Eugenio Calabi, an exceptionally inventive mathematician renowned among his peers for his transformative originality, passed away at the age of 100 on September 25. His former student, Xiuxiong Chen, described him as a mathematician who thrived on initiating new subjects rather than concluding existing ones.
During his six-decade tenure at the University of Pennsylvania, Calabi’s unique perspective and preference for unconventional choices in mathematics led to the exploration of novel and intriguing questions, often with unforeseen and lasting consequences. Calabi’s most notable contribution came in 1953 when he proposed a conjecture related to a distinct class of manifolds.
Manifolds, mathematical spaces or surfaces that can exist in various dimensions, have the distinctive property that small neighbourhoods around each point appear flat. Calabi became intrigued by Kähler manifolds during his time at Princeton University, smooth manifolds that only exist in even dimensions, such as 2, 4, and 6.
These manifolds can have varying curvature, unlike a sphere, which has constant curvature. Calabi focused on the Ricci curvature, a measure of curvature, and suggested that Kähler manifolds could exhibit zero Ricci curvature at every point while adhering to specific topological constraints. Many mathematicians doubted the feasibility of such shapes, believing them to be too good to be true.
Shing-Tung Yau, initially sceptical of Calabi’s conjecture, stumbled upon it in 1970 while a graduate student at the University of California, Berkeley. To prove the conjecture, one needed to solve a complex equation that had never been tackled before. Yau initially claimed to have found counterexamples in 1973, but Calabi later requested clarification. Yau’s reevaluation revealed an error in his calculations, suggesting that the conjecture might indeed be correct.
Yau spent the next three years proving the existence of the manifolds proposed by Calabi. In 1976, he, Calabi, and another mathematician confirmed the validity of his proof, establishing the existence of what is now known as Calabi-Yau manifolds. Yau’s groundbreaking work earned him a Fields Medal in 1982, marking the culmination of Calabi’s visionary concept.
Around the same time, physicists exploring theories of nature’s fundamental forces realized that the universe might contain hidden dimensions within Calabi-Yau manifolds, as proposed by string theory. These manifolds were seen as the ideal fit for string theory, given their compatibility with a property called supersymmetry. By 1984, Yau had already identified thousands of different six-dimensional Calabi-Yau shapes.
Calabi-Yau manifolds continue to be a focus of scientific investigation, with physicists and mathematicians publishing numerous papers examining their properties each year. While their presence in our observable world remains uncertain, their significance in advancing scientific theories is undeniable
Calabi’s impact on mathematics extended far beyond his conjecture’s influence on physics. His journey to becoming a mathematician began with a background in chemical engineering and a brief stint as a U.S. Army translator during World War II. Eventually, he transitioned to mathematics and earned his doctorate at Princeton before establishing a long-term position at the University of Pennsylvania in 1964.
Calabi remained passionate about mathematics throughout his life, conducting research well into his 90s. He was known for his enthusiasm in sharing mathematical ideas, often engaging in lengthy conversations with colleagues and students, scribbling formulas on napkins, envelopes, and scraps of paper.
Reflecting on his life’s work, Calabi regarded mathematics as his favourite hobby and considered himself extraordinarily fortunate to have pursued it as a profession. As he once said, “To follow your hobbies as a profession is the extraordinary luck I’ve had in my life.”
