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Biographical Memoirs Volume 61 (1992) / Chapter Skim
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Norbert Wiener
Pages 388-437

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From page 389...
... Cannon, Alan Turing, and others. This synthesis, which he called "cybernetics," has sinre - ...
From page 390...
... Leo Wiener stuc3ied engineering in Berlin anc! medicine in Warsaw.
From page 391...
... By other accounts, Leo Wiener was exceptionally original, imaginative, and productive intellectually. At the same time, he was a fine teacher ant!
From page 392...
... His brother Frederick was born in the same year. In 1909, Norbert was graduated from Tufts College with a cum laude A.B.
From page 393...
... ~ of OUI because OI Koyce-s ctlmlnlsnlng nealtn, worked for his doctorate with Karl Schmidt of Tufts, who as a young professor was interested in mathematical logic. In 1913, Norbert was gracluatecl from Harvard with a Ph.D.
From page 394...
... But however much Wiener's professional career clepencled on his prowess in pure mathematics, his later work was to clisplay quite convincingly and effectively a profound concern for issues of external relevance. With Harcly's support, Wiener was to become better known for his work on Tauberian theorems than for his earlier and probably more innovative work on Brownian motion, which was outside the mainstream of mathematics cluring the twenties.
From page 395...
... He stuclied there with John Dewey, among others. Following this, Norbert received a junior position at Harvard University.
From page 396...
... He remained at the Institute up to the time of his death, and his scientific interaction with it was to prove a great mutual benefit. CENTRAL DECADE I: 1919-29 Wiener's early mathematical papers concerned mathematical logic and its relations to space, time, and measurement.
From page 397...
... This was the beginning of his work on a mathematical theory of Brownian motion, essentially the theory of "Wiener space," as others have called it, and the prime example for the moclern theory of functions of functions, or for functions of infinitely many variables. The physical theory of Brownian motion had earlier been studied by Einstein and Smoluchowski and was proposed to Wiener as a topic for investigation by Russell; coincidentally and again serenclipitously, the problem of integration in function space had been proposed to Wiener by T
From page 398...
... During the early 1920s, Wiener sank his roots creeper into functional integration, while at the same time making significant contributions in a variety of other parts of analysis. The novelty of his Brownian motion theory was such that it was not at all wiclely appreciated at the time, and the few who diet, such as H
From page 399...
... An even longer memoir, "Generalized Harmonic Analysis," which appeared in 1930, reflected in part his work with Bohr, proviclecl an alternative approach to his Brownian motion theory, and connected this with the spectral analysis of functions on the line. CENTRAL DECADE II: 1930-40 The Wiener family visited Cambridge University during 1931-32, and at Harcly's invitation, Wiener lectured on har
From page 400...
... is ultimately seen to be invariant under the group of unitary operators on a Hilbert space. The connection was to lead to "The Homogeneous Chaos," one of Wiener's most seminal papers, which facilitated harmonic analysis in Wiener space and related to the mathematical theory of Bose-Einstein quantum fields.
From page 401...
... Paley's book, Fourier Transforms in the Complex Domain (1934,31. The extension of real harmonic analysis to the complex domain was one of Wiener's major secondary themes.
From page 402...
... R Pitt came to Cambricige and worked with Wiener on analytic functions of absolutely convergent Laplace-StieTtjes transforms, in extension of the core of Wiener's Tauberian theory.
From page 403...
... the only one comparable In depth ant! originality to his earlier work on Brownian motion, on real harmonic analysis, anti on potential theory, was on what he termed "the Homogeneous Chaos." This work in the late thirties related ergo(lic theory to Wiener space Wand to harmonic analysis.
From page 404...
... wave representations of a quantized Bose-Einstein field the In are what are known as the e-particle subspaces in this connection- as mathematically formulatect in the fifties but icleationally going back to the beginnings of quantum field theory in the highly heuristic form given by Dirac, accorcling to which "a Bose-Einstein f~elct is equivalent to an assembly of harmonic oscillators." It was characteristic of Wiener's extraordinary scientific intuition that he was able to construct a basic part of quantum fielcl theory on the basis of pure thought, starting from his theory of functional integration. However, in this work, as well as in later, relatect work with A
From page 405...
... The basic theory was given in a report published during the war; this was effectively a ctraft of his monograph, "Extrapolation, Interpolation, and Smoothing on Stationary Time Series," published in 1950. This work represents a special case of the stucly of mechanisms as crevices that effect an input-output transfer, with regard to smoothing, feeclback, and stability, independently of internal dynamics.
From page 406...
... But important factors in his becoming a mathematician were Hardy's leadership and, probably, Russell's declining interest in mathematical logic. ~ , ~ By the middle and late thirties, Wiener had attained pure mathematical eminence, indeed a virtual world preeminence in a major part of mathematical analysis.
From page 407...
... of development that would concern him in the forties and thereafter. His work on Fourier analysis, and especially that in the complex domain, providecI a general mathematical theory that was clearly most relevant to theoretical network ant!
From page 408...
... to important mathematical papers, but when the war came he was quick to turn to applications along cybernetic lines, such as prediction theory anct fire control. He never again returned to mathematical work at the intense ant!
From page 409...
... Probably the most important was his book Nonlinear Problems in Rando~n Theory, which made the ideas of his theory of Brownian motion and the homogeneous chaos accessible to engineers concerned with time series. He continued his earlier mathematical collaborations with E
From page 410...
... In more recent years, functional integration has played a major part in quantum field theory although in a quite heuristic form in the physical literature as in the path integral formalism originated by Feynman. In any event, Wiener's work in the early twenties appears, with some hindsight, to foreshadow somewhat the important and influential formulation in rigorous mathematical terms of the theory of probability by Kolmogorov in ~ 933.
