The most important mathematician you’ve (probably) never heard of

When discussing the greatest physicists of the 20^th century, most will name Albert Einstein. In contrast, the field of mathematics often lacks such universally recognized figures, which is why it is essential to shed light on Alexander Grothendieck.

Einstein revolutionized physics with his groundbreaking theories of relativity and contributed significantly to quantum mechanics, achieving fame that transcended the scientific community. Conversely, Grothendieck transformed the landscape of mathematics through his profound work, yet he withdrew from both academia and public life before his death, remaining a somewhat enigmatic figure.

Grothendieck’s personality, coupled with the complexity of his work, often made him less appealing in the spotlight compared to the charismatic Einstein. While Einstein had a gift for storytelling, making intricate concepts relatable (like the famous twin paradox), Grothendieck’s abstract notions require considerable background knowledge to understand fully.

At the core of Grothendieck’s influence lies his redefinition of algebraic geometry, a domain exploring the intricate relationship between algebraic equations and geometric shapes. For instance, the equation x² + y² = 1 describes a circle. The jump from foundational concepts introduced by Descartes in the 17th century to Grothendieck’s advanced methodologies showcases his genius in seeking broad generalizations in mathematics.

For Grothendieck, the relationship between algebra and geometry was just the beginning. He recognized that equations like x² = 0 and x = 0, which yield the same algebraic variety (a singular solution), mask essential distinctions in their nature. In 1960, he introduced the concept of a “scheme” to capture the additional information lost in traditional algebraic geometry.

Understanding schemes requires grasping another mathematical idea: a ring. In this context, a ring is a collection of numbers that remains consistent through operations such as addition and multiplication — for instance, the integers. This framework allowed Grothendieck to enrich algebraic varieties with layers of complexity, thus transforming problems from diverse mathematical fields into geometrical questions while preserving crucial structural information.

The introduction of schemes had profound implications for numerous mathematical problems. Notably, Grothendieck and his collaborators leveraged schemes to prove three of the Weil conjectures in 1965, which focused on counting solutions for specific types of algebraic varieties—a challenge that had eluded mathematicians for years. Testaments to Grothendieck’s influence are the significant breakthroughs that followed, including Pierre Deligne’s eventual proof of the final conjecture in 1974.

Schemes also played a critical role in solving Fermat’s Last Theorem, a puzzle in number theory that persisted for centuries. Andrew Wiles, who finally cracked the theorem in 1995, utilized Grothendieck’s concepts of algebraic geometry, translating the problem into inquiries involving elliptic curves accessible through schemes.

Grothendieck’s impact extends far beyond these examples; his work laid the foundational tools for countless areas of mathematics. His innovative generalization of “space” to the concept of a “topos” enriches analyses by integrating various dimensions of information critical for solving complex mathematical problems.

Despite his monumental contributions to mathematics, Grothendieck remains a lesser-known figure outside of specialized circles, largely due to the complexity of his work and his preference for a life away from the limelight. A committed pacifist, he notably declined the Fields Medal award in 1966 in protest against the military actions of the Soviet Union. “Fertility is measured by offspring, not by honours,” he stated, emphasizing his desire for the quality of his work to speak for itself.

In 1970, he severed ties with academia and retreated into isolation, disillusioned with the mathematical community and its associations. His later writings, including his autobiography, Harvests and Sowings, and a philosophical manuscript, The Key of Dreams, offer glimpses into his intricate thoughts on mathematics and existence. Over time, he withdrew further into seclusion, almost legend-like in his hermitism, ultimately passing away in 2014, yet leaving behind a legacy that continues to shape the mathematical landscape.