01 · A life in mathematics
The person behind the work
Alexander Grothendieck was born in Berlin in 1928 and reshaped algebraic geometry through a new conceptual framework. At the IHES, where he was a permanent professor from 1958 to 1970, he developed schemes, new cohomological methods and far-reaching relationships between geometry and arithmetic. He received the 1966 Fields Medal. Although the IMU’s retrospective list labels his institution as the University of Paris, the IHES documents his appointment during that period. His collaborative seminars and extensive foundational writing changed the questions mathematicians could formulate about spaces defined by equations.
The contribution, explained
Classical algebraic geometry studies shapes cut out by polynomial equations, often over an algebraically closed field. Grothendieck’s schemes extend the underlying language. One builds local geometric pieces from commutative rings and glues them using compatible overlaps. This lets arithmetic rings and families varying over a base be treated geometrically, while retaining algebraic information that a set of visible points can miss. The framework works together with sheaves and cohomology to compare local and global structure. Its power comes from making many previously separate situations instances of one carefully defined language.
Why it matters
Schemes and their associated tools became foundations of modern algebraic geometry and arithmetic geometry, enabling subsequent breakthroughs such as the proof of the Weil conjectures.