Subcategories

  • Geo-Sci-Info

    Organisers:

    Michel Ledoux (Université de Toulouse, France) Mokshay Madiman (University of Delaware, USA)

    Mini-course:

    Mokshay Madiman (University of Delaware, USA)

    Speakers:

    Thomas Courtade (University of California, Berkeley, USA) Nathael Gozlan (Université Paris-Est, France) Oliver Johnson (University of Bristol, UK) Jan Maas (IST, Austria) Karl-Theodor Sturm (Institut für Angewandte Mathematik, Germany)

    Abstract:
    This session will explore the geometry of particular instances as well as general classes of metric measure spaces as captured using the notion of entropy. In the general context, the ground-breaking work of Lott-Villani-Sturm has provided a notion of Ricci curvature lower bounds that make sense for a large class of non-discrete metric measure spaces. Particular contexts such as discrete and Euclidean spaces benefit from separate development, either because the Lott-Villani-Sturm notion does not apply (discrete) or because much more can be said than in the general setting (Euclidean). The session will explore the swirl of ideas that have seen rapid development in these areas in recent years, including optimal transport, concentration of measure, and isoperimetry for general settings, and entropy power inequalities, Stein’s method, and connections/parallels with convex geometry in particular settings.


  • Pierre Baudot, Daniel Bennequin, Michel Boyom, Herbert Gangl, Matilde Marcolli, John Terrila

    Geo-Sci-Info

    Organisers: Pierre Baudot, Daniel Bennequin, Michel Boyom, Herbert Gangl, Matilde Marcolli, John Terilla

    List of speakers (TBA):

    Daniel Bennequin
    Université Paris Diderot-Paris 7, UFR de Mathématiques, Equipe Géométrie et Dynamique, Paris, France. José Ignacio Burgos Gil
    ICMAT (CSIC), Madrid, Spain Michel Boyom
    Université du Languedoc-Montpellier II, France. Philippe Elbaz-Vincent
    Institut Fourier, Grenoble, France. Tom Leinster
    School of Mathematics, University of Edinburgh, Edinburgh. Matilde Marcolli
    Mathematics Department, Caltech, Pasadena, USA. John Terilla, Queens College, USA.

    Abstract:
    A classical result of Cencov [1] in information geometry established that the Fisher-Rao metric is the unique metric (up to a constant) on the space of finite probability densities that is invariant under the action of the diffeomorphism group. This result was extended to infinite sample spaces via n-tensorisation by Ay, Jost, Le and Schwachhofer [2]. Entropy first appeared in the computation of the degree one homology of the discrete group SL2 over C with coefficients in the adjoint action by choosing a pertinent definition of the derivative of the Bloch-Wigner dilogarithm [3]. It could be shown that the functional equation with 5-terms of the dilogarithm implies the functional equation of information with 4-terms. Then it was discovered that a finite truncated version of the logarithm appearing in cyclotomic studies also satisfied the functional equation of entropy, suggesting a higher degree generalisation of information, analog to polylogarithm [4]. This information generalisation was achieved by algebraic means that yet holds over finite fields [5], and further developed into framework where information functions appears as derivation [6]. a more geometric version in terms of algebraic cycles was found [7], introducing the notion of additive dilogarithm with respect to the notion of multiplicative structures [8]. On another side, after that entropy appeared in tropical and idempotent semi-ring analysis in the study of the extension of Witt semiring to the characteristic 1 limit [9], thermodynamic semiring and entropy operad could be constructed as deformation of the tropical semiring [10]. Introducing Rota-Baxter algebras, It allowed to derive a renormalisation procedure. Further completing the entropic landscape in number theory, entropy was encountered as height-degree function in the context of Arakelov geometry while following the program of relating the arithmetic geometry of toric varieties and convex analysis [11]. Defining the category of finite probability and using Fadeev axiomatization, it could be shown that the only family of function which has functorial property is Shannon information loss [12]. Introducing a deformation theoretic framework, and chain complex of random variables, a homotopy probability theory could be constructed for which the cumulants coincide with the morphisms of the homotopy algebras [13]. Using the geometrical and combinatorial structure of probability and random variable seen as partitions, the basement of the cohomology, topos and of a (quasi)-operad of information could be constructed in a probabilistic setting, allowing to retrieve entropy uniquely as the first cohomology group. Mutual-informations arise from the consideration of the trivial action, and appear as differential operator, while their "non-positive" extrema are giving rise to homotopy links [14]. In a series of papers and courses [15], M.Gromov reviews results on the subject and introduces to some geometrical formulation of information inequalities, as well as a complete program in current development rooted on Morse cohomology and homotopy.

    It is appealing and a very good sign that an important part of those developments has been achieved in parallel and independently with very different approaches and methods; the session will be one of the first occasion to gather the actors of the domain, and hence to promote the convergence and discuss open problems. Moreover, following these important extensions of the information and probability framework, It appears natural to ask now for a proper logical foundation of information theory, extending the Boolean world of information digit. Given those developments, motivic integration, topos and homotopy type, that are inherently geometrical logics, provide some candidate for such a program that we propose to discuss.

