From Frank Ramsey to René Thom: beyond optimisation

• Ivar Ekeland (CEREMADE, Université Paris-Dauphine, Paris-Sciences-Lettres)

Room: Centre de conférences Marilyn et James Simons I will introduce a class of optimisation problems in the calculus of variations arising from economic theory, and I will show why the optimal solution makes no sense. A new concept of solution must be introduced, leading to an implicit ODE in the sense of Thom.

Cobordism and Spaces of Manifolds

• Oscar Randal-Williams (Univ. de Cambridge)

Room: Centre de conférences Marilyn et James Simons Thom's most celebrated work developed the notion of cobordism of manifolds, and translated the problem of classifying manifolds up to cobordism into a problem in homotopy theory: this translation is mediated by his Transversality Theorem. I will describe a different point of view on this topic, arising from the modern theory of cobordism categories.

Doors opened by Thom in Morse theory and towards the h-principle

• Emmanuel Giroux (CNRS et ENS)

Room: Centre de conférences Marilyn et James Simons We will review the content and the influence of two papers by R. Thom. The first is his 1949 note "sur une partition en cellules associée à une fonction sur une variété" where he exhibits the cell decomposition provided by any gradient of a Morse function ; we will also discuss the two unpublished manuscripts dated 1957 where Thom applies Morse Theory to analyze the topology of complex affine algebraic/analytic manifolds. The second article is the 1959 paper "remarques sur les problèmes comportant des inéquations différentielles globales" where Thom scatters seeds for the general study of open differential relations, a theme that Gromov will later embrace in the vast theory of the h-principle.

Broken/open book decompositions and generic dynamics of Reeb vector fields

• Ana Rechtman (Univ. Grenoble-Alpes)

Room: Centre de conférences Marilyn et James Simons Every non degenerate Reeb vector field on a closed 3-manifold is carried by a broken book decomposition. A broken book provides a system of transverses surfaces to the flow that allow to code the dynamics, allowing to prove several generic properties of Reeb vector fields. After explaining these results, I will present recent work on the existence of homoclinic orbits for Reeb vector fields. This is work in collaboration with V. Colin, P. Dehornoy and U. Hryniewicz.

The geometric ideas of René Thom related to manifolds with singularities

• Dennis Sullivan (City Univ. of New York, Graduate Center)

Room: Centre de conférences Marilyn et James Simons This talk is about two papers of Rene Thom. Firstly, the famous one in the fifties in “Commentarii Mathematici Helvetici “ celebrated for its methods of transversality and cobordism, but where the originating problem or question related to the title is much less well known . Secondly, the one in the sixties “BAMS “ where the mysterious term “manifolds with singularities “ was completely revealed in terms of manifold strata with locally constant transversal structure. This , with examples and ideas revealing the appropriate context is rougher than smooth and as least as good as topological.

The World of Shadows: from Plato to Thom

• Mikhail Gromov (IHES)

Room: Centre de conférences Marilyn et James Simons

Comment René Thom a changé la biologie moléculaire

• Antoine Danchin (Institut Pasteur)

Room: Centre de conférences Marilyn et James Simons

The gap between theory and experience: evolution of a problem from morphogenesis to semiophysics

• Sara Franceschelli (ENS Lyon)

Room: Centre de conférences Marilyn et James Simons Catastrophe theory is introduced by René Thom as an “art of models” for the study of morphogenesis. In this talk I will start from the observation that in Thom’s founding writings on morphogenesis the relation of mathematical models with experience is thought through the mediation of images of Waddington’s epigenetic landscape. I will then track the survival of images or properties of the epigenetic landscape in the evolution of Thom’s thinking from catastrophe theory to semiophysics. In so doing I will highlight the use and role of analogies by Thom, showing both aspects of continuity and of innovation with respect the morphological tradition he inserts in. I will finally argue that Thom’s use of analogy challenges the main contemporary philosophical analysis of analogy and analogical reasoning in science. This unnoticed practice represents, in my opinion, one of the aspects of the actuality of René Thom’s thinking.

