Question:  What American mathematician who made fundamental contributions to game theory, differential geometry, and the study of partial differential equations, had a film based on his the life called, A Beautiful Mind?

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Nicolai V. Krylov: Nicolai Vladimirovich Krylov (Russian: Никола́й Влади́мирович Крыло́в ; born 5 June 1941) is a Russian mathematician specializing in partial differential equations, particularly stochastic partial differential equations and diffusion processes.  Krylov studied at Lomonosov University, where he in 1966 under E. B. Dynkin attained a doctoral candidate title (similar to a PhD) and in 1973 a Russian doctoral degree (somewhat more prestigious than a PhD).  He taught from 1966 to 1990 at the Lomonosov University and is since 1990 a professor at the University of Minnesota.  At the beginning of his career (starting from 1963) he, in collaboration with Dynkin, worked on nonlinear stochastic control theory, making advances in the study of convex, nonlinear partial equations of 2nd order ("i.e." Bellman equations), which were examined with stochastic methods.  This led to the Evans-Krylov theory, for which he received with Lawrence C. Evans in 2004 the Leroy P. Steele Prize of the American Mathematical Society (for work done simultaneously and independently by both Krylov and Evans).  They proved the second order differentiability (Hölder continuity of the second derivative) of the solutions of convex, completely nonlinear, second order elliptical partial differential equations and thus the existence of "classical solutions" (Theorem of Evans-Krylov).  He was in 1978 at Helsinki and in 1986 at Berkeley an Invited Speaker for the ICM.  He received the Humboldt Research Award in 2001.  In 1993 he was elected a member of the American Academy of Arts and Sciences (1993).  He should not be confused with the mathematician Nikolay M. Krylov.

John E. Osborn (mathematician): John E. Osborn (July 12, 1936 – May 30, 2011) was an American mathematician.  He obtained B.S. (1958), M.S. (1963), and Ph.D. (1965) degrees at the University of Minnesota, Minneapolis.  His Ph.D. adviser was Hans Weinberger.  Osborn made fundamental contributions to computational mathematics, especially to the theory of numerical solution of partial differential equations, eigenvalue approximations, and the finite element method.  He also co-authored several textbooks on differential equations and numerical computation with the goal of introducing computation into sophomore level differential equations courses.

Andrew Ronald Mitchell: Andrew Ronald Mitchell (22 June 1921 – 22 November 2007), popularly known as Ron Mitchell, was a British applied mathematician and numerical analyst.  He was a professor of mathematics at the University of St Andrews, Dundee, Scotland.  He was known for his contributions to the field of numerical analysis of partial differential equations in general and finite difference method and finite element method in particular.  Mitchell has authored several influential books on numerical solution of partial differential equations, including "The Finite Element Analysis in Partial Differential Equations" with Richard Wait and "The Finite Difference Method in Partial Differential Equations" with David F. Griffiths.

A Beautiful Mind (film): A Beautiful Mind is a 2001 American biographical drama film based on the life of John Nash, a Nobel Laureate in Economics.  The film was directed by Ron Howard, from a screenplay written by Akiva Goldsman.  It was inspired by a bestselling, Pulitzer Prize-nominated 1998 book of the same name by Sylvia Nasar.  The film stars Russell Crowe, along with Ed Harris, Jennifer Connelly, Paul Bettany, Adam Goldberg, Judd Hirsch, Josh Lucas, Anthony Rapp, and Christopher Plummer in supporting roles.  The story begins in Nash's days as a graduate student at Princeton University.  Early in the film, Nash begins to develop paranoid schizophrenia and endures delusional episodes while painfully watching the loss and burden his condition brings on wife Alicia and friends.

John Forbes Nash Jr.: John Forbes Nash Jr. (June 13, 1928 – May 23, 2015) was an American mathematician who made fundamental contributions to game theory, differential geometry, and the study of partial differential equations. Nash's work has provided insight into the factors that govern chance and decision-making inside complex systems found in everyday life.

Stability theory: In mathematics, stability theory addresses the stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions.  The heat equation, for example, is a stable partial differential equation because small perturbations of initial data lead to small variations in temperature at a later time as a result of the maximum principle.  In partial differential equations one may measure the distances between functions using Lp norms or the sup norm, while in differential geometry one may measure the distance between spaces using the Gromov–Hausdorff distance.

Eberhard Hopf: Eberhard Frederich Ferdinand Hopf (April 17, 1902, Salzburg, Austria-Hungary – July 24, 1983, Bloomington, Indiana) was a mathematician and astronomer, one of the founding fathers of ergodic theory and a pioneer of bifurcation theory who also made significant contributions to the subjects of partial differential equations and integral equations, fluid dynamics, and differential geometry.  The Hopf maximum principle is an early result of his (1927) that is one of the most important techniques in the theory of elliptic partial differential equations.

Arthur Milgram: Arthur Norton Milgram (3 June 1912, in Philadelphia – 30 January 1961) was an American mathematician.  He made contributions in functional analysis, combinatorics, differential geometry, topology, partial differential equations, and Galois theory.  Perhaps one of his more famous contributions is the Lax–Milgram theorem—a theorem in functional analysis that is particularly applicable in the study of partial differential equations.  In the third chapter of Emil Artin's book "Galois Theory", Milgram also discussed some applications of Galois theory.  Milgram also contributed to graph theory, by co-authoring the article "Verallgemeinerung eines graphentheoretischen Satzes von Rédei" with Tibor Gallai in 1960.

Symbol of a differential operator: In mathematics, the symbol of a linear differential operator associates to a differential operator a polynomial by, roughly speaking, replacing each partial derivative by a new variable.  The symbol of a differential operator has broad applications to Fourier analysis.  In particular, in this connection it leads to the notion of a pseudo-differential operator.  The highest-order terms of the symbol, known as the principal symbol, almost completely controls the qualitative behavior of solutions of a partial differential equation.  Linear elliptic partial differential equations can be characterized as those whose principal symbol is nowhere zero.  In the study of hyperbolic and parabolic partial differential equations, zeros of the principal symbol correspond to the characteristics of the partial differential equation.  Consequently, the symbol is often fundamental for the solution of such equations, and is one of the main computational devices used to study their singularities.

Bäcklund transform: In mathematics, Bäcklund transforms or Bäcklund transformations (named after the Swedish mathematician Albert Victor Bäcklund) relate partial differential equations and their solutions.  They are an important tool in soliton theory and integrable systems.  A Bäcklund transform is typically a system of first order partial differential equations relating two functions, and often depending on an additional parameter.  It implies that the two functions separately satisfy partial differential equations, and each of the two functions is then said to be a Bäcklund transformation of the other.

Supporting paragraphs: A Beautiful Mind (film) John Forbes Nash Jr.
Answer: John Forbes Nash Jr.