Hillel Furstenberg
Hebrew: הלל פירסטנברג
 
Birthdate:  (80) 
Birthplace:  Berlin, Berlin, Germany 
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About Hillel Furstenberg
Born: 29 Sept 1935 in Berlin, Germany 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 antiSemitic 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 :
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 humankindness, 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:
... while still an undergraduate, I was exposed to a series of highlevel 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, ab1. 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 = ab1. If in addition (c*b)*(c*a) = a*b for all a, b, c in G, then G is abelian. In analogy with semigroups, a "halfgroup" is a system G satisfying (1) and (2). (Not every halfgroup is a group.) A structure theorem for halfgroups 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 timesequences 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 firstrate and highly original dissertation on a very difficult subject.
After a year 195859 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 measuretheoretic 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
Hillel Furstenberg's Timeline
1935 
1935

Berlin, Berlin, Germany


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