    [1] Cencov, N.N. Statistical Decision Rules and Optimal Inference. Translations of Mathematical Monographs. 1982.
    [2] Ay, N. and Jost, J. and Lê, H.V. and Schwachhöfer, L. Information geometry and sufficient statistics. Probability Theory and Related Fields 2015 PDF
    [3] Cathelineau, J. Sur l’homologie de sl2 a coefficients dans l’action adjointe, Math. Scand., 63, 51-86, 1988. PDF
    [4] Kontsevitch, M. The 1+1/2 logarithm. Unpublished note, Reproduced in Elbaz-Vincent & Gangl, 2002, 1995 PDF
    [5] Elbaz-Vincent, P., Gangl, H. On poly(ana)logs I., Compositio Mathematica, 130(2), 161-214. 2002. PDF
    [6] Elbaz-Vincent, P., Gangl, H., Finite polylogarithms, their multiple analogues and the Shannon entropy. Vol. 9389 Lecture Notes in Computer Science. 277-285, Archiv.
    [7] Bloch S.; Esnault, H. An additive version of higher Chow groups, Annales Scientifiques de l’École Normale Supérieure. Volume 36, Issue 3, May–June 2003, Pages 463–477. PDF
    [8] Bloch S.; Esnault, H. The Additive Dilogarithm, Documenta Mathematica Extra Volume : in Kazuya Kato’s Fiftieth Birthday., 131-155. 2003. PDF
    [9] Connes, A., Consani, C., Characteristic 1, entropy and the absolute point. preprint arXiv:0911.3537v1. 2009.
    [10] Marcolli, M. & Thorngren, R. Thermodynamic Semirings, arXiv 10.4171 / JNCG/159, Vol. abs/1108.2874, 2011.
    Marcolli, M. & Tedeschi, R. Entropy algebras and Birkhoff factorization, arXiv, Vol. abs/1108.2874, 2014.
    [11] Burgos Gil J.I., Philippon P., Sombra M., Arithmetic geometry of toric varieties. Metrics, measures and heights, Astérisque 360. 2014 . PDF
    [12] Baez, J.; Fritz, T. & Leinster, T. A Characterization of Entropy in Terms of Information Loss Entropy, 13, 1945-1957, 2011. PDF
    Baez J. C.; Fritz, T. A Bayesian characterization of relative entropy. Theory and Applications of Categories, Vol. 29, No. 16, p. 422-456. 2014. PDF
    [13] Drummond-Cole, G.-C., Park., J.-S., Terrila, J., Homotopy probability theory I. J. Homotopy Relat. Struct. November 2013. PDF
    Drummond-Cole, G.-C., Park., J.-S., Terrila, J., Homotopy probability theory II. J. Homotopy Relat. Struct. April 2014. PDF
    Park., J.-S., Homotopy theory of probability spaces I: classical independence and homotopy Lie algebras. Archiv . 2015
    Tomasic, I., Independence, measure and pseudofinite fields. Selecta Mathematica, 12 271-306. Archiv. 2006.
    [14] Baudot P., Bennequin D. The homological nature of entropy. Entropy, 17, 1-66; 2015. PDF
    [15] Gromov, M. In a Search for a Structure, Part 1: On Entropy, unpublished manuscript, 2013. PDF
    Gromov, M. Symmetry, probability, entropy. Entropy 2015. PDF
    Gromov, M. Morse Spectra, Homology Measures, Spaces of Cycles and Parametric Packing Problems, april 2015. PDF
    [16]McMullen, C.T., Entropy and the clique polynomial, 2013. PDF

  • Geo-Sci-Info

    Organisers: Frédéric Barbaresco, Joël Bensoam

    List of speakers (TBA):

    Frédéric Barbaresco
    Thales Air Systems. Limours, France. Joël Bensoam
    Ircam, centre G. Pompidou, CNRS UMR 9912, Paris, France. François Gay-Balmaz
    LMD - Ecole Normale Supérieure de Paris, Paris, France. Frédéric Hélein
    Institut de Mathématiques de Jussieu, Paris, France. Bernhard Maschke
    Claude Bernard University Lyon 1, LAGEP, Villeurbanne, France.

    Abstract:
    This session will explore a unifying framework for Classical/Stochastic Geometric Mechanics and Statistical Physics, based on Symplectic and Multi-Symplectic Geometry and Lie Group Theories. Jean- Marie Souriau has introduced a “Lie Group Thermodynamics” in Statistical Physics and “Continuous Medium Thermodynamics” that could beneficiate of most recent works from Geometric Mechanics and its most recent stochastic developments. In particular, the conservation laws (or more generally the circulation invariants: Noether’s currents) that derive from the underlying symmetries of Lie groups allow reducing the Partial Differential Equations (PDEs) problems. Other approaches will be considered as Euler-Poincaré reduction theorem for stochastic processes taking values in a Lie group, which is a generalization of the Lagrangian version of reduction and its associated variational principles. The objective is to embrace common principles for all the changes of state of the body , both the change of place and the change of physical qualities extended to the case of incomplete knowledge and random uncertainties.

  • Geo-Sci-Info

    Organiser: Dominique Spehner
    List of Speakers :

    Madalin Guta, University of Nottingham, UK. Dominique Spehner, Institut Fourier, Saint Martin d'Hères, France. Karol Zyczkowski, Jagiellonian University, Cracow, Poland.

    Abstract:
    One of the central questions in quantum information theory is to characterize quantum correlations in composite quantum systems. The latter play the role of resources in quantum information processing tasks and are believed to be at the origin of the higher efficiencies with respect to classical information processors.
    A promising approach to the characterization and quantification of these correlations has emerged in recent years. It is based on Riemannian geometries on the convex cone of quantum states. In particular, the amount of quantum correlations in a given state has been quantified by the distance between this state and a distinguished subset of states, formed e.g. by all disentangled states or by all classical states. The properties and examples of Riemannian metrics such that good measures of entanglement and quantum discord are obtained, have been studied. The explicit determination of