György Hajós (February 21, 1912,

Budapest Budapest is the Capital city, capital and List of cities and towns of Hungary, most populous city of Hungary. It is the List of cities in the European Union by population within city limits, tenth-largest city in the European Union by popul ...

– March 17, 1972,

) was a Hungarian

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...

who worked in

group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ( ...

graph theory In mathematics and computer science, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph ...

, and

geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

Biography

Hajós was born February 21, 1912, in

; his great-grandfather, Adam Clark, was the famous Scottish engineer who built the

Chain Bridge A chain bridge is a historic form of suspension bridge for which chains or eyebars were used instead of wire ropes to carry the bridge deck. A famous example is the Széchenyi Chain Bridge in Budapest. Construction types are, as for other suspensi ...

in Budapest. He earned a teaching degree from the

University of Budapest A university () is an institution of tertiary education and research which awards academic degrees in several academic disciplines. ''University'' is derived from the Latin phrase , which roughly means "community of teachers and scholars". Univ ...

in 1935. He then took a position at the

Technical University of Budapest The Budapest University of Technology and Economics ( or in short ), official abbreviation BME, is a public research university located in Budapest, Hungary. It is the most significant university of technology in the country and is considered t ...

, where he stayed from 1935 to 1949. While at the Technical University of Budapest, he earned a doctorate in 1938. He became a professor at the

Eötvös Loránd University Eötvös Loránd University (, ELTE, also known as ''University of Budapest'') is a Hungarian public research university based in Budapest. Founded in 1635, ELTE is one of the largest and most prestigious public higher education institutions in ...

in 1949 and remained there until his death in 1972. Additionally he was president of the

János Bolyai Mathematical Society The János Bolyai Mathematical Society (Bolyai János Matematikai Társulat, BJMT) is the Hungarian mathematical society, named after János Bolyai, a 19th-century Hungarian mathematician, a co-discoverer of non-Euclidean geometry. It is the profe ...

from 1963 to 1972.

Research

Hajós's theorem In group theory, Hajós's theorem states that if a finite abelian group is expressed as the Cartesian product of simplexes, that is, sets of the form \ where e is the identity element, then at least one of the factors is a subgroup. The theorem wa ...

is named after Hajós, and concerns factorizations of

Abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commu ...

s into Cartesian products of subsets of their elements. This result in group theory has consequences also in geometry: Hajós used it to prove a conjecture of

Hermann Minkowski Hermann Minkowski (22 June 1864 – 12 January 1909) was a mathematician and professor at the University of Königsberg, the University of Zürich, and the University of Göttingen, described variously as German, Polish, Lithuanian-German, o ...

that, if a

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...

of any dimension is tiled by

hypercube In geometry, a hypercube is an ''n''-dimensional analogue of a square ( ) and a cube ( ); the special case for is known as a ''tesseract''. It is a closed, compact, convex figure whose 1- skeleton consists of groups of opposite parallel l ...

s whose positions form a

lattice Lattice may refer to: Arts and design * Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material * Lattice (music), an organized grid model of pitch ratios * Lattice (pastry), an or ...

, then some pair of hypercubes must meet face-to-face. Hajós used similar group-theoretic methods to attack

Keller's conjecture In geometry, Keller's conjecture is the conjecture that in any Tessellation, tiling of -dimensional Euclidean space by identical hypercubes, there are two hypercubes that share an entire -dimensional face with each other. For instance, in any til ...

on whether cube tilings (without the lattice constraint) must have pairs of cubes that meet face to face; his work formed an important step in the eventual disproof of this conjecture. Hajós's conjecture is a

conjecture In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis or Fermat's conjecture (now a theorem, proven in 1995 by Andrew Wiles), ha ...

made by Hajós that every graph with chromatic number contains a subdivision of a complete graph . However, it is now known to be false: in 1979, Paul A. Catlin found a counterexample for , and

Paul Erdős Paul Erdős ( ; 26March 191320September 1996) was a Hungarian mathematician. He was one of the most prolific mathematicians and producers of mathematical conjectures of the 20th century. pursued and proposed problems in discrete mathematics, g ...

and Siemion Fajtlowicz later observed that it fails badly for

random graph In mathematics, random graph is the general term to refer to probability distributions over graphs. Random graphs may be described simply by a probability distribution, or by a random process which generates them. The theory of random graphs l ...

s. The

Hajós construction In graph theory, a branch of mathematics, the Hajós construction is an operation on graphs named after that may be used to construct any critical graph or any graph whose chromatic number is at least some given threshold. The construction Let ...

is a general method for constructing graphs with a given

chromatic number In graph theory, graph coloring is a methodic assignment of labels traditionally called "colors" to elements of a graph. The assignment is subject to certain constraints, such as that no two adjacent elements have the same color. Graph coloring i ...

, also due to Hajós.. As cited by .

Awards and honors

Hajós was a member of the

Hungarian Academy of Sciences The Hungarian Academy of Sciences ( , MTA) is Hungary’s foremost and most prestigious learned society. Its headquarters are located along the banks of the Danube in Budapest, between Széchenyi rakpart and Akadémia utca. The Academy's primar ...

, first as a corresponding member beginning in 1948 and then as a full member in 1958. In 1965 he was elected to the

Romanian Academy of Sciences The Romanian Academy of Sciences was an institution established in Romania by a group of 26 scientists, dissatisfied with the imperfect organization of the Scientific Section of the Romanian Academy, which was left in the background, with only 12 ...

, and in 1967 to the

German Academy of Sciences Leopoldina The German National Academy of Sciences Leopoldina (), in short Leopoldina, is the national academy of Germany, and is located in Halle (Saale). Founded on 1 January 1652, based on academic models in Italy, it was originally named the ''Academi ...

. He won the Gyula König Prize in 1942, and the

Kossuth Prize The Kossuth Prize (, ) is a state-sponsored award in Hungary, named after the Hungarian politician and revolutionist Lajos Kossuth. The Prize was established in 1936, by the Hungarian National Assembly, to acknowledge outstanding personal and grou ...

in 1951 and again in 1962.György Hajós
in the Hungarian Biographical Lexicon (Ágnes Kenyeres. Magyar Életrajzi Lexikon. Budapest: Akadémiai Kiadó, 1994. 9789630524971), freely available on www.mek.iif.hu

References

{{DEFAULTSORT:Hajos, Gyorgy Hungarian people of Scottish descent Members of the Hungarian Academy of Sciences 20th-century Hungarian mathematicians Geometers Algebraists 1912 births 1972 deaths Mathematicians from Austria-Hungary