Mathematics is an actively studied field in Hungary. For example, in the Alfréd Rényi Institute of Mathematics more than hundred researchers work on various areas like number theory, graph theory or low dimensional topology. This also shows that mathematics has long traditions in the country. In this short article, we present the work of a few Hungarian mathematicians who became widely known abroad as well (some of them also lived abroad for several years or decades), and whose research had significant impact on the development of mathematics in Hungary.

At EGMO, the problems are usually from the areas of geometry, combinatorics, number theory and algebra. Hence, although it was not very common in the 20th century that a mathematician worked in exactly one of these fields, we go through these topics to see the contributions of Hungarian mathematicians.

**Geometry**

As we have all learned at school, and use it in mathematical problem solving from time to time, the sum of the angles of a triangle is equal to 180°. However, as the example of a Hungarian military officer from the 19th century shows, interesting mathematics can start with questioning statements looking as trivial as this one.

János Bolyai (Kolozsvár/Cluj, 1802 – Marosvásárhely/Mureș-Oșorhei, 1860) was born in Transylvania. He could learn calculus and analytical mechanics from his father, who was also a mathematician, and worked as a teacher in Marosvásárhely. János studied for four years at the Imperial and Royal Military Academy in Vienna, Austria. Later, he served the military as an engineer for more than ten years, until his injury and retirement.

Already during his studies in Vienna, it became clear that calculating the trajectory of a cannonball is not the kind of mathematics he is really interested in. He started to work on an old question related to Euclid’s axioms of geometry, namely, on the question about the parallel postulate. In Euclid’s work Elements (written around 300 BC), one of the axioms is equivalent to the following: given a line and a point outside of it, there is at most one line which goes through the point and which is parallel to the line. This is also equivalent to the fact that the sum of the angles in a triangle is equal to 180°. For Bolyai, the following question was exciting and gripping: is this true in every „consistent” geometric model of the plane?

This question was studied by other mathematicians at that time. János Bolyai (independently of the Russian mathematician Nikolai Ivanovich Lobachevsky) settled this by constructing a geometric model where Euclid’s every other axiom is satisfied, but there are several different parallel lines going through the same point outside the original line. This also meant that the sum of the angles in a triangle can be less than 180°.

This was one of the first models of non-Euclidean geometry. Together with other similar models, it became very important in the study of surfaces, for example, in the work of Bernhard Riemann. As the structure of space is a basic question in physics and astronomy, these models were also key elements in the cosmological theory of Hermann Minkowski, and had significant effect on the special relativity theory of Albert Einstein. This also shows that the work of János Bolyai had a huge impact on the way we think of geometry, surfaces and space.

**Number theory**

Another famous story of Hungarian mathematics is that of Pál (Paul) Erdős. He strongly believed that finding the right questions is at least as important as the solution – and he proved this by proposing hundreds of problems to his colleagues, during his intensive social activities and often unexpected visits around the world. He liked offering payments (from 25 dollars to several thousand dollars) for colleagues who first solve his favourite problems in number theory, combinatorics, graph theory, probability theory. Many of these questions were settled throughout the years, others are still open and serve as a basis for active research.

Pál Erdős (Budapest, 1913 – Warsaw, 1996) was the son of a couple of two teachers. During his years in high school, he regularly worked on the problems of KöMaL, a (still published) monthly journal inteded to help students to develop their mathematical knowledge and problem-solving skills. He was a very active member of a group of young mathematicians and students, who regularly met in Városliget, the largest park of Budapest, and worked on various open problems. Later, Erdős spent several years abroad (in Manchester and Princeton, for example), but from 1955, he could return to Budapest. He continuously growed the group of mathematicians he worked together with, and, with the help of his collaborators, regarding the number of published papers, he became the second most prolific mathematician on the world (with around 1500 papers, just after the Swiss mathematician Leonhard Euler, listed here).

He had various results in number theory, combinatorics and other topics, and his work had a huge impact on the research of many mathematicians, both in Hungary and around the world. To mention one particular result from number theory: with Ginsburg and Ziv, he proved that for every integer *n*, among *2n-1* integers, we can always find exactly *n* numbers whose sum is divisible by *n*.

**Combinatorics**

The area of combinatorics is very popular among Hungarian mathematicians, partially due to the effect of the work of Pál Erdős. Hence it is not by chance that two out of the three Hungarian mathematicians who have been awarded the Abel prize (Péter Lax, László Lovász, Endre Szemerédi), have done significant research in the area of combinatorics.

László Lovász was born in Budapest, in 1948. Both within and outside the area of combinatorics, his research is often motivated by problems coming from computer science (notice that 1948 was the year when „Manchester Baby” run its first program, which meant that it was not necessary any more to rewire or restructure the machines to modify the set of instructions, and that the memory of a computer could be used as we more or less use it nowadays as well). Among other results, Lovász proposed widely used algorithms (e.g. the LLL lattice reduction algorithm), and elaborated the algorithmic theory of convex bodies and grids. He also had a leading role in the foundations of graph limit theory, as his book „Large networks and graph limits” shows. This topic was partially motivated by the evolving interest in the structure of large networks like the internet, or online social networks. His work showed that a new viewpoint on these large discrete structures can lead to various new results by the application of tools from mathematical analysis and probability theory.

Endre Szemerédi (born in Budapest, in 1940) also worked in the fields of combinatorics and computer science. The Szemerédi regularity lemma analyses the structure of large graphs (networks), by stating that we can partition the vertices in such a way that we see a simpler, homogeneous structure with a small random error. Another very famous result of Szemerédi is about arithmetic progressions, which is an important topic of arithmetic combinatorics. Namely, in 1975, he proved a conjecture of Erdős and Turán, by showing that if a subset of the natural numbers contains at least 1 % of the first *n* numbers for infinitely many numbers *n*, then, for every *k*, the subset contains infinitely many arithmetic progressions of length *k*. Of course, 1% can be replaced by any positive ratio in this statement.

**Algebra**

John von Neumann (Budapest, 1903 – Washington, 1957) was born in Hungary, but he studied in Germany and Switzerland, and spent most of his adult life in the United States. The mathematical object called von Neumann algebra proves that, among many other areas, he worked in algebra. More precisely, his interest in quantum mechanics lead him to studying such abstract algebraic notions. Also related to physics and quantum mechanics, as many of the leading scientists at that time, he was involved in the Manhattan Project, that is, in the development of nuclear weapons.

He was also one of the leading scientists in computer science. He was working on constructing some of the first electronic computers, and his ideas about how a computer should work are used nowadays as well. In particular, he was a member of the team who built EDVAC, which was finished in 1949, and, similarly to „Manchester Baby”, it was among the first computers that could store the program codes in their own memory. John von Neumann also worked on game theory, a quickly developing area of economics at that time.

In addition, if you are interested in what it was like in the second half of the 20th century to work as an economist and university professor, corporate director, advisor of government committees – as a women in communities consisting mostly of men, we highly recommend the book of Marina von Neumann Whitman, the daughter of John von Neumann (*The Martian’s Daughter. A Memoir*, 2012).

Source of photos: Wikipedia, homepages of Eötvös Loránd University