Von Neumann had perhaps the widest coverage of any mathematician of his time,[9] integrating pure and applied sciences and making major contributions to many fields, including mathematics, physics, economics, computing, and statistics.

He was a pioneer in building the mathematical framework of quantum physics, in the development of functional analysis, and in game theory, introducing or codifying concepts including cellular automata, the universal constructor and the digital computer.

[40][41] In 1926, he graduated as a chemical engineer from ETH Zurich and simultaneously passed his final examinations summa cum laude for his Ph.D. in mathematics (with minors in experimental physics and chemistry) at the University of Budapest.

[44] Hermann Weyl remembers how in the winter of 1926–1927 von Neumann, Emmy Noether, and he would walk through "the cold, wet, rain-wet streets of Göttingen" after class discussing hypercomplex number systems and their representations.

[107] In a series of papers published in 1932, von Neumann made foundational contributions to ergodic theory, a branch of mathematics that involves the states of dynamical systems with an invariant measure.

Anyone wishing to get an unforgettable impression of the razor edge of von Neumann's mind, need merely try to pursue this chain of exact reasoning for himself—realizing that often five pages of it were written down before breakfast, seated at a living room writing-table in a bathrobe.

[179] Birkhoff noted in his posthumous article on von Neumann that most of these results were developed in an intense two-year period of work, and that while his interests continued in lattice theory after 1937, they became peripheral and mainly occurred in letters to other mathematicians.

[181] This ratio was applied to the residuals from regression models and is commonly known as the Durbin–Watson statistic[182] for testing the null hypothesis that the errors are serially independent against the alternative that they follow a stationary first order autoregression.

[182] Subsequently, Denis Sargan and Alok Bhargava extended the results for testing whether the errors on a regression model follow a Gaussian random walk (i.e., possess a unit root) against the alternative that they are a stationary first order autoregression.

[197][198][199][200] Other minor results from his early career include a proof of a maximum principle for the gradient of a minimizing function in the field of calculus of variations,[201][202][203][204] and a small simplification of Hermann Minkowski's theorem for linear forms in geometric number theory.

He argued that the mathematics of quantum mechanics allows the collapse of the wave function to be placed at any position in the causal chain from the measurement device to the "subjective consciousness" of the human observer.

He intended it to be a joint synthesis of formal logic and probability theory and when he attempted to write up a paper for the Henry Joseph Lecture he gave at the Washington Philosophical Society in 1945 he found that he could not, especially given that he was busy with war work at the time.

[240][241] Von Neumann made fundamental contributions in the field of fluid dynamics, including the classic flow solution to blast waves,[242] and the co-discovery (independently by Yakov Borisovich Zel'dovich and Werner Döring) of the ZND detonation model of explosives.

During World War II, he approached R. H. Kent, the director of the US Army's Ballistic Research Laboratory, with a computer program for calculating a one-dimensional model of 100 molecules to simulate a shock wave.

[251][252] In earlier work led by Oswald Veblen von Neumann helped develop basic ideas involving spinors that would lead to Roger Penrose's twistor theory.

[255] From this work he wrote a paper with A. H. Taub and Veblen extending the Dirac equation to projective relativity, with a key focus on maintaining invariance with regards to coordinate, spin, and gauge transformations, as a part of early research into potential theories of quantum gravity in the 1930s.

[261] Von Neumann's functional-analytic techniques—the use of duality pairings of real vector spaces to represent prices and quantities, the use of supporting and separating hyperplanes and convex sets, and fixed-point theory—have been primary tools of mathematical economics ever since.

Dantzig then listened dumbfounded while von Neumann provided an hourlong lecture on convex sets, fixed-point theory, and duality, conjecturing the equivalence between matrix games and linear programming.

[314] Positive results of Norman A. Phillips in 1955 prompted immediate reaction and von Neumann organized a conference at Princeton on "Application of Numerical Integration Techniques to the Problem of the General Circulation".

Once again he strategically organized the program as a predictive one to ensure continued support from the Weather Bureau and the military, leading to the creation of the General Circulation Research Section (now the Geophysical Fluid Dynamics Laboratory) next to the JNWPU.

"[318][319] His research into weather systems and meteorological prediction led him to propose manipulating the environment by spreading colorants on the polar ice caps to enhance absorption of solar radiation (by reducing the albedo).

[39] Von Neumann made his principal contribution to the atomic bomb in the concept and design of the explosive lenses that were needed to compress the plutonium core of the Fat Man weapon that was later dropped on Nagasaki.

[343] During several meetings of the advisory board of the US Air Force, von Neumann and Edward Teller predicted that by 1960 the US would be able to build a hydrogen bomb light enough to fit on top of a rocket.

[344] Schriever enlisted Trevor Gardner, who in turn visited von Neumann several weeks later to fully understand the future possibilities before beginning his campaign for such a weapon in Washington.

[359][353] Von Neumann entered government service primarily because he felt that, if freedom and civilization were to survive, it would have to be because the United States would triumph over totalitarianism from Nazism, Fascism and Soviet Communism.

[371] In the early 1940s, Ulam had concocted for him a doctoral-style examination to find weaknesses in his knowledge; von Neumann was unable to answer satisfactorily a question each in differential geometry, number theory, and algebra.

[366] He was a quick speaker: Banesh Hoffmann found it very difficult to take notes, even in shorthand,[379] and Albert Tucker said that people often had to ask von Neumann questions to slow him down so they could think through the ideas he was presenting.

"It seems fair to say that if the influence of a scientist is interpreted broadly enough to include impact on fields beyond science proper, then John von Neumann was probably the most influential mathematician who ever lived," wrote Miklós Rédei.

"[408] While consulting for non-academic projects von Neumann's combination of outstanding scientific ability and practicality gave him a high credibility with military officers, engineers, and industrialists that no other scientist could match.

[427] Alongside his later papers were informal essays targeted at colleagues and the general public, such as his 1947 The Mathematician,[428] described as a "farewell to pure mathematics", and his 1955 Can we survive technology?, which considered a bleak future including nuclear warfare and deliberate climate change.