Riemann was a phenomenal genius whose work was exceptionally deep, creative and rigorous; he made revolutionary contributions in many areas of pure mathematics, and also inspired the development of physics. He had poor physical health and died at an early age, yet is still considered to be among the most productive mathematicians ever. He was the master of complex analysis, which he connected to both topology and number theory. He applied topology to analysis, and analysis to number theory, making revolutionary contributions to all three fields. He took non-Euclidean geometry far beyond his predecessors. He introduced the Riemann integral which clarified analysis. Riemann's other masterpieces include differential geometry, tensor analysis, the theory of functions, and, especially, the theory of manifolds. He generalized the notions of distance and curvature and, therefore, described new possibilities for the geometry of space itself. Several important theorems and concepts are named after Riemann, e.g. the Riemann-Roch theorem, a key connection among topology, complex analysis and algebraic geometry. He was so prolific and original that some of his work went unnoticed (for example, Weierstrass became famous for showing a nowhere-differentiable continuous function; later it was found that Riemann had casually mentioned one in a lecture years earlier). Like his mathematical peers (Gauss, Archimedes, Newton), Riemann was intensely interested in physics. His theory unifying electricity, magnetism and light was supplanted by Maxwell's theory; however modern physics, beginning with Einstein's relativity, relies on Riemann's notions of the geometry of space.
Riemann's teacher was Carl Gauss, who helped steer the young genius towards pure mathematics. Gauss selected "On the hypotheses that Lie at the Foundations of Geometry" as Riemann's first lecture; with this famous lecture Riemann advanced Gauss' initial effort in differential geometry, extended it to multiple dimensions, and introduced the new and important theory of differential manifolds. Five years later, to celebrate his election to the Berlin Academy, Riemann presented a lecture "On the Number of Prime Numbers Less Than a Given Quantity," for which "Number" he presented and proved an exact formula, albeit weirdly complicated and seemingly intractable. Numerous papers have been written on the distribution of primes, but Riemann's contribution is incomparable, despite that his Berlin Academy lecture was his only paper ever on the topic, and number theory was far from his specialty. In the lecture he posed the "Hypothesis of Riemann's zeta function" which is now considered the most important and famous unsolved problem in mathematics. (Asked what he would first do, if he were magically awakened after centuries, David Hilbert replied "I would ask whether anyone had proved the Riemann Hypothesis.") ζ() was defined for convergent cases in Euler's mini-bio, which Riemann extended via analytic continuation for all cases. The Riemann Hypothesis "simply" states that in all solutions of ζ(s = a+bi) = 0, either s has real part a=1/2 or imaginary part b=0.

Despite his great creativity (Gauss praised Riemann's "gloriously fertile originality"), Riemann once said: "If only I had the theorems! Then I should find the proofs easily enough."

David Hilbert (1862-1943) Prussia, Germany

Hilbert, often considered the greatest mathematician of the 20th century, was unequalled in many fields of mathematics, including axiomatic theory, invariant theory, algebraic number theory, class field theory and functional analysis. He proved many new theorems, including the fundamental theorems of algebraic manifolds, and also discovered simpler proofs for older theorems. 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. His Nullstellensatz theorem laid the foundation of algebraic geometry. 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 "Finite Basis Theorem," now regarded as one of the most important results of general algebra. His mentor, Paul Gordan, had sought the proof for many years, and rejected Hilbert's proof as non-constructive. Later, Hilbert produced the first constructive proof of the Finite Basis Theorem, as well. In number theory, he proved Waring's famous conjecture which is now known as the Hilbert-Waring theorem.
Any one man can only do so much, so the greatest mathematicians should help nurture their colleagues. Hilbert provided a famous List of 23 Unsolved Problems, which inspired and directed the development of 20th-century mathematics. Hilbert was warmly regarded by his colleagues and students, and contributed to the careers of several great mathematicians and physicists including Georg Cantor, Hermann Minkowski, Hermann Weyl, John von Neumann, Emmy Noether, Alonzo Church, and Albert Einstein.

Eventually Hilbert turned to physics and made key contributions to classical and quantum physics and to general relativity. He published the "Einstein Field Equations" independently of Einstein (though his writings make clear he treats this as strictly Einstein's invention.)