Harry Furstenberg, Mathematical Reviews MR0077541 (17,1053b).
  5. M Brennan, YU Alumnus Dr Hillel Furstenberg Wins the Abel Prize, the "Math Nobel", YU News, Yeshiva University (25 March 2020).
  6. G Burstyn, Professor Hillel Furstenberg: In His Prime, aish (14 October 2020).
  7. G Burstyn, In His Prime, Mishpacha (14 October 2020).
  8. G Burstyn, Professor Hillel Furstenberg: In His Prime, aish (14 October 2020).
  9. B I Dundas and C Skau, Interview with Abel Laureate 2020 Hillel Furstenberg, Notices of the American Mathematical Society 68 (7) (2021), 1189-1196..
  10. B I Dundas and C Skau, Interview with Abel Laureate 2020 Hillel Furstenberg, European Mathematical Society Newsletter 118 (2020), 45-51.
  11. L Fuller-Wright, Mathematics Ph.D. alumnus Furstenberg receives Abel Prize, Princeton University (19 March 2020).
  12. H Furstenberg, Inspiring the Love of Mathematics: The Legacy of Jekuthiel Ginsburg at Yeshiva, The Commentator: The Official Newspaper of Yeshiva College (16 May 2005).
  13. Furstenberg and Smale Receive 2006-2007 Wolf Prize, Notices Amer. Math. Soc. 54 (5) (2007), 631-632.
  14. Furstenberg, Hillel, Encyclopedia.com.
  15. Furstenberg and Margulis awarded 2020 Abel prize, Notices of the American Mathematical Society 67 (6) (2020), 908-911.
  16. L Gierczak, Prix Abel 2020: Hillel Furstenberg et Gregori Margulis, Pour la Science 511 (5) (2020), 13b-13b.
  17. Harvey Prize 1993 Hillel Furstenberg, TECHNION Israel Institute of Technology.
  18. Hillel Furstenberg, American Academy of Arts and Sciences (September 2023).
  19. Hillel Furstenberg, a graduate student *58 of Salomon Bochner at Princeton, shared the 2020 Abel Prize with Gregory Margulis, Department of Mathematics, Princeton University (2020).
  20. Hillel (Harry) Furstenberg, Wolf Prize Laureate in Mathematics 2006/7, The Wolf Prize (11 December 2018).
  21. Hillel H Furstenberg, National Academy of Sciences.
  22. M Keane, Review: Recurrence in ergodic theory and combinatorial number theory, by H Furstenberg, Mathematical Reviews MR0603625 (82j:28010).
  23. P Masani, Review: Stationary processes and prediction theory, by Harry Furstenberg, Mathematical Reviews MR0140151 (25 #3573).
  24. MASID, Yeshiva College (June 1955).
  25. Paul Turán Memorial Lectures 2006 Hillel Furstenberg, Hungarian Academy of Sciences.
  26. Professor Hillel Furstenberg, Israel Academy of Sciences and Humanities.
  27. Professor Hillel Furstenberg, Einstein Institute of Mathematics.
  28. Professor Hillel Furstenberg, The EMET Prize 2004, IsraCast (7 October 2007).
  29. M Rosenblatt, Review: Stationary processes and prediction theory, by Harry Furstenberg, Quarterly of Applied Mathematics 19 (2) (1961), 174-175.
  30. SIAM, Review: Stationary processes and prediction theory, by Harry Furstenberg, SIAM Review 3 (1) (1961), 77.
  31. L Sloman, From Systems in Motion, Infinite Patterns Appear, Quanta Magazine (5 December 2022).
  32. The Elchanite, Talmudical Academy (1951), 22.
  33. The Abel Prize 2020, Citation, Abel Prize, Norwegian Academy of Science and Letters (2020).
  34. The Abel Prize 2020, Press Release, Abel Prize, Norwegian Academy of Science and Letters (2020).
  35. The Israel Academy is proud to congratulate its Member, Prof Hillel Furstenberg upon receiving the prestigious Abel Prize for 2020, Israel Academy of Sciences and Humanities (18 March 2020).
  36. B Weiss, Life and work of Hillel Furstenberg, Nieuw Arch. Wiskd. (5) 22 (1) (2021), 14-20.
  37. G Weiss, Review: Stationary processes and prediction theory, by Harry Furstenberg, Science, New Series 133 (3458) (1961), 1069.

Additional Resources (show)

Other pages about Hillel Furstenberg:

  1. Inspiring the Love of Mathematics
  2. The Abel Prize 2020

Other websites about Hillel Furstenberg:

  1. Mathematical Genealogy Project
  2. MathSciNet Author profile
  3. zbMATH entry

Honours (show)

Cross-references (show)

Written by J J O'Connor and E F Robertson
Last Update December 2023