György Hajós
Encyclopedia
György Hajós was a Hungarian mathematician
who worked in group theory
, graph theory
, and geometry
.
; his grandfather, Adam Clark, was the lead engineer of the Chain Bridge in Budapest. He earned a teaching degree from the University of Budapest
in 1929, and taught mathematics at the secondary school level from 1929 until 1935. He then took a position at the Technical University of Budapest, 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 in 1949 and remained there until 1972. Additionally he was president of the János Bolyai Mathematical Society
from 1963 to 1972.
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
that, if a Euclidean space
of any dimension is tiled by hypercube
s whose positions form a lattice
, then some pair of hypercubes must meet face-to-face. Hajós used similar group-theoretic methods to attack Keller's conjecture
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
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, P. A. Catlin found a counterexample for , and Paul Erdős
and Siemion Fajtlowicz
later observed that it fails badly for random graph
s. The Hajós construction
is a general method for constructing graphs with a given chromatic number, also due to Hajós.
, 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, and in 1967 to the German Academy of Sciences Leopoldina. He won the Gyula König Prize in 1942, and the Kossuth Prize
in 1951 and again in 1962.
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
who worked in group theory
Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
, graph theory
Graph theory
In mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...
, and geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
.
Biography
Hajós was born February 21, 1912, in BudapestBudapest
Budapest is the capital of Hungary. As the largest city of Hungary, it is the country's principal political, cultural, commercial, industrial, and transportation centre. In 2011, Budapest had 1,733,685 inhabitants, down from its 1989 peak of 2,113,645 due to suburbanization. The Budapest Commuter...
; his grandfather, Adam Clark, was the lead engineer of the Chain Bridge in Budapest. He earned a teaching degree from the University of Budapest
University of Budapest
The Eötvös Loránd University or ELTE, founded in 1635, is the largest university in Hungary, located in Budapest.-History:The university was founded in 1635 in Nagyszombat by the archbishop and theologian Péter Pázmány. Leadership was given over to the Jesuits...
in 1929, and taught mathematics at the secondary school level from 1929 until 1935. He then took a position at the Technical University of Budapest, 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 in 1949 and remained there until 1972. Additionally he was president of the János Bolyai Mathematical Society
János Bolyai Mathematical Society
The János Bolyai Mathematical Society is the Hungarian mathematical society, named after János Bolyai, a 19th century Hungarian mathematician, a co-discoverer of non-Euclidean geometry. It is the professional society of the Hungarian mathematicians, applied mathematicians, and mathematics teachers...
from 1963 to 1972.
Research
Hajós's theorem is named after Hajós, and concerns factorizations of Abelian groupAbelian group
In abstract algebra, 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 their order . Abelian groups generalize the arithmetic of addition of integers...
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
Hermann Minkowski was a German mathematician of Ashkenazi Jewish descent, who created and developed the geometry of numbers and who used geometrical methods to solve difficult problems in number theory, mathematical physics, and the theory of relativity.- Life and work :Hermann Minkowski was born...
that, if a Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
of any dimension is tiled by hypercube
Hypercube
In geometry, a hypercube is an n-dimensional analogue of a square and a cube . It is a closed, compact, convex figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, perpendicular to each other and of the same length.An...
s whose positions form a lattice
Lattice (group)
In mathematics, especially in geometry and group theory, a lattice in Rn is a discrete subgroup of Rn which spans the real vector space Rn. Every lattice in Rn can be generated from a basis for the vector space by forming all linear combinations with integer coefficients...
, then some pair of hypercubes must meet face-to-face. Hajós used similar group-theoretic methods to attack Keller's conjecture
Keller's conjecture
In geometry, Keller's conjecture is the conjecture introduced by that in any tiling of Euclidean space by identical hypercubes there are two cubes that meet face to face. For instance, as shown in the illustration, in any tiling of the plane by identical squares, some two squares must meet edge to...
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
Conjecture
A conjecture is a proposition that is unproven but is thought to be true and has not been disproven. Karl Popper pioneered the use of the term "conjecture" in scientific philosophy. Conjecture is contrasted by hypothesis , which is a testable statement based on accepted grounds...
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, P. A. Catlin found a counterexample for , and Paul Erdős
Paul Erdos
Paul Erdős was a Hungarian mathematician. Erdős published more papers than any other mathematician in history, working with hundreds of collaborators. He worked on problems in combinatorics, graph theory, number theory, classical analysis, approximation theory, set theory, and probability theory...
and Siemion Fajtlowicz
Siemion Fajtlowicz
Siemion Fajtlowicz is a Polish mathematician, currently a professor at the University of Houston. He is known for developing the conjecture-making computer program Graffiti....
later observed that it fails badly for random graph
Random graph
In mathematics, a random graph is a graph that is generated by some random process. The theory of random graphs lies at the intersection between graph theory and probability theory, and studies the properties of typical random graphs.-Random graph models:...
s. The Hajós construction
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:...
is a general method for constructing graphs with a given chromatic number, also due to Hajós.
Awards and honors
Hajós was a member of the Hungarian Academy of SciencesHungarian Academy of Sciences
The Hungarian Academy of Sciences is the most important and prestigious learned society of Hungary. Its seat is at the bank of the Danube in Budapest.-History:...
, 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, and in 1967 to the German Academy of Sciences Leopoldina. He won the Gyula König Prize in 1942, and the Kossuth Prize
Kossuth Prize
The Kossuth Prize is a state-sponsored award in Hungary, named after the Hungarian politician and revolutionary Lajos Kossuth. The Prize was established in 1948 by the Hungarian National Assembly, to acknowledge outstanding personal and group achievements in the fields of...
in 1951 and again in 1962.