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René Thom
René Thom
René Thom
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René Frédéric Thom (French: [ʁəne tɔm]; 2 September 1923 – 25 October 2002) was a French mathematician, who received the Fields Medal in 1958.

Key Information

He made his reputation as a topologist, moving on to aspects of what would be called singularity theory; he became world-famous among the wider academic community and the educated general public for one aspect of this latter interest, his work as the founder of catastrophe theory (later developed by Christopher Zeeman).[1][2][3][4][5]

Life and career

[edit]

René Thom grew up in a modest family in Montbéliard, Doubs and obtained a Baccalauréat in 1940. After the German invasion of France, his family took refuge in Switzerland and then in Lyon. In 1941 he moved to Paris to attend Lycée Saint-Louis and in 1943 he began studying mathematics at École Normale Supérieure, becoming agrégé in 1946.[6]

He received his PhD in 1951 from the University of Paris. His thesis, titled Espaces fibrés en sphères et carrés de Steenrod (Sphere bundles and Steenrod squares), was written under the direction of Henri Cartan.[7]

After a fellowship at Princeton University Graduate College (1951–1952), he became Maître de conférences at the Universities of Grenoble (1953–1954) and Strasbourg (1954–1963), where he was appointed Professor in 1957. In 1964 he moved to the Institut des Hautes Études Scientifiques, in Bures-sur-Yvette, where he worked until 1990.[8]

In 1958, Thom received the Fields Medal at the International Congress of Mathematicians in Edinburgh for the foundations of cobordism theory, which were already present in his thesis.[9] He was invited speaker at the International Congress of Mathematicians two more times: in 1970 in Nice[10] and 1983 in Warsaw (which he did not attend).[11]

He was awarded the Brouwer Medal in 1970,[12] the Grand Prix Scientifique de la Ville de Paris in 1974, and the John von Neumann Lecture Prize in 1976.[13] He became the first president, together with Louis Néel, of the newly established Fondation Louis-de-Broglie In 1973 [14] and was elected Member of the Académie des Sciences of Paris in 1976.[15]

Salvador Dalí paid homage to René Thom with the paintings The Swallow's Tail and Topological Abduction of Europe.[16]

Research

[edit]

While René Thom is most known to the public for his development of catastrophe theory between 1968 and 1972,[17] his academic achievements concern mostly his mathematical work on topology.[18][19]

In the early 1950s, it concerned what are now called Thom spaces, characteristic classes, cobordism theory, and the Thom transversality theorem. Another example of this line of work is the Thom conjecture, versions of which have been investigated using gauge theory. From the mid 1950s he moved into singularity theory, of which catastrophe theory is just one aspect, and in a series of deep (and at the time obscure) papers between 1960 and 1969 developed the theory of stratified sets and stratified maps, proving a basic stratified isotopy theorem describing the local conical structure of Whitney stratified sets, now known as the Thom–Mather isotopy theorem. Much of his work on stratified sets was developed so as to understand the notion of topologically stable maps, and to eventually prove the result that the set of topologically stable mappings between two smooth manifolds is a dense set.

Thom's lectures on the stability of differentiable mappings, given at the University of Bonn in 1960, were written up by Harold Levine and published in the proceedings of a year long symposium on singularities at Liverpool University during 1969–70, edited by C. T. C. Wall. The proof of the density of topologically stable mappings was completed by John Mather in 1970, based on the ideas developed by Thom in the previous ten years. A coherent detailed account was published in 1976 by Christopher Gibson, Klaus Wirthmüller, Andrew du Plessis, and Eduard Looijenga.[20]

During the last twenty years of his life Thom's published work was mainly in philosophy and epistemology, and he undertook a reevaluation of Aristotle's writings on science. In 1992, he was one of eighteen academics who sent a letter to Cambridge University protesting against plans to award Jacques Derrida an honorary doctorate.[21]

Beyond Thom's contributions to algebraic topology, he studied differentiable mappings, through the study of generic properties. In his final years, he turned his attention to an effort to apply his ideas about structural topography to the questions of thought, language, and meaning in the form of a "semiophysics".

