Felix Klein-"In sense, mathematics has been most advanced by those who distinguish themselves by intuition rather than by rigorous proofs"
Formula: He showed that that surface was a curve in projective space, that its equation was x³y + y³z + z³x = 0

Christian Felix Klein (25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work in group theory, complex analysis, non-Euclidean geometry, and on the connections between geometry and group theory. His 1872 Erlangen Program, classifying geometries by their underlying symmetry groups, was a hugely influential synthesis of much of the mathematics of the day.

Klein's dissertation, on line geometry and its applications to mechanics, classified second degree line complexes using Weierstrass's theory of elementary divisors.
Klein's first important mathematical discoveries were made in 1870. In collaboration with Sophus Lie, he discovered the fundamental properties of the asymptotic lines on the Kummer surface. They went on to investigate W-curves, curves invariant under a group of projective transformations. It was Lie who introduced Klein to the concept of group, which was to play a major role in his later work. Klein also learned about groups from Camille Jordan.

A hand-blown Klein Bottle

Klein devised the bottle named after him, a one-sided closed surface which cannot be embedded in three-dimensional Euclidean space, but it may be immersed as a cylinder looped back through itself to join with its other end from the "inside". It may be embedded in Euclidean space of dimensions 4 and higher.
In the 1890s, Klein turned to mathematical physics, a subject from which he had never strayed far, writing on the gyroscope with Arnold Sommerfeld. In 1894 he launched the idea of an encyclopedia of mathematics including its applications, which became the Enzyklopädie der mathematischen Wissenschaften. This enterprise, which ran until 1935, provided an important standard reference of enduring value.

John von Neumann-"If people do not believe that mathematics is simple, it is not only because they do not realize how complicated life is." 
Formula: Von Neumann's model of an expanding economy considered the matrix pencil  A − λB with nonnegative matrices A and B; von Neumann sought probability vectors p and q and a positive number λ that would solve the complementarity equation

${\displaystyle p^{T}(A-\lambda B)q=0}$John von Neumann (/vɒn ˈnɔɪmən/; Hungarian: Neumann János Lajos, pronounced [ˈnɒjmɒn ˈjaːnoʃ ˈlɒjoʃ]; December 28, 1903 – February 8, 1957) was a Hungarian-American pure and applied mathematician, physicist, inventor, computer scientist, and polymath. He made major contributions to a number of fields, including mathematics (foundations of mathematics, functional analysis, ergodic theory, geometry, topology, and numerical analysis), physics (quantum mechanics, hydrodynamics and quantum statistical mechanics), economics (game theory), computing (Von Neumann architecture, linear programming, self-replicating machines, stochastic computing), and statistics.
He was a pioneer of the application of operator theory to quantum mechanics, in the development of functional analysis, and a key figure in the development of game theory and the concepts of cellular automata, the universal constructor and the digital computer. He published over 150 papers in his life: about 60 in pure mathematics, 20 in physics, and 60 in applied mathematics, the remainder being on special mathematical subjects or non-mathematical ones. His last work, an unfinished manuscript written while in the hospital, was later published in book form as The Computer and the Brain.
His analysis of the structure of self-replication preceded the discovery of the structure of DNA. In a short list of facts about his life he submitted to the National Academy of Sciences, he stated "The part of my work I consider most essential is that on quantum mechanics, which developed in Göttingen in 1926, and subsequently in Berlin in 1927–1929. Also, my work on various forms of operator theory, Berlin 1930 and Princeton 1935–1939; on the ergodic theorem, Princeton, 1931–1932."

 

 Nikolay Ivanovich Lobachevsky-"There is no branch of mathematics, however abstract, which may not some day be applied to phenomena of the real world"

 Formula:For planar algebra, non-Euclidean geometry arises in the other cases. When ε² = +1, then z is a split-complex number and conventionally j replaces epsilon. Then

${\displaystyle zz^{\ast }=(x+y\mathbf {j} )(x-y\mathbf {j} )=x^{2}-y^{2}\!}$

and {z | z z* = 1} is the unit hyperbola.
When ε² = 0, then z is a dual number

 
Nikolay Ivanovich Lobachevsky was the first mathematician to publish an account of non-Euclidean geometry.
Born on December 1, 1792 in Novgorod, Russia. When he was only 7, Lobachevsky's father died. As such, he moved to Kazan, Siberia where he eventually studied at the University of Kazan. There, he studied physics and mathematics under Johann Bartels who had also instructed Karl Gauss. By 1811, he had a master's in mathematics and physics and by 1814 he was appointed a lectureship. By 1816, Lobachevsky became a professor.
In 1829, he published his Non-Euclidean Geometry which was the first account on the matter to be published in the world. Basically, Lobachevsky stopped the common practice of turning Euclid's Fifth Postulate into a theorem. Rather, he studied a geometry apart from Euclid's final postulate. In other terms, he considered Euclid's postulate as a special case of a more simplified, general geometry.