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A small history of how mathematicians were trying to find as many pi decimals as they could :) Enjoy
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The story about Fermat’s Last Theorem is incredibly old. The problem states that no 3 positive integers a, b and c satisfy the an + bn = cn for any integer value of n strictly greater than 2. The theorem was first stated by Pierre de Fermat in 1637 in the margin of a copy of Arithmetica where he also claimed he had a proof that was to large to fit in the margin. After this mathematicians tried in vain to find a proof for it for approximately 360 years. Moreover, the theorem was in the Guinness Book of World Records as the “the most difficult mathematical problem” due to the fact that it has the largest number of unsuccessful proofs.

The first successful proof was released in 1994 by Andrew Wiles and published in 1995 – 358 years of pure struggle, trial and error from mathematicians all over the world. Andrew Wiles, an English mathematician with a childhood fascination with Fermat’s Last Theorem, and a prior study area of elliptical equations, decided to commit himself to accomplishing the second half: proving a special case of the modularity theorem (then known as the Taniyama–Shimura conjecture) for semistable elliptic curves. Wiles worked on that task for six years in near-total secrecy, covering up his efforts by releasing prior work in small segments as separate papers and confiding only in his wife. The unsolved problem stimulated the development of algebraic number theory in the 19th century and the proof of the modularity theorem in the 20th century.
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The Abel Prize Award is the mathematical corespondent of the Nobel Prize. The Prize is named after the Norwegian mathematician Niels Henrik Abel (1802 – 1829).

The Abel Prize was first proposed in 1899, especially after it was known that the Nobel Prize is not going to include a prize in mathematics. The prize was proposed to be part of the 1902 celebration of 100th anniversary of Abel’s birth by mathematician Sophus Lie. Things started to be put into order for organizing such an event, but after Sophus Lie died and other problems (mostly political between Sweden and Norway) appeared, the first attempt to create the Prize ended. Things started to move after almost 100 years, in 2001 when a group was formed to develop a proposal, which was presented to the Prime Minister of Norway. After this the Norwegian government announced that the prize would be awarded beginning in 2002 – the 200th anniversary of Abel’s birth (exactly 100 years after the first proposal in 1902). If you want to read more about the history of the event and all the laureates during the time I totally advice to check http://www.abelprize.no/
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In 1942–45, Samuel Eilenberg and Saunders Mac Lane introduced categories, functors, and natural transformations as part of their work in topology, especially algebraic topology. Eilenberg and Mac Lane later wrote that their goal was to understand natural transformations. That required defining functors, which required categories.

Category theory is also, in some sense, a continuation of the work of Emmy Noether (one of Mac Lane’s teachers) in formalizing abstract processes; Noether realized that understanding a type of mathematical structure requires understanding the processes that preserve that structure. To achieve this understanding, Eilenberg and Mac Lane proposed an axiomatic formalization of the relation between structures and the processes that preserve them.

Categorical logic is now a well-defined field based on type theory for intuitionistic logics, with applications in functional programming and domain theory. At the very least, category theoretic language clarifies what exactly these related areas have in common (in some abstract sense).

Category theory has been applied in other fields as well. For example, John Baez has shown a link between Feynman diagrams in Physics and monoidal categories. Another application of category theory, more specifically: topos theory, has been made in mathematical music theory, see for example the book The Topos of Music, Geometric Logic of Concepts, Theory, and Performance by Guerino Mazzola.

More recent efforts to introduce undergraduates to categories as a foundation for mathematics include those of William Lawvere and Rosebrugh (2003) and Lawvere and Stephen Schanuel (1997) and Mirroslav Yotov (2012).
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Topology developed as a field of study out of geometry and set theory, through analysis of such concepts as space, dimension, and transformation. Such ideas go back to Gottfried Leibniz, who in the 17th century envisioned the geometria situs (Greek-Latin for "geometry of place") and analysis situs (Greek-Latin for "picking apart of place"). Leonhard Euler's Seven Bridges of Königsberg Problem and Polyhedron Formula are arguably the field's first theorems. The term topology was introduced by Johann Benedict Listing in the 19th century, although it was not until the first decades of the 20th century that the idea of a topological space was developed. By the middle of the 20th century, topology had become a major branch of mathematics.
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Chaos theory is the field of study in mathematics that studies the behavior and condition of dynamical systems that are highly sensitive to initial conditions—a response popularly referred to as the butterfly effect.

The history of Chaos theory is embedded in physics and computer science. An early proponent of chaos theory was Henri Poincaré. In the 1880s, while studying the three-body problem, he found that there can be orbits that are nonperiodic, and yet not forever increasing nor approaching a fixed point. In 1898 Jacques Hadamard published an influential study of the chaotic motion of a free particle gliding frictionlessly on a surface of constant negative curvature. Early studies were all directly inspired by physics: the three-body problem in the case of Birkhoff, turbulence and astronomical problems in the case of Kolmogorov, and radio engineering in the case of Cartwright and Littlewood. Despite initial insights in the first half of the twentieth century, chaos theory became formalized as such only after mid-century, when it first became evident to some scientists that linear theory, the prevailing system theory at that time, simply could not explain the observed behavior of certain experiments.

The main catalyst for the development of chaos theory was the electronic computer. Much of the mathematics of chaos theory involves the repeated iteration of simple mathematical formulas, which would be impractical to do by hand. Electronic computers made these repeated calculations practical, while figures and images made it possible to visualize these systems.
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Joseph-Louis Lagrange, born on 25 January 1736, was an Italian Enlightenment Era mathematician and astronomer. He made significant contributions to the fields of analysis, number theory, and both classical and celestial mechanics.

Lagrange was one of the creators of the calculus of variations, deriving the Euler–Lagrange equations for extrema of functionals. He also extended the method to take into account possible constraints, arriving at the method of Lagrange multipliers. Lagrange invented the method of solving differential equations known as variation of parameters, applied differential calculus to the theory of probabilities and attained notable work on the solution of equations. He proved that every natural number is a sum of four squares. His treatise "Theorie des fonctions analytiques" laid some of the foundations of group theory, anticipating Galois. In calculus, Lagrange developed a novel approach to interpolation and Taylor series. He studied the three-body problem for the Earth, Sun and Moon (1764) and the movement of Jupiter’s satellites (1766), and in 1772 found the special-case solutions to this problem that yield what are now known as Lagrangian points. Also, he has transformed Newtonian mechanics into a branch of analysis, Lagrangian mechanics as it is now called, and presented the so-called mechanical "principles" as simple results of the variational calculus.