Bibliography

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See also

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References

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René Thom

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from Grokipedia
René Thom (1923–2002) was a French renowned for his pioneering contributions to , singularity theory, and the development of , which applies topological methods to model abrupt changes in dynamical systems. He received the in 1958 for his creation of cobordism theory and proof of the Thom-Hirzebruch theorem, which revolutionized the using . Born on September 2, 1923, in , , Thom demonstrated early aptitude in , earning his in in 1940 and in philosophy in 1941. He entered the in 1943, graduating in 1946, and completed his doctorate in 1951 at the under , focusing on topological problems. Influenced by philosophers like and Albert Lautman during his studies, Thom's early career blended rigorous with broader intellectual interests. Thom's academic career began with a research position at the CNRS in Strasbourg in 1946, followed by teaching roles at the University of Grenoble (1953–1954) and (1954–1963), where he became a in 1957. In 1964, he joined the () in Bures-sur-Yvette, remaining there until his retirement. His seminal work in the 1950s included the Thom transversality theorem, which provided tools for studying generic singularities in smooth maps, laying groundwork for modern singularity theory. The , awarded at the 1958 in , recognized his foundational advancements in . In the 1960s and 1970s, Thom shifted toward , introduced in his influential book Structural Stability and (1972), which classified elementary catastrophes arising from stable singularities and applied them to fields like , physics, and . This work, while controversial for its interdisciplinary ambitions, earned him the Grand Prix Scientifique de la Ville de in 1974. Later in his career, Thom increasingly explored , drawing on Aristotelian ideas to address , linguistics, and in works like Mathematical Models of (1974) and Semiophysics: A Sketch (1988). He became an honorary member of the London Mathematical Society in 1990 and passed away on October 25, 2002, in Bures-sur-Yvette.

Biography

Early Life and Education

René Thom was born on September 2, 1923, in , a town in the department of eastern , to parents who owned a . His father possessed a solid , including knowledge of Latin, and composed , which contributed to Thom's early exposure to and open-minded thinking. Thom's childhood was profoundly shaped by , as fell under German occupation from 1940 to 1945. To escape the conflict, he and his brother were evacuated southward, eventually reaching , where they experienced hardship including near-starvation before returning to occupied . During this tumultuous period, Thom developed a strong interest in through self-directed exploration, particularly in and , which he encountered around age 13 or 14. His formal education began at the primary school in in 1931, where he earned a , followed by attendance at the local Collège Cuvier. Influenced by dedicated teachers, Thom excelled in and , obtaining his in from in 1940 and in philosophy from in June 1941. In 1942, Thom began preparing for entry to the (ENS) in at the Lycée Saint-Louis, though wartime disruptions delayed his admission until 1943 after an initial failure the previous year. At ENS, his studies from 1943 to 1946 were guided by prominent mathematicians and , and he was deeply influenced by the rigorous Bourbaki group's approach to mathematics. He passed the examination in 1946, earning his master's equivalent, and completed his doctoral thesis in 1951 at the under Cartan's supervision. The thesis, titled Espaces fibrés en sphères et carrés de Steenrod, examined fiber bundles over spheres and the application of Steenrod squares in , laying foundational work in cobordism theory.

