Invited and Tutorial speakers

Prof. Mikhail Gromov (Abel Prize, IHES, Bures-sur-Yvette, France)
On the Structure of Entropy
Abstact: Mathematics is about "interesting structures". What make a structure interesting is an abundance of interesting problems; we study a structure by solving these problems. The worlds of science, as well as of mathematics itself, is abundant with gems (germs?) of simple beautiful ideas. When and how may such an idea direct you toward beautiful mathematics? I  present in this talk a 20th century mathematician's perspective on Boltzmann's idea of entropy.
Biography: Mikhail Leonidovich Gromov. a French–Russian mathematician.  He studied in  the  Leningrad University  where he was a student of Vladimir Rokhlin. He has been  working on the Riemannian Geometry,  Geometric PDE,  Symplectic Geometry, Geometric Theory of Groups  and also on mathematical formalizations of ideas coming from biology, psychology and linguistic. 

Prof. Daniel Bennequin (Denis Diderot University, Paris, France)
Topological forms of information
Abstract: This talk will present recent joint works with Pierre Baudot, where we propose a general definition of categories of informations, and study a natural cohomology for associated information quantities, and homotopical derived notions. Entropies of Shannon, Kullback and Von Neumann, appear as first fundamental classes for classical and quantum setting respectively. The decomposition of entropy in higher mutual information functions appears as an homotopical structure, and generates a new kind of topology. Possible applications to the study of large statistical data and dynamics of neuronal systems will be mentioned
Biography: Born the 3 January 1952. Graduated from Ecole Normale Supérieure, he has defended his PhD in 1982 with Alain Chenciner at Paris VII University (Doctorat d'Etat, Entrelacements et équations de Pfaff). He was Professor at Strasbourg University and was member of Bourbaki. He is currently Professor at Denis Diderot University and member of  l'Institut Mathématique de Jussieu. He has made many major contributions to contact geometry during 80's and was initiator of Contact Topology with Yakov Eliashberg. During the years 1990, he worked, with his students and colleagues in Strasbourg, on integrable systems and geometrical structures of Mathematical Physics. Since 2000 he is working in Neuroscience (mainly in LPPA directed by A.Berthoz, C-d-F, Paris); he made contributions to the study of geometrical invariance in human movements duration, dynamical structure of vestibular end sensors,  organization of vestibular information flow, eye movements preparation, and  gaze functions during locomotion.

Prof. Roger Balian (French Academy of Sciences Member, Scientific Consultant of CEA)
The entropy-based quantum metric
Abstract: The von Neumann entropy S(^D) generates in the space of quantum density matrices ^D the Riemannian metric ds² = - d²S(^D), which is physically founded and which characterizes  the amount of quantum information lost by mixing ^D and ^D + d^D. A rich geometric structure is thereby implemented in quantum mechanics. It includes a canonical mapping between the spaces of states and of observables, which involves the Legendre transform of S(^D). The Kubo scalar product is recovered within the space of observables. Applications are given to equilibrium and non-equilibrium quantum statistical mechanics. There the formalism is specialized to the relevant space of observables and to the associated reduced states issued from the maximum entropy criterion, which result from the exact states through an orthogonal projection. Von Neumann’s entropy specializes into a relevant entropy. Comparison is made with other metrics. The Riemannian properties of the metric ds² = - d²S(^D) are derived. The curvature arises from the non-Abelian nature of quantum mechanics; its general expression and its explicit form for q-bits are given.
Biography: Roger Balian has been working at the "Institut de Physique Théorique" of Saclay (CEA), which he has directed (1979-1987). He was also Professor of Statistical Physics at Ecole Polytechnique (1972-1998), and Director of the "Ecole d'Eté de Physique Théorique des Houches" (1972-1980). His research works have addressed various topics, often related to statistical  physics: superfluid Helium 3, signal theory, information/entropy, waves and complex trajectories, foundations of quantum mechanics, Casimir effect,  quantum liquids,  nuclear structure, gauge theories, distribution of galaxies...
Dr. Stefano Bordoni (Bologna University, Italy)
Duhem’s abstract thermodynamics.
Abstract: In the second half of the nineteenth century, two different traditions of research emerged from Clausius’ thermodynamics. Maxwell and Boltzmann pursued the integration of thermodynamics with the kinetic theory of gases, whereas others relied on a macroscopic and more abstract approach, which set aside specific mechanical models. Massieu, Gibbs, and Helmholtz exploited the structural analogy between mechanics and thermodynamics, the young Planck and J.J. Thomson aimed at filling the gap between thermodynamics and the theory of elasticity, and Oettingen developed a dual mathematical structure for heat and work. Starting from 1891, Pierre Duhem put forward the most original and systematic reinterpretation of abstract thermodynamics, and at the same time the boldest upgrade of Analytical mechanics. He developed and transformed the second tradition: his design of a generalized mechanics based on thermodynamics led to an astonishing mathematical unification between physics and chemistry. Purely mechanical phenomena and chemical reactions represented the opposite poles in Duhem’s Energetics.
Biography: He graduated in physics, and received a PhD in History of science and then a PhD in Philosophy. He has recently gained a qualification as associate professor of Logic, philosophy and history of science. He has published papers and books on the history of science, and in particular history of physics. He has given seminars and lectures in some Italian universities and at the Max-Planck-Institut für Wissenschaftsgeschichte in Berlin. He has lectured in History of physics, History of science, and Mathematics.


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