Barry Simon, professor of mathematics and theoretical physics at Caltech, publishes in December 1983 "Holonomy, the Quantum Adiabatic Theorem, and Berry's Phase", showing that the "geometric phase factor" Michael Berry had discovered a few months earlier in his study of the quantum adiabatic theorem is precisely the holonomy of a Hermitian line bundle, since the adiabatic theorem itself naturally defines a connection on that bundle. Simon not only clarifies the mathematical origin of the Berry phase and provides simpler calculation formulas, but also establishes for the first time the formal connection between Berry's work and the topological invariant Thouless, Kohmoto, Nightingale, and den Nijs (TKNN) had introduced in 1982 to explain the quantization of the quantum Hall effect: both turn out to be, at bottom, descriptions of the same mathematical object, today known as the Chern number. Simon himself explains in his acknowledgments that he learned of Berry's work directly from Berry, and that it was B. Souillard who first suggested, via Berry, that a connection to the TKNN work must exist. This formal bridge, published in December 1983 — just months before Berry's formal publication in March 1984 — proved foundational to the modern topological understanding of condensed matter.