Academic Career

After graduating from the ENS in 1946 with the , Thom took a research position at the CNRS in , allowing him to continue working under 's supervision while preparing his doctoral thesis. Following his doctoral studies under , Thom spent a postdoctoral year at in 1951–1952, where he engaged with leading topologists such as Norman Steenrod and attended seminars by and . Upon returning to France, Thom taught at the University of Grenoble during the 1953–1954 academic year. He then moved to the , where he held a teaching position from 1954 to 1963 and was promoted to full professor in 1957. At Strasbourg, building on his earlier research experiences under Cartan and Charles Ehresmann, Thom advanced his work in within a vibrant mathematical environment. In 1963, Thom joined the newly established (IHÉS) in Bures-sur-Yvette as a permanent professor, a role he held until his retirement in 1990. At IHES, he directed the seminar, co-leading sessions with Bernard Malgrange starting in 1964, and organized interdisciplinary seminars that bridged mathematics with biology and philosophy, fostering discussions on and qualitative dynamics. These activities allowed Thom to evolve his toward broader applications while mentoring emerging mathematicians in a collaborative setting.

Mathematical Contributions

Topology and Cobordism Theory

In the early 1950s, René Thom introduced as a key construct for analyzing , particularly those over spheres. The of a ξ\xi over a base space BB is formed by taking the one-point compactification of the total space and identifying the complement of the zero section with the base point, yielding a space that encodes the topological properties of the bundle in a compact form. This development facilitated the study of stable equivalence classes of bundles and their associated characteristic classes. Central to Thom's framework is the Thom isomorphism theorem, which relates the groups of the to those of the base space. Specifically, it establishes that the reduced of the Thom space Th(ξ)\mathrm{Th}(\xi) is isomorphic to the of the base BB with coefficients twisted by the orientation sheaf of the bundle, shifted by the rank of ξ\xi. Proved in his seminal 1952 paper "Espaces fibrés en sphères et carrés de Steenrod," this theorem provided a bridge between bundle geometry and , enabling computations of via bundle data and influencing the understanding of Steenrod squares as operations on these spaces. The highlighted the role of the Thom class, a fundamental class generating the module structure, and laid groundwork for subsequent advances in theory. Thom's work extended profoundly into cobordism theory, where he addressed the classification of manifolds up to cobordism equivalence. In his 1954 paper "Quelques propriétés globales des variétés différentiables," Thom showed that oriented manifolds are determined up to cobordism by their Pontryagin numbers, establishing that the oriented cobordism ring Ω∗SO\Omega_*^{SO} is generated by these invariants. This result was later refined by John Milnor, who proved in 1960 that the ring is isomorphic to a polynomial algebra Z[x1,x2,… ]\mathbb{Z}[x_1, x_2, \dots] freely generated by the bordism classes of the complex projective spaces CP2,CP4,…\mathbb{CP}^{2}, \mathbb{CP}^{4}, \dots, where the xix_i correspond to universal Pontryagin classes via the associated map. Thom achieved this by associating cobordism groups to the homotopy groups of Thom spaces for the universal oriented bundle over the Grassmannian, and computing these via a spectral sequence that converges to the cohomology of the Thom space, prefiguring the Atiyah-Hirzebruch spectral sequence. His approach combined geometric insights with algebraic tools, demonstrating that unoriented cobordism groups vanish in odd dimensions and are finite in even dimensions below a certain threshold. A pivotal tool in Thom's cobordism calculations was the transversality theorem, which he established in the same 1954 work and later refined. The theorem states that for a smooth map f:M→Nf: M \to N between manifolds and a submanifold S⊂NS \subset N, there exists a generic perturbation of ff such that ff is transverse to SS, meaning the preimage f−1(S)f^{-1}(S) is a of MM of dim⁡M−dim⁡N+dim⁡S\dim M - \dim N + \dim S whenever this value is nonnegative. This result relies on the density of transverse maps in the space of smooth maps, proved using properties of jet spaces and the openness of transversality conditions. The theorem ensured that representative manifolds in classes could be chosen with controlled intersections, simplifying computations and extending to multi-jet transversality for higher-order phenomena. By 1959, Thom had applied these ideas more broadly in lectures and subsequent expositions, solidifying its role as a cornerstone of generic singularity analysis. Thom's advancements in and have had lasting applications to the of smooth manifolds. His ring provides a complete set of invariants via Pontryagin numbers, allowing manifolds to be distinguished or shown equivalent based on these algebraic quantities rather than direct geometric comparison. For instance, it implies that simply connected oriented 4-manifolds are by their signature and , derived from . This framework shifted manifold from ad hoc methods to a systematic . Furthermore, Thom's constructions profoundly influenced . The Thom spaces of universal bundles give rise to Thom spectra, such as the oriented cobordism spectrum MSO\mathrm{MSO}, whose homotopy groups recover the cobordism ring Ω∗SO\Omega_*^{SO}. This identification links geometric problems of manifold bordism to algebraic computations in stable homotopy, enabling the use of to relate to cobordism invariants. Thom's geometric perspective on these spectra facilitated breakthroughs in computing stable stems and understanding periodicity phenomena, such as the Adams applications in the 1960s. His work thus bridged and , inspiring ongoing research in equivariant and generalized theories.