René Thom and semantic information

• Daniel Bennequin (IMJ-PRG)

Room: Centre de conférences Marilyn et James Simons René Thom was not satisfied by the mathematical formulation of the notion of information, mainly restricted to a statistical theory without reference to meaning (note that Shannon in 1948 also expressed his unsatisfaction), and he suggested several leads to remediate (cf. Stabilité Structurelle et Morphogénèse, 1972, 1977). Part of the work of Thom in the eighties, can be considered as a development of these ideas about information (or “more exactly signification”), based on forms appearing in topological dynamics and singularity theory, for applications to Biology, Linguistic and Philosophy (cf. Esquisse d’une Sémiophysique, 1988). This work of Thom aimed at the same time, to understand what is intelligibility and invention in human individuals and societies. In fact, part of the exigences of Thom are realized by variants of Shannon’s theory, going back to Carnap and Bar-Hillel, 1952, but now without any use of probability and statistics. In this approach, semantic information relies on semantic structures (present in formal languages), through the consideration of their homotopical invariants, that can be themselves topological spaces. As evoked by René Thom himself, the vanishing homology of singular functions is a prototype of “information theory”; moreover, it gives an indication for the origin of semantic structures in general. Recent researches come from the necessity to understand what is a “semantic functioning” in artificial neural networks, and what are its obstructions (Belfiore and Bennequin, Topos and stacks of deep neural networks, 2021). An example will be given in relation with psychological experiments about simple concepts, their learning and use (Belfiore and Bennequin, Spaces of semantic information, 2023).

René Thom’s Catastrophe Theory and the Quantum Theory of Atoms in Molecules

• Cherif Matta (Mount Saint Vincent Univ.)

Room: Centre de conférences Marilyn et James Simons John Dalton’s law of multiple proportions of 1803 marks the beginnings of the modern atomic theory of matter, while Avogadro introduced in 1811 the idea of the molecule as a composite of “atoms-in-molecules” that behaves collectively as the fundamental unit of a given chemical compound. Half a century later, by mid (and second half) of the nineteenth century, chemists like Kekulé, Couper, Loschmidt, and others began what has become the chemical tradition of “drawing” chemical diagrams by connecting atomic symbols by lines introducing the first “chemical graphs” to be used as predictive tool of chemical properties and reactivity. The topology of a given collection of interacting atoms, a chemical compound, is all the difference between, say, acetic acid (the acid constituent of vinegar), oxiranol (an intermediate in interstellar chemistry), and the toxic refrigerant methyl formate! All three are composed of 2 carbon atoms, 4 hydrogens, and two oxygen atoms, but what a difference topology can make! Later, early in the 20th century, Gilbert Newton Lewis introduced the concept of the shared electron pair as being the physical mechanism of this “chemical bonding”. However, and up to the sixties of the last century, never has anyone ever seen a line connecting any atoms in space. That line connecting atoms in real three-dimensional space, the network of bonds connecting atomic symbols, has remained a mere summary of experimental experience until Richard F. W. Bader came along with a theory now termed the Quantum Theory of Atoms in Molecules (or QTAIM for short). QTAIM starts from the electron density of molecules, complexes, and crystals, and analyzes the three-dimensional topography of this scalar field. In doing that, Bader came up with a theory of chemical structure and its structural stability [1] based, in some of its core concepts, on René Thom’s Catastrophe Theory [2]. Bader discovered what is known as the “bond path” which is a line of maximal electron density connecting the nuclei of atoms that are known to be chemically bonded. In the grand majority of cases, the bond paths topology maps perfectly the bonded structure deduced from experimental chemistry – but with some exceptions and those exceptions can be among the most interesting cases. The bond paths network provides a rigorous and unambiguous definition of a “chemical structure” and the catastrophe points where there is a change in the topology (and hence change in chemical structure). (The word “structure” is used carefully and not to be confused with “geometry” as it is often done in the literature). These ideas let to the development of an entire quantum mechanics of open electronic systems, that is of atoms-in-molecules. Since the electron density is an experimental observable and beginning in the eighties with the advent of modern detectors, faster computers, low-temperature crystallographic experiments, and bright X-ray sources – the environment was set to go beyond the spherical approximation in refining X-ray diffraction data to the aspherical (multipolar) refinement. That has resulted in that, now, bond paths are routinely observed experimentally – what has remained an abstract summary of experiment up to these developments. Experiment and theory have been brought together by QTAIM through the definition of chemical structure and structural stability in the precise language of René Thom’s Catastrophe Theory. [1] Bader, R. F. W. Atoms in Molecules: A Quantum Theory. Oxford, U.K.: Oxford University Press; 1990. [2] Thom, R. Structural Stability and Morphogenesis: An Outline of a General Theory of Models (English Translation). Massachusetts: Adison-Wesley Publishing Company; 1972.

