Jean-Pierre Serre

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]