Vladimir Arnold, while still Kolmogorov's student, defended a 1959 dissertation containing his resolution of Hilbert's 13th problem, earning his degree in 1961

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Fascitiello's 2022 historical essay on Andrey Kolmogorov's circle states that his student Vladimir Arnold defended his dissertation in 1959 "with the dissertation containing his famous resolution of Hilbert's 13th problem," and received his degree in 1961. David Hilbert had posed the 13th problem in his 1900 address to the International Congress of Mathematicians; in the continuous-function form the Kolmogorov school addressed, it asks whether every continuous function of several variables can be expressed as a composition of continuous functions of two variables (Hilbert framed it around the solution of the general seventh-degree equation and expected the answer to be no). Arnold's resolution — building on Kolmogorov's 1956 reduction of the question to continuous functions of three variables — showed that every continuous function of three variables can be so expressed through functions of two variables, closing the problem in the affirmative against Hilbert's expectation. Kolmogorov's own companion theorem of the same year strengthened the representation further: any continuous multivariate function can be written as a finite composition of continuous single-variable functions and addition. Arnold did this work while still a student under Kolmogorov's supervision, a detail Fascitiello's essay records as part of the broader culture of Kolmogorov's mathematical school in Moscow.

This pair of 1957 results — commonly cited by publication date rather than the later dissertation-defense date this note records — is now jointly known as the Kolmogorov-Arnold representation theorem; it sat as a pure mathematics result for over six decades before being cited as the direct structural basis of a 2024 neural-network architecture (claim-liu-2024-kan-paper-names-architecture-after-kolmogorov-arnold-theorem).