Eugenio Calabi, American mathematician and professor emeritus at the University of Pennsylvania, specializing in differential geometry. Konrad Jacobs, Erlangen
The Italian mathematician Eugenio Calabi died on September 25, 2023 at the age of 100 in Beaumont, Bryn Mawr (USA). This year saw numerous tributes around the world to celebrate his impressive legacy and important contributions to geometry. It is unusual that the 100th birthday of an eminent mathematician with more than 70 years of scientific heritage and three generations of descendants is celebrated under his watchful eye – this happened at one of the conferences in Hefei (China) – .
Calabi was born in Milan, Italy in May 1923 and moved to the United States with his family at a young age. He studied at the Massachusetts Institute of Technology, funded by a prestigious Putnam Scholarship, which was also received by others such as Richard P. Feynman, Nobel Prize in Physics, and John Milnor, Fields Medal. In 1950, he read his dissertation at Princeton University on the properties of certain geometric spaces known as Kähler manifolds. After working as a professor at the University of Minnesota, Calabi moved to the University of Pennsylvania in 1964. A few years later, he received the prestigious Thomas A. Scott Professorship in Mathematics, which he held until his retirement in 1994, when he became Emeritus at the same institution.
His work has left a deep mark on modern geometry. His obsession was to give bare space a “preferred” shape, as if someone were sculpting a piece of clay with their hands, looking for a hidden figure that he had never before imagined. For example, if you place a rope tied at its ends on a flat surface, what shape can it best take? The answer of many will be a circumference, because he is “the same everywhere,” or perhaps because he is “the most perfect figure.” A mathematician might add that this perception has to do with a variational property of this curve: it is the one that maximizes the total area contained within it.
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A mathematical method to find these preferred curves in the plane – called flow of average curvature – is as follows: we start with an arbitrary curve (that does not intersect itself) and make it “evolve” like this that it loses area to a constant speed and its circumference decreases as quickly as possible. Over time, the curve becomes convex, tending to form a circle of smaller and smaller radius until it collapses at a point. Just before this collapse, the preferred shape of the curve can be observed with the naked eye on a very small scale.
If the original curve intersects itself, a singularity or “spike” may occur as it evolves, changing the preferred shape of the curve. By placing oneself in the place of the singularity just before it arises, one observes, through a change of scale, the “self-similar” evolution of a curve that comes from an infinite past: a curve that does not change shape as it evolves with it currently. In this case, the curve also moves through translations, that is, all of its points move in a fixed direction at a constant speed. Eugenio Calabi discovered this solution to curvature flow in the 1980s and called it La Parca (the Grim Reaper).
Each sharp curve discovered by Calabi within La Parca (marked in blue in the image) will collapse to a point and disappear before reaching him.Mario García Fernández
It seems that Calabi made this discovery during a tea break, in the middle of a conversation, surrounded by his colleagues. The Grim Reaper turns out to be the only solution, defined from an infinite past of medium curvature flow that evolves through translations: an essential characteristic that would only be understood many years later. This is perhaps one of Calabi's most unique characteristics: his influence on the work of his colleagues often came through long informal conversations with sharp observations and key examples that later became fundamental parts of future theories. Mathematics. In the words of Edoardo Vesentini (researcher at the Scuola Normale Superiore di Pisa): “In the most intimidating theories and in the theorems that tormented me the most, Calabi's simple explanations arrived.”
In his case, these explanations seemed to arise from intuition or aesthetic taste. As Calabi himself explained during a visit to Spain in September 2000: “The main source of geometric intuition is ultimately linked to our sensory perception of the world.” As we move into more abstract areas, one obviously has to interpret what sensory experience means. I tried to make this as visible as possible to convey this idea.”
The joy of pure discovery and the beauty of geometry were actually two driving forces of Calabi's mathematics. However, his work has turned out to have important implications for other application areas such as theoretical physics. As Calabi himself described, “mathematicians invent imaginary worlds, and scientists decide much later whether these can accommodate real scientific theories.” One of these worlds imagined by Calabi arose from the study of the preferred shape of an important class of geometric spaces known as complex manifolds are. These objects are made rigid by giving them a concept of distance (the so-called Kähler metric). The preferred shape of this space results from the selection of all possible dimensions, which makes the space curve more homogeneous. A special case of this problem is known as the Calabi problem. For more than 20 years, great mathematicians tried to deal with it and came up with contradictory solutions. Finally, in 1978, Shing-Tung Yau solved the problem and gave rise to the varieties popularly known as Calabi Yau varieties. For this important achievement, the international mathematical community awarded the Chinese mathematician the Fields Medal in 1982. The general problem initially raised by Calabi remains open to this day and had a major influence on the development of complex geometry in the 20th century. and beginnings of the 21st century. Much of the activity for years has focused on the study of geometries known as Kähler-Einstein, of which the Calabi-Yau manifolds are a special case.
Years later it turned out that Calabi's criterion for determining the preferred spatial shape had a close connection with the field equations of general relativity introduced by Albert Einstein. In these equations, the distribution of matter and energy in space determines its curvature. In the absence of matter or when we establish a homogeneous distribution of it, space takes on the preferred shape imagined by Eugenio Calabi. Surprisingly, far from being a mere analogy, Calabi-Yau spaces with their beautiful geometric shapes play a key role in some modern physical theories dealing with the problem of quantum gravity, such as the well-known superstring theory.
Mario García Fernandez He is a Ramón y Cajal researcher at the Autonomous University of Madrid and a member of the Institute of Mathematical Sciences (ICMAT).
Oscar Garcia-Prada He is a research professor at the Higher Council for Scientific Research and a member of ICMAT.
Coffee and theorems is a field dedicated to mathematics and the environment in which it arises, coordinated by the Institute of Mathematical Sciences (ICMAT). In which researchers and members of the Center describe the latest advances in this discipline, share points of contact between mathematics and other social and cultural expressions and remember those who shaped its development and knew how to turn coffee into theorems. The name is reminiscent of the definition of the Hungarian mathematician Alfred Rényi: “A mathematician is a machine that converts coffee into theorems.”
Editorial and coordination: Ágata A. Timón G Longoria (ICMAT).
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