From page 411...
... To quote from Wiener: The Brownian motion was nothing new as an object of study by physicists. There were fundamental papers by Einstein and Smoluchowski that covered it, but whereas these papers concerned what was happening to any given particle at a specific time, or the long-time statistics of many particles, they did not concern themselves with the mathematical properties of the curve followed by a single particle.
From page 412...
... dx~tJ as a random variable, or measurable function, on W Such integrals arise in generalized form in the modern theory of stochastic differential equations, which has been principally clevelopec!
From page 413...
... which in the case of a finite-ctimensional Hilbert space was equivalent to conventional harmonic analysis of square-integrable functions in Euclidean space. The invariant Gaussian measure g on a real Euclidean space H is not countably aciclitive when H is infinite dimensional, i.e., a Hilbert space.
From page 414...
... , a correspondence emphasized by Wiener in special cases in the form of an action on W For example, if Brownian motion is considerect on the entire real line instead of the interval to,]
From page 415...
... However, it was not until more than a clecacle after Wiener's work that quantum field theory was subsumable uncler a clear mathematical theory, especially in its wave aspects, to which the Wiener space formulation corresponded. A heuristic treatment of particle aspects had been given by V
From page 416...
... The development of the theory of harmonic analysis on general locally compact commutative groups, basically complete by the micl-forties, confirmed this insight. In a way the analog of the Plancherel theory on Wiener space, which was connected with Wiener's Gaussian approach to finiteclimensional Plancherel theory, could be construct!
From page 417...
... in contrast to [2 harmonic analysis in infinitely many dimensions, or analysis on Wiener space. [2, the space of all square-integrable functions, is simple in that it is invariant under Fourier transformation, but L~ is not; this is the nub of the difficulty for Wiener's prototypical result on the invertibility of an absolutely convergent Fourier series that nowhere vanishes.
From page 418...
... For an arbitrary function f in Hi, every function in the closed ideal it generates evidently vanishes wherever the Fourier transform offdoes; but are all such functions in this ideal? Alternatively, one may ask whether the finite linear combinations of the fix + cJ are dense in the subset of L~ consisting of functions whose Fourier transforms vanish where that of f does—the "spectral synthesis" question.
From page 419...
... real and complex harmonic analysis, and aspects of ergoclic and potential theory, on the pure mathematical sicle, and for cybernetics and rigorous excursions into statistical mechanics and the theory of light on the applied sicle. PERSONAL AND SOCIAL LIFE Wiener was at the opposite end of the spectrum from ivory tower scientists or academic philosophers.
From page 420...
... 420 BIOGRAPHICAL MEMOIRS friends in attempts to obtain for me similar honors elsewhere.
From page 421...
... Wiener would on occasion become absorbed in intricate questions of technique, as shown for example by the counterexample he developer! with Pitt to the invertibility of an arbitrary nonvanishing absolutely convergent Laplace-Stieltjes transform.
From page 422...
... His work on relativity, quantum theory, light, and statistical mechanics for the most part display topical imagination more than mature scholarship. But some of this work, such as his theory of the coherency of light, has been quite significant.
From page 423...
... , largely mark the- end of the innovative phase of his scientific career, apart from his continuing work on prediction theory and some unsystematic excursions. His main motive in writing his unusually personal yet philosophical autobiographical books was that I wish to think out to myself what my career has meant and to come to that emotional peace which only a thorough consideration and understanding of one's past bring.
From page 424...
... from the Harvard graduate school; the Bocher Prize (1933) of the American Mathematical Society, for outstanding research in analysis, jointly with Marston Morse; the Lorct and Taylor American Design Award (1949~; and the ASTME Research Mecial (1946~.
From page 425...
... ~ -7 - -- r ~ `;oIlcern gnat it nacr gone too tar, not only for relevance outside of pure mathematics but for optimal growth of this subject itself. In 193S, at the height of his pure mathematical achievements, he expressed himself as follows: It is a falsification of the history of mathematics to represent pure mathematics as a self-contained science drawing inspiration from itself alone and morally taking in its own washing.
From page 426...
... overall as an outstanding mocle! in this century for a life of synthesis of pure intellectual penetration with external relevance.
From page 427...
... Army Aberdeen Proving Ground, Maryland 1919 Editorial Writer, Boston Herald 1919-24 MIT, Instructor in Mathematics 1925-29 MIT, Assistant Professor of Mathematics 1929-32 MIT, Associate Professor of Mathematics; Bocher Prize, American Mathematical Society ~ 1933) 1932-59 MIT, Professor of Mathematics; Lord and Taylor American Design Award (1949~; Hon.
From page 428...
... 428 BIOGRAPHICAL MEMOIRS 1959-60 MIT, Institute Professor; ASTME Research Medal ~ 1960) 1960-64 MIT, Institute Professor Emeritus; National Medal of Science ~ 1963)
From page 429...
... NORBERT WIENER SELECTED BIBLIOGRAPHY 429 1913 On a method of rearranging the positive integers in a series of ordinal numbers greater than that of any given fundamental sequence of omegas. Messenger Math.
From page 430...
... A new vector method in integral equations.
From page 431...
... 2:118-23. Une methode nouvelle pour la demonstration des theoremes de M
From page 432...
... Notes on the theory and application of Fourier transforms I, II. Trans.
From page 433...
... Fourier Transforms in the Complex Domain.
From page 434...
... Harmonic analysis and ergodic theory.
From page 435...
... The prediction theory of multivariate stochastic processes, Part I
From page 436...
... . Cambridge, Mass.: The MIT Press; New York: Wiley.


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