Mathematicians conduct research in fields such as logic, theory, abstract algebra, numerical analysis, topology, geometry, dynamical systems, combinatorics, optimization, computation, probability and statistics. These fields comprise both pure mathematics and applied mathematics, along with links between the two. Theoretical mathematicians advance the subject by developing new principles and recognize previously unknown relationships between existing principles of mathematics. Applied mathematicians use theories and techniques, such as mathematical modeling and computational methods, to formulate and solve practical problems. In some cases, scientists who research other fields such as Sir Isaac Newton are also considered mathematicians if their research provides insights into the subject.
Mathematics is one of the oldest and most fundamental sciences. It is generally considered that its origins stem from Ancient Greece through the work of Archimedes, Euclid and Pythagorus. Archimedes (c. 287-212 BCE) is generally considered to be the greatest mathematician of ancient times, discovering the lever, the principle of buoyancy, and coming close to inventing calculus. He also founded mathematical physics. Euclid (c. 330-260 BCE) is one of the world's most famous mathematicians. Euclid brought logic to number theory and geometry, and his ideas are still used. Pythagoras (c. 580 - 500 BCE) is credited with developing the theory of functions, the significance of numbers, and Pythagorean theorem.
The Egyptian mathematician Hypatia (c. 370- 415) is noted as being the first female to work within advanced mathematics. Chu Shih-Chieh (1280-1303) is one of the greatest Chinese mathematicians of all time. Chu's major contribution was the theory of equations. René Descartes (1596 – 1650) invented analytic geometry, which is a combination of Euclidean geometry and algebra. This foundation was crucial in the later discovery of infinitesimal calculus and analysis.
The early 1700s saw intense competition between two highly respected mathematicians; Gottfried Leibniz and Sir Isaac Newton. Leibniz (1646-1716) was a German mathematician, philosopher, historian, and physicist. He devised the calculating machine and the concept of binary numbers used in computing. Leibniz is also credited with discovering the series for pi known as Leibniz's theorem. Both men are credited with the invention of calculus at around the same time, with some argument as to who was officially first. Later, Karl Friedrich Gauss (1777-1855) was a German who worked on electricity, magnetism, and planetary orbits, while Baron Augustin Cauchy (1789-1857) was a French mathematician whose work concentrated on analysis, the theory of numbers, and the theory of substitution groups. Leonhard Euler (1707-1783) was a Swiss mathematician, physicist, and one of the founders of pure mathematics.
In the 1900s the work of Évariste Galois (1811 – 1832) of France laid the foundations for Galois theory, a major branch of abstract algebra, and the sub-field of Galois connections. Jules Henri Poincaré (1854 – 1912) was known by peers as The Last Universalist, since he excelled in all fields of mathematics. Georg Friedrich Riemann (1826-1866) was a German mathematician whose work had a great influence on geometry and analysis. His work on the geometry of space greatly influenced modern theoretical physics. Meanwhile another German, Georg Cantor (1845 -1918), is best known as the inventor of set theory, which has become a fundamental theory in mathematics.
The early twentieth century saw German mathematician David Hilbert (1862 – 1943) come to great prominence. He discovered and developed invariant theory and the axiomatization of geometry. He also formulated the theory of Hilbert spaces, one of the foundations of functional analysis. Hilbert is known as one of the founders of proof theory and mathematical logic, as well being the first to distinguish between mathematics and metamathematics. British philosopher Bertrand Russell (1872-1970) built on this with his vital work in mathematical logic and is considered to be one of the brightest intellectuals of the twentieth century. Perhaps the most colorful of present day mathematicians is John Horton Conway (1937 -), who is best known for his analyses of games and puzzles and important contributions to other areas of pure math such as group and number theory, and geometry.
The best-known award in mathematics is the Fields Medal. Established in 1936 and awarded every four years, the Fields Medal is often considered to be the equivalent of science's Nobel Prize, and is awarded for a particular body of work, whether it is an innovation, or resolution of an outstanding mathematical problem. Grigori Perelman (1966 -) was awarded $1 million in 2010 for proving the Poincaré Conjecture but had refused to accept the 2006 Fields Medal, saying: "If the proof is correct then no other recognition is needed."