Trois remarques sur les modèles morphologiques de René Thom

• Jean Petitot (EHESS)

Room: Centre de conférences Marilyn et James Simons Nous proposerons trois remarques sur le statut des modèles morphologiques de René Thom. 1. Une remarque d'histoire des sciences concernant leurs relations avec les modèles de réaction-diffusion. 2. Une remarque philosophique sur leur lien avec la conceptualité aristotélicienne de l'hylémorphisme. 3. Une remarque sur l'implémentation neuronale de certaines structures géométriques ayant intéressé René Thom en perception visuelle (l'exemple du cut-locus).

René Thom's archetypal morphologies of human language and his semiophysics applied to visual art and music

• Wolfgang Wildgen (Univ. de Brême)

Room: Centre de conférences Marilyn et James Simons René Thom made his major contributions to linguistics and semiotics, first, in his book “Stabilité Structurelle et Morphogenèse”, in the chapter “L’homo loquens” (The Speaking Man) in 1972, and second, in his book “Esquisse d’une sémiophysique”, 1988, chapter 2, "Le Langage" (Language). In the first, he presented a list of 16 "morphologies archétypes" (Archetypal Morphologies); in the second, he proposed to bridge the gap between the sciences of culture and communication and the natural sciences. The present contribution describes how Thom's linguistic conjectures and his program of "sémiophysique" have been further developed and modified in ongoing research until today. The last section exemplifies the application of his topological and dynamic view to visual art and music. Although a final evaluation of Thom's journey into linguistics and semiotics cannot be given, its impact on contemporary and future thought in linguistics and semiotics can be estimated.

From Thom’s gradient cell decomposition to the curvature of Milnor fibers and the log-canonical threshold

• Bernard Teissier (IMJ-PRG)

Room: Centre de conférences Marilyn et James Simons In the early 1970’s Thom tried to use a gradient cell decomposition for the Milnor fiber of an isolated complex hypersurface singularity to prove the finiteness of the monodromy. It did not work but he noted the geometric significance of the polar curve associated to the singularity and a general hyperplane direction. I will describe some of the several uses of the polar curve to obtain algebraic, topological and Lipschitz-geometric information about isolated hypersurface singularities.

Variants of Lojasiewicz's gradient inequality

• Krzysztof Kurdyka (Univ. Savoie Mont Blanc)

Room: Centre de conférences Marilyn et James Simons The celebrated Lojasiewicz gradient inequality (L) has an important consequence; the length of a gradient trajectory, between two levels, is uniformly bounded. So, a trajectory has a limit when approaching a critical level. It has inspired Ren ́e Thom to formulate the conjecture that the trajectory has a tangent at the limit. I will describe various versions and extensions of inequality (L). I will state a variant of this inequality, which is valid for a map with values in a finite-dimensional vector space. It yields the boundedness of the volume of a submanifold transverse to the kernels of differentials of the map. This is an analogue of the boundedness of the length of gradient trajectories.

Déploiement, stratification, monodromy group

• Norbert A’Campo (Univ. De Bâle)

Room: Centre de conférences Marilyn et James Simons A main question addressed in this talk is: How to compute the geometric monodromy group of a complex hypersurface singularity? A seminal example is given by the Pham spine in the Milnor fibre of a Pham singularity: From the spine one computes the integral local homology monodromy. We show how this spine computes by a tête-à-tête construction the symplectic local monodromy diffeomorphism as a generalized Dehn twist with core the spine, [AFPP]. In case of curve singularities on normal surfaces one gets a characterisation of all possible local monodromy diffeomorphisms as a mixed tête-à-tête diffeomorphism by the work of Portilla and Sigurdson, [PSi]. By a real morsification one obtains a generating set of monodromy mapping classes for the monodromy group of the local versal unfolding. Recently Portilla and Salter have described this group as a framed mapping class group, [PSa]. In the case of plane curve singularities of type An a real analytic stratification of the space of smooth fibres will be presented. The top dimensional strata are cells and the monodromy is given by wall crossing data. We propose as conjecture a généralisation to all plane curve singularities based on the work of Portilla and Salter.

Separating actions from angles

• Alain Chenciner (IMCCE/Observatoire de Paris)

Room: Centre de conférences Marilyn et James Simons One of the simplest problems of normalization concerns analytic local diffeomorphisms of the plane in the neighborhood of a weakly attracting elliptic fixed point at which the derivative is a non-periodic rotation. Defining "geometric normalization" as an analytic local change of coordinates that transforms the local diffeomorphism into one leaving invariant the foliation by circles centered at the fixed point, I shall describe works with David Sauzin, Shanzhong Sun, and Qiaoling Wei which address the existence of normalizations and geometric normalizations.