A mathematician is a person whose primary area of study and research is the field of mathematics.

Problems in mathematics

The publication of new discoveries in mathematics continues at an immense rate in hundreds of scientific journals. One of the most exciting recent developments was the proof of Fermat's Last Theorem by Andrew Wiles, following 350 years of the brightest mathematical minds attempting to settle the problem.

There are many famous open problems in mathematics, many dating back tens, if not hundreds, of years. Some examples include the Riemann hypothesis (from 1859) and Goldbach's conjecture (1742). The Millennium Prize Problems highlight longstanding, important problems in mathematics and offers a US$1,000,000 reward for solving any one of them. One of these problems, the Poincaré conjecture (1904), was proven by Russian mathematician Grigori Perelman in a paper released in 2003; peer review was completed in 2006, and the proof was accepted as valid.

Motivation

Mathematicians are not necessarily number theorists (a misconception among many people) but are people who research mathematics fields in general such as topology, modern algebra (which is not simple algebra), differential topology, functional analysis etc... Most problems and theorems come from within mathematics itself, or are inspired by theoretical physics. To a lesser extent, problems have come from economics, games and computer science. Some problems are simply created for the challenge of solving them. Mathematics is a very interesting field and in modern day has several applications in physics, computer science, chemistry and many other such areas.

There are no Nobel Prizes awarded to mathematicians. The award that is generally viewed as having the highest prestige in mathematics is the Fields Medal. This medal, sometimes described as the "Nobel Prize of Mathematics", is awarded once every four years to as many as four young (under 40 years old) awardees at a time, who are talented mathematicians. Other prominent prizes include the Abel Prize, the Nemmers Prize, the Wolf Prize, the Schock Prize, and the Nevanlinna Prize.

Differences

Mathematics differs from natural sciences in that physical theories in the sciences are tested by experiments, while mathematical statements are supported by proofs which may be verified objectively by mathematicians. If a certain statement is believed to be true by mathematicians (typically because special cases have been confirmed to some degree) but has neither been proven nor dis-proven, it is called a conjecture, as opposed to the ultimate goal: a theorem that is proven true. Physical theories may be expected to change whenever new information about our physical world is discovered. Mathematics changes in a different way: new ideas don't falsify old ones but rather are used to generalize what was known before to capture a broader range of phenomena. For instance, calculus (in one variable) generalizes to multivariable calculus, which generalizes to analysis on manifolds. The development of algebraic geometry from its classical to modern forms is a particularly striking example of the way an area of mathematics can change radically in its viewpoint without making what was proved before in any way incorrect. While a theorem, once proved, is true forever, our understanding of what the theorem really means gains in profundity as the mathematics around the theorem grows. A mathematician feels that a theorem is better understood when it can be extended to apply in a broader setting than previously known. For instance, Fermat's little theorem for the nonzero integers modulo a prime generalizes to Euler's theorem for the invertible numbers modulo any nonzero integer, which generalizes to Lagrange's theorem for finite groups.