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The Abel Prize is an international award for outstanding work in mathematics, named after Niels Henrik Abel. It was first awarded in 2003 to Jean-Pierre Serre. The Norwegian Academy of Science and Letters administers the selection.^[ref]

The mathematical work behind an award is more informative than the honour alone. The 2009 award recognized Mikhail Gromov's contributions to geometry and topology, including global geometric questions whose behaviour cannot be read from local coordinates alone.^[ref] The 2021 award to László Lovász and Avi Wigderson recognized contributions connecting discrete mathematics and theoretical computer science.^[ref] The prize's scope therefore spans several mathematical traditions.

By Jean-Pierre Serre. Annals of Mathematics, second series, 54(3), 425–505 (November 1951).

Institution-hosted original: www.college-de-france.fr/media/jean-pierre-serre/UPL7235285843586540944_Serre_The_se.pdf

Verified locator: printed page 425, Introduction; loop space, contractible total space and the relation between successive homotopy groups. Chapter II constructs the singular-homology spectral sequence.

Jean-Pierre Serre developed methods connecting algebraic topology, algebraic geometry and number theory. He received the first Abel Prize in 2003. ^[ref]

His 1951 work used singular homology to study loop spaces and homotopy groups. A loop space turns paths in a topological space $X$ that start and end at a fixed base point into points of a new space $\Omega X$. Serre treated $\Omega X$ as the fibre of a fibration whose total space is contractible, linking the topology of the fibre to that of $X$. The resulting Serre spectral sequence organizes the calculation into successive algebraic stages.^[ref]

The dimension shiftmakes the strategy useful: information about families of loops becomes information about higher-dimensional homotopy groups. For example, a loop in $\Omega X$ describes a continuously varying family of loops in $X$, introducing a second parameter. Turning a geometric question into a calculation about another space is a characteristic strength of this approach.^[ref]