In May 1940, while Germany was invading France, 16-year-old René Thom was finishing high school in Montbéliard, his hometown. Six years later – after finishing his career at the prestigious Ecole Normale Supérieure, where he was rejected on his first attempt at admission – he received his doctorate in mathematics from the University of Strasbourg. There he met several members of the Bourbaki group and then devoted himself to their task of rewriting modern mathematics. And it was there that he began his work in geometry, which led him to win the Fields Medal a decade later.
His work allowed us to effectively classify the different types of geometric shapes that exist. We can explain classification in mathematics by analogy with taxonomic classification in biology, as Thom himself would do. Every geometric shape, like every animal, belongs to a species, and these are divided into genera, which in turn are divided into families. Higher up we have Order, Edge, Class and Kingdom. In biology, this classification involves identifying certain characteristics – such as hair or scales – that can be used to distinguish animals from different groups. Mathematics is about doing something similar.
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In 1854, Bernhard Riemann defined with some rigor the concept of species in the context of geometry, which we now call “Riemann variety”. Over the next 100 years, several mathematicians discovered certain geometric features that allowed different species to be distinguished. For example the Euler characteristic or Betti numbers.
Likewise, other geometric concepts analogous to gender or family were introduced. But just as we are far from knowing all the biological species or genera that exist on Earth, neither were we expected to know all the geometric species or genera; there were too many. Everything changed when, in the early 1950s, René Thom introduced a classification similar to that of the Kingdom and much higher up the taxonomic hierarchy.
Thom stated that two geometric shapes are in the same geometric range if one can be transformed into the other through a certain process. For example, when a cell divides, its cell membrane initially has the shape of a sphere, but after division it takes the shape of two spheres. The geometric process by which the first sphere becomes the other two is called cobordism and keeps the objects in the same area. The figure of the bull is also in the same area as the figure of the sphere, although the geometric method used is different from that mentioned.
Not satisfied with this, Thom found and classified all possible geometric kingdoms. This achievement, unprecedented in the history of geometry, sparked a revolution that continues to this day. Such was the impact that in the following three editions of the Fields Medal, a geometer whose work was based on Thom’s new classification won: John Milnor (1962), Stephen Smale (1966) and Sergei Novikov (1970).
Shortly after receiving the Fields Medal, Thom received a permanent position at the French Institut des Hautes Études Scientifiques, where he created a new mathematical basis for biology or semiotics: catastrophe theory. He wanted to address processes in which continuity is interrupted, as happens during the development of an embryo.
In his autobiography, Thom connects his radical change in research topic with his encounter with the mathematician Alexandre Grothendieck: “His technical superiority was overwhelming. His seminar attracted all the mathematicians in Paris, while I had nothing new to offer. This led me to abandon the world of pure mathematics and turn to more general concepts such as the theory of morphogenesis, a topic that interested me more and led me to a very general form of philosophical biology.
A follower of Heraclitus and Aristotle, his philosophical viewpoint was that the obsession with accuracy was an obstacle to understanding. “If I had to choose between rigor and understanding, I would undoubtedly choose the latter,” he explained. He preferred qualitative explanations to quantitative ones and was very critical of modern mathematics – promoted primarily by his former colleagues at Bourbaki – which, in his view, saw only algebra as a source of accuracy – while Thom preferred geometry.
Paradoxically, Thom’s greatest contribution – his classification of geometric shapes – helped reduce geometry to algebra, as the French mathematician and Bourbaki co-founder Jean Dieudonné sharply noted in a 1972 open letter to Thom: “It is ironic that Thom clearly believes that He had an aversion to algebra and is remembered for its original use in his theory of cobordism.
Federico Cantero Moran He is a professor at the Autonomous University of Madrid 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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