Singularity Theory

Singularity theory emerged in the 1960s through René Thom's pioneering work on the local analysis of singularities in smooth mappings between manifolds, focusing on those that are generic, meaning they occur in an open dense subset of the space of all such mappings. In the 1960s and 1970s, Thom's ideas were developed further by collaborators including John Mather and , leading to comprehensive classifications and stability results. Thom formulated this field as the study of stable singularities, where stability is defined relative to equivalence under diffeomorphisms: two mappings are equivalent if there exist diffeomorphisms of the source and target manifolds that conjugate one to the other, preserving the topological structure. This approach emphasized the qualitative behavior near critical points, where the differential of the map fails to be invertible, bridging and geometry by classifying singularities based on their persistence under small perturbations. A central achievement in singularity theory, building on Thom's work, was Vladimir I. Arnold's classification of simple singularities into the ADE series—A_k (folds and cusps for k ≥ 1), D_k (beaks and for k ≥ 4), and E_k (umbilics for k = 6,7,8)—derived from the topology of the discriminant variety, which parameterizes the bifurcation set of the mapping. These singularities are finite in number for low codimensions and represent the generic forms that arise in mappings from manifolds of dimensions up to 7, with the A_k series corresponding to corank-1 singularities like the fold f(x,y)=x2+yf(x,y) = x^2 + y and cusp f(x,y)=x3+xyf(x,y) = x^3 + xy. The relies on normal forms, where each type is distinguished by its versal unfolding and the topology of the associated Milnor , providing a finite list of types that dominate the local structure. Thom developed the concept of versal unfoldings to capture the essential deformations of a singularity, defining a versal unfolding of a germ f:(Rn,0)→(Rp,0)f: (\mathbb{R}^n, 0) \to (\mathbb{R}^p, 0) as a family F(x,λ)F(x, \lambda) with parameters λ∈Rq\lambda \in \mathbb{R}^q (where q is the ) such that every nearby unfolding is equivalent to a of F via a in the parameter space. This minimal parameter space unfolds the singularity transversally, revealing all possible nearby behaviors while ensuring ; for instance, the versal unfolding of the A_2 cusp is F(x,y;λ)=x3+xy+λF(x,y;\lambda) = x^3 + x y + \lambda, generating the swallowtail catastrophe in higher . Versality ensures that the unfolding is universal, encompassing the of all deformations without redundancy. In collaboration with and Bernard Malgrange, Thom advanced the understanding of singularity stability, culminating in the theorem that stable mappings form an open in the of smooth maps between manifolds when the target dimension is sufficiently low (specifically, for mappings into dimensions less than 7). This result, building on Thom's transversality ideas and Malgrange's preparation theorem for analytic functions, establishes that generic smooth maps exhibit only finitely many singularity types, which are stable under perturbation and dense in the Whitney C^\infty topology. Arnold extended this to detailed classifications, while Malgrange's work provided analytic foundations for the smooth case, confirming Thom's on the prevalence of stable configurations.

