Nationality : German

Date of birth : January 23, 1862

Date of death : February 14, 1943

Fields :Mathematician and Philosopher

Known for Hilbert's basis theorem, Hilbert's axioms, Hilbert's program, Hilbert's problems, Einstein–Hilbert action and Hilbert space

Hilbert make great efforts in many fields of mathematics, including axiomatic theory, invariant theory, algebraic number theory, class field theory and functional analysis His examination of calculus led him to the invention of "Hilbert space," considered one of the key concepts of functional analysis and modern mathematical physics. He was a founder of fields like metamathematics and modern logic. He was also the founder of the "Formalist" school which opposed the "Intuitionism" of Kronecker and Brouwer. He developed a new system of definitions and axioms for geometry, replacing the 2200 year-old system of Euclid. As a young Professor he proved his "Finiteness Theorem," now regarded as one of the most important results of general algebra. The methods he used were so novel that, at first, the "Finiteness Theorem" was rejected for publication as being "theology" rather than mathematics! In number theory, he proved Waring's famous conjecture which is now known as the Hilbert-Waring theorem.

Around 1909, Hilbert dedicated himself to the study of differential and integral equations; his work had direct consequences for important parts of modern functional analysis. In order to carry out these studies, Hilbert introduced the concept of an infinite dimensional Euclidean space, later called Hilbert space. His work in this part of analysis provided the basis for important contributions to the mathematics of physics in the next two decades, though from an unanticipated direction. Later on, Stefan Banach amplified the concept, defining Banach spaces. Hilbert space is the most important single idea in the area of functional analysis, particularly of the spectral theory of self-adjoint linear operators, that grew up around it during the 20th century.

Hilbert unified the field of algebraic number theory with his 1897 treatise Zahlbericht (literally "report on numbers"). He also resolved a significant number theory problem formulated by Waring in 1770. As with the finiteness theorem, he used an existence proof that shows there must be solutions for the problem rather than providing a mechanism to produce the answers. He then had little more to publish on the subject; but the emergence of Hilbert modular forms in the dissertation of a student means his name is further attached to a major area.

He made a series of conjectures on class field theory. The concepts were highly influential, and his own contribution is seen in the names of the Hilbert class field and the Hilbert symbol of local class field theory. Results on them were mostly proved by 1930, after work by Teiji Takagi that established Tagaki as Japan's first mathematician of international stature.

Hilbert did not work in the central areas of analytic number theory, but his name has become known for the Hilbert–Pólya conjecture, for reasons that are anecdotal.

He made a series of conjectures on class field theory. The concepts were highly influential, and his own contribution is seen in the names of the Hilbert class field and the Hilbert symbol of local class field theory. Results on them were mostly proved by 1930, after work by Teiji Takagi that established Tagaki as Japan's first mathematician of international stature.

Hilbert did not work in the central areas of analytic number theory, but his name has become known for the Hilbert–Pólya conjecture, for reasons that are anecdotal.

Eventually Hilbert turned to physics and made key contributions to classical and quantum physics and to general relativity. He may have published the "Einstein Field Equations" independently of Einstein. (Since he had already learned of the theory's intuition from personal lectures by Einstein, it is wrong, as some do, to claim Hilbert's publication diminishes Einstein's greatness.

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