### Endre Szemerédi

Born 21 August 1940, Budapest, Hungary

Abel Prize (2012) “for his fundamental contributions to discrete mathematics and theoretical computer science, and in recognition of the profound and lasting impact of these contributions on additive number theory and ergodic theory.”

Endre Szemerédi’s talent for mathematics was put to good use in primary school – if he helped the toughest boy in his class with exams, he in return received protection from other pupils’ attacks. Szemerédi’s father planned a medical career for his son, and so the young Endre focused on biology, rather than attending one of Hungary’s elite high schools for mathematics. However after six months at university, Szemerédi decided medicine was not for him. He then worked for nearly two years in a machine-making factory, before enrolling in Mathematics at the Eötvös Lorànd University in Budapest (M.Sc. 1965). After that Szemerédi wanted to write his doctoral thesis at Moscow State University. Inspired by his mentor in Budapest, Paul Erdös, Szemerédi was particularly interested in number theory, and hoped to work under the guidance of number theorist Alexander Gelfond in Moscow. But a misspelling gave him instead Israel Gelfand, an expert in functional analysis, as supervisor. Strict bureaucracy made it impossible for him to swap and so he took his Ph.D. in 1970 with Gelfand. After graduating, Szemerédi went to the Alfred Renyi Institute of Mathematics at the Hungarian Academy of Sciences in Budapest, where he still works today. Since 1986, he has also been New Jersey Professor of Computer Science at Rutgers University in New Jersey (USA). In addition, he was often a visiting scholar at prestigious universities: 1974 at Stanford, 1980 at McGill University in Montreal (Canada), from 1981 to 1983 at the University of South Carolina, and from 1985 to 1986 at the University of Chicago. From 1987 to 1988, Szemerédi was Fairchild Distinguished Scholar at the California Institute of Technology, and in 2008 Eisenbud Professor at the Mathematical Science Research Institute in Berkeley.

Endre Szemerédi was first a corresponding member (1982) and then from 1987 a full member of the Hungarian Academy of Sciences, he is a member of the American National Academy of Sciences (2010), the Norwegian Academy of Science and Letters (2012) and the Academia Europaea. He received an honorary doctorate from Charles University in Prague in 2010.

He has received numerous awards, including the Pólya Prize of the Society of Industrial and Applied Mathematics (1975), the Leroy P. Steele Prize (2008), and the Rolf Schock Prize of the Swedish Academy of Sciences (2008).

Szemerédi works at the juncture of randomness and structure. In the early 1970s he completed his most important achievement, the proof of a conjecture by Pál Turán and Paul Erdős from the 1930s. It concerned the question of whether a sequence of natural numbers having positive density in natural numbers, contains arbitrarily long arithmetic progressions. Klaus Friedrich Roth, Fields medalist in 1958, had shown in the 1950s that such sets must contain at least three equally-spaced numbers.

As an example, the set of even numbers has positive density. Szemerédi’s results tell us that such a set contains an arbitrarily long sequence of numbers, which all have the same distance from each other. In our example this would be 2,4,6,8, … – all of them have a distance of 2. In other cases, the question is much more difficult to answer.

Szemerédi’s main tool was his so-called ‘regularity lemma’, a mathematical statement that became the basis of hundreds of works in number theory and discrete mathematics. Szemerédi’s work on arithmetic progressions played some role in the works of both Ben Green, and 2006 Fields medalist Terence Tao. 1998 Fields medalist Timothy Gowers explored and furthered Szemerédi’s theorem. And in 1977, the Israeli mathematician Hillel Fürstenberg gave an alternative proof of the theorem with methods of ergodic theory (stochastics). Thus Szemerédis work became the starting point for a connection between number theory, combinatorics and ergodic theory.