Catastrophe Theory

Catastrophe theory, developed by René Thom in the late , provides a mathematical framework for modeling abrupt changes in the behavior of dynamical systems where small, continuous variations in control parameters lead to sudden jumps or discontinuities in the system's state. This approach extends singularity theory to time-dependent processes, focusing on the topological structure of potential functions that govern equilibria. Thom identified key elementary catastrophes—such as , cusp, swallowtail, and —as universal models for these transitions, emphasizing their role in capturing generic behaviors in natural phenomena without requiring detailed mechanistic descriptions. The elementary catastrophes are classified according to their corank, which measures the dimension of the kernel of the Hessian matrix at the singularity (typically 1 or 2 for low-dimensional cases), and their codimension, which indicates the minimum number of control parameters needed to unfold the singularity stably (usually up to 4 for physical relevance). For instance, the cusp catastrophe, with corank 1 and codimension 2, exemplifies and , where the system can switch between two states depending on values, leading to path-dependent outcomes. Its standard form is given by the equilibrium equation x3+ax+b=0x^3 + a x + b = 0, where xx represents the and a,ba, b are control parameters; for certain regions in the (a,b)(a, b)-plane, the equation yields three real roots, enabling sudden jumps as parameters cross bifurcation lines. Thom applied to diverse fields, including , where fold and cusp singularities describe caustics formed by light rays concentrating at focal points; , modeling in structures under gradual loading that results in sudden collapse via swallowtail-like instabilities; and , particularly , where these models explain embryonic development processes such as tissue folding and organ formation through metastable transitions. These applications are elaborated in his seminal 1972 book Stabilité Structurelle et Morphogenèse, which integrates topological insights to interpret biological forms as stable attractors emerging from catastrophic bifurcations. The mathematical foundation of catastrophe theory rests on the recognition , which provides criteria for identifying when a local singularity in a system's potential function is equivalent to one of the elementary catastrophe forms, ensuring that observed behaviors in generic systems align with these topological models. This , building on Thom's and later rigorized by Mather, guarantees for unfoldings of finite , allowing reliable prediction of qualitative dynamics from singularity types without exhaustive parameter exploration.

Philosophical and Interdisciplinary Work

Epistemology and Philosophy of Science

In the 1970s, René Thom shifted his focus from pure mathematics to philosophy and the philosophy of science, driven by a period of personal and intellectual reevaluation following a decline in his mathematical output and a desire to apply topological concepts to natural phenomena such as biology. This transition is evident in his seminal work Structural Stability and Morphogenesis (1972), where he began exploring the qualitative dimensions of scientific inquiry beyond formal proofs. Thom critiqued structuralism and set theory as fundamentally inadequate for capturing the qualitative essence of natural forms, arguing that they reduce complex realities to discrete, abstract constructions that overlook the continuous and morphological aspects of the world; instead, he advocated a return to Aristotelian hylomorphism, viewing objects as composites of matter and form where form provides the unifying principle of stability and intelligibility. Thom's epistemological views emphasized the inherent limitations of mathematical modeling in physics, particularly its overreliance on quantitative precision and deterministic predictions. In his essay "Mathématiques et Prévision," he contended that scientific prediction is not a smooth, continuous process but fundamentally topological and discontinuous, involving singularities and qualitative jumps that —used here as a philosophical lens rather than a strict mathematical tool—best illuminates through its focus on and morphological changes. He argued that models in physics often fail to account for these discontinuities, treating the as a deterministic machine while ignoring the qualitative tendencies that govern real-world , such as the emergence of forms from underlying continua. Extending these ideas to and physics, Thom proposed that forms and their stability are ontologically prior to functions, challenging reductionist accounts that prioritize mechanistic explanations or genetic . Drawing on a reevaluation of 's biological writings, he identified proto-topological intuitions in concepts like boundaries and continuity, portraying as a qualitative where stable forms—modeled via attractors and bifurcations—precede and constrain functional adaptations, as seen in his interpretations of Aristotelian notions of homeomerous and anhomeomerous bodies. This perspective positioned as a precursor to modern , emphasizing the ceaseless creation and destruction of forms in nature over empirical quantification. Thom's philosophical commitments manifested in public stances against what he saw as threats to scientific , most notably his participation in a 1992 open letter to The Times protesting Jacques Derrida's proposed honorary doctorate from Cambridge University. As one of eighteen signatories, including prominent analytic philosophers, Thom endorsed the letter's condemnation of Derrida's deconstructive approach as an obscure assault on reason, truth, and scholarly standards, associating it with the undermining of objective inquiry in favor of relativistic literary or political agendas.

