Theory and Application of Harmonic Integrals

work by Hodge

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July 20, 2026

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The "Theory and Application of Harmonic Integrals" is a book by British mathematician William Hodge, published in 1941. This work introduced what is now known as the Hodge conjecture. The conjecture proposes that for certain "nice" spaces, specifically projective algebraic varieties, their complex shapes can be approximated by simpler geometric components called algebraic cycles.12

Harmonic functions, related to this field, are mathematical functions where the value at any point equals the average of values around that point. They satisfy Laplace's equation and are crucial in describing equilibrium conditions in physics, such as temperature or electrical charge distribution. Harmonic analysis, a related mathematical procedure, breaks down complex periodic phenomena into simpler components, often using Fourier series.34

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Sir William Hodge
…Hodge formulated in his book Theory and Application of Harmonic Integrals what became known as the Hodge conjecture: that for certain “nice” spaces (projective algebraic varieties), their complicated shapes can be covered (approximated) by a collection of simpler geometric pieces called algebraic cycles. The conjecture received little attention before he…
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…Hodge formulated in his book Theory and Application of Harmonic Integrals what became known as the Hodge conjecture: that for certain “nice” spaces (projective algebraic varieties), their complicated shapes can be covered (approximated) by a collection of simpler geometric pieces called algebraic cycles. The conjecture received little attention before he…
Read More