Riemann

Bernhard Riemann (1826–1866) was a German mathematician whose work laid the foundations for analysis, geometry, and mathematical physics. He studied at the University of Göttingen and the University of Berlin, and later held a professorship at Göttingen, where his ideas would influence generations of mathematicians.

In analysis, he introduced the Riemann integral, providing a rigorous basis for the notion of integration. His work on complex analysis led to the concept of Riemann surfaces, which allow rigorous treatment of multi-valued complex functions. He also formulated the Riemann mapping theorem, a central result in geometric function theory that characterizes conformal equivalence between simply connected planar domains and the unit disk.

In geometry, Riemann initiated the development of what is now called Riemannian geometry, introducing ideas of curvature and metric for curved spaces. These concepts form the core of modern differential geometry and underpin the geometric framework of general relativity.

In number theory, his 1859 work proposed the Riemann hypothesis, asserting that the nontrivial zeros of the Riemann zeta function lie on a critical line. This hypothesis has profound implications for the distribution of prime numbers and remains one of the most important unsolved problems in mathematics.

Riemann's ideas shaped many areas of mathematics and physics, and his influence extends across analysis, geometry, topology, and theoretical physics. He died in 1866 at the age of 39, leaving a lasting legacy that continues to inform contemporary research.