Semiophysics and Applications

In the 1980s, René Thom coined the term "semiophysics" to describe an interdisciplinary framework that synthesizes —the study of signs and meaning—with the formal rigor of physics, aiming to model qualitative aspects of human experience through topological and dynamical systems. This approach leverages to represent semantic bifurcations, where small changes in parameters lead to abrupt shifts in meaning, and archetypal forms, which Thom viewed as stable morphological structures underlying perception and communication. He elaborated these ideas in his 1989 book Sémiophysique, later translated as Semio Physics: A Sketch (1990), where semiophysics is presented as a "" for analyzing significant forms across domains. Thom applied semiophysics extensively to , developing topological models for meanings and structures that emphasize discontinuous transitions over linear processes. For instance, he modeled the "to give" as a cusp catastrophe, a three-parameter singularity where the roles (giver, gift, receiver) emerge through bifurcations in a stability landscape, capturing the qualitative valency of the action. In Sémiophysique, he outlined 16 archetypal forms derived from catastrophe germs, such as the or swallowtail, to describe plot dynamics in stories, integrating morphological archetypes with syntactic elements for a proto-semantic . These models prioritize the perceptual salience of linguistic forms, treating meaning as emergent from gestaltic configurations rather than arbitrary symbols. Extensions of semiophysics to art and music involved analyzing visual and auditory forms through stability landscapes and morphological archetypes, revealing underlying dynamical patterns in creative expression. Thom examined paintings by , interpreting elements like melting clocks or hyperbolic shapes as manifestations of catastrophe-induced discontinuities, influenced by Dalí's own engagement with Thom's theory in works such as The Swallow's Tail (1983). In music, he applied these concepts to motifs, such as the descending scale in Leonard Cohen's "Hallelujah," as a swallowtail catastrophe tracing emotional bifurcations across verses, where stability basins represent harmonic resolutions and shifts denote narrative tension. Through collaborations with cognitive scientists like Jean Petitot and Wolfgang Wildgen, Thom proposed that human fundamentally depends on discontinuous perceptual shifts, modeled as bifurcations in morphodynamic systems, rather than smooth, continuous information processing. This view posits as a series of qualitative restructurings via singularities, such as cusp models for in or visual contours, aligning with empirical findings in where thresholds produce abrupt Gestalt reorganizations. These ideas, rooted in Thom's topological semantics, influenced later work in cognitive morphodynamics, emphasizing the brain's reliance on attractor dynamics for meaning formation.

Legacy

Awards and Honors

René Thom received the in 1958 at the in , recognizing his pioneering contributions to , particularly the development of cobordism theory. In 1970, he was awarded the Brouwer Medal by the Royal Dutch Mathematical Society for his lifetime achievements in . He received an honorary degree from the in 1970. In 1974, Thom was awarded the Grand Prix Scientifique de la Ville de . The Society for Industrial and Applied Mathematics honored Thom with the Lecture in 1976, acknowledging his influential work on singularity theory. That same year, Thom was elected to membership in the Académie des Sciences in the mathematics section. He became an honorary member of the London Mathematical Society in 1990.

Influence and Criticisms

Thom's work in singularity theory profoundly influenced subsequent developments in , particularly through the efforts of and his school, who extended Thom's ideas on stable mappings and versal unfoldings to classify singularities in complex spaces. This foundational role is evident in the establishment of singularity theory as a distinct field, where Thom's topological approaches provided tools for analyzing geometric objects with singular points, inspiring applications in and deformation theory. In , his concepts have found utility in , notably for path planning algorithms that model configuration spaces using singularity resolution to avoid obstacles and ensure smooth trajectories in high-dimensional environments. In the biological sciences, , as articulated in Thom's Structural Stability and Morphogenesis, was adopted to model , describing how stable forms emerge from dynamic processes like and embryonic development through cusp catastrophes representing bifurcations in morphogenetic fields. Similarly, in physics, the theory offered qualitative frameworks for phase transitions, such as those between and states, by classifying abrupt changes via elementary catastrophes, though its adoption remained limited due to challenges in empirical validation and the need for precise fitting in experimental data. These interdisciplinary extensions highlighted Thom's emphasis on qualitative dynamics over quantitative , yet they often faced scrutiny for insufficient testable hypotheses. Thom's philosophical legacy revitalized qualitative morphology in cognitive science and semiotics, where his semiotic arrow and topological models of meaning influenced analyses of perceptual forms and linguistic structures as dynamic, form-based processes rather than linear symbol systems. This approach resonated in dynamical systems psychology, inspiring models of cognitive transitions and emotional states as morphological shifts, akin to catastrophes in mental organization, and extending to semiotics through applications in art and music interpretation. His semiophysics, blending topology with phenomenology, promoted a "physics of meaning" that prioritized intuitive forms in understanding complex phenomena. Criticisms of Thom's theories centered on accusations of overgeneralization, particularly in 's popular applications, such as Christopher Zeeman's models for biological and social behaviors, which extended the framework beyond its mathematical rigor and sparked a "catastrophe theory backlash" in the late 1970s and 1980s through critiques like those in Science labeling it as mathematically unsubstantiated . Debates also portrayed semiophysics as overly speculative, critiqued by figures like for lacking empirical grounding and formal precision in bridging and philosophy. On a personal note, Thom's daughter, Françoise Thom, has distinguished herself as a and Sovietologist, teaching contemporary Russian at and authoring works on post-Soviet politics that reflect an analytical depth echoing her father's interdisciplinary mindset. The centenary of Thom's birth in 2023 was marked by a three-day conference at the , underscoring his enduring interdisciplinary appeal across , , and the sciences through discussions on his morphological legacies.

References

  1. https://mathshistory.st-andrews.ac.uk/Biographies/Thom/
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  4. https://www.theguardian.com/news/2002/nov/14/guardianobituaries.highereducation1
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  7. https://www.math.stonybrook.edu/~dennis/publications/PDF/DS-pub-0116.pdf
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  10. https://uberty.org/wp-content/uploads/2015/12/Thom-Structural-Stability-and-Morphogenesis.compressed.pdf
  11. https://www.fermentmagazine.org/Stories/Thom/Predire.pdf
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  17. http://ontology.buffalo.edu/smith/derridaletter.htm
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  19. http://jeanpetitot.com/ArticlesPDF/Petitot_CognitiveMorphodynamics.pdf
  20. https://mathshistory.st-andrews.ac.uk/Honours/BrouwerMedal/
  21. https://warwick.ac.uk/services/gov/hongrads/list/
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  23. https://cths.fr/an/savant.php?id=112053
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  27. https://www.uni-bremen.de/fileadmin/user_upload/fachbereiche/fb10/fb10/Materialien/Wildgen/pdf/Rene_Thom_s_archetypal_morphologies_of_human_language_and_his_semiophysics_conference_September_2023.pdf
  28. https://www.encyclopedia.com/science/science-magazines/does-catastrophe-theory-represent-major-development-mathematics
  29. https://www.soviethistorylessons.com/francoise-thom-interview
  30. https://www.ihes.fr/en/centenary-thom/
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