David Hilbert
Jan 23 1862 - Feb 14 1943Born Königsberg, Germany (now Kaliningrad, Russia).Died Göttingen, Germany.
Hilbert received his Ph.D. from the University of Königsberg and was a member of staff there from 1886 to 1895 In 1895 he was appointed to the chair of mathematics at the University of Göttingen, where he continued to teach for the rest of his life.
Hilbert's first work was on invariant theory, in 1888 he proved his famous Basis Theorem. First he gave an existence proof but, after Cayley, Gordan, Lindemann and others were baffled, in 1892 Hilbert produced a constructive proof which satisfied everyone.
In 1893 while still at Königsberg he began a work "Zahlbericht" on algebraic number theory. The "Zahlbericht" (1897) is a brilliant synthesis of the work of Kummer, Kronecker and Dedekind but contains a wealth of Hilbert's own ideas. The ideas of the present day subject of 'Class field theory' are all contained in this work.
Hilbert's work in geometry had the greatest influence in that area after Euclid. A systematic study of the axioms of Euclidean geometry led Hilbert to propose 21 such axioms and he analysed their significance.
He published "Grundlagen der Geometrie" in 1899 putting geometry on a formal axiomatic setting. His famous 23 Paris problems challenged (and still today challenges) mathematicians to solve fundamental questions.
In 1915 Hilbert discovered the correct field equation for general relativity before Einstein but never claimed priority.
In 1934 and 1939 two volumes of "Grundlagen der Mathematik" were published which were intended to lead to a 'proof theory' a direct check for the consistency of mathematics. Göde's paper of 1931 showed that this aim is impossible.
Hilbert contributed to many branches of mathematics, including invariants, algebraic number fields, functional analysis, integral equations, mathematical physics, and the calculus of variations.


Karl Gustav Jacob Jacobi
Dec 10 1804 - Feb 18 1851Born Potsdam, Germany. Died Berlin, Germany.

Jacobi founded the theory of elliptic functions.
Jacobi's father was a banker and his family were prosperous so he received a good education at the University of Berlin. He obtained his Ph.D. in 1825 and taught mathematics at the University of Königsberg from 1826 until his death, being appointed to a chair in 1832.
He founded the theory of elliptic functions based on four theta functions. His "Fundamenta nova theoria functionum ellipticarum" in 1829 and its later supplements made basic contributions to the theory of elliptic functions.
In 1834 Jacobi proved that if a single-valued function of one variable is doubly periodic then the ratio of the periods is imaginary. This result prompted much further work in this area, in particular by Liouville and Cauchy.
Jacobi carried out important research in partial differential equations of the first order and applied them to the differential equations of dynamics.
He also worked on determinants and studied the functional determinant now called the Jacobian. Jacobi was not the first to study the functional determinant which now bears his name, it appears first in a 1815 paper of Cauchy. However Jacobi wrote a long memoir "De determinantibus functionalibus" in 1841 devoted to the this determinant. He proves, among many other things, that if a set of n functions in n variables are functionally related then the Jacobian is identically zero, while if the functions are independent the Jacobian cannot be identically zero.
Jacobi's reputation as an excellent teacher attracted many students. He introduced the seminar method to teach students the latest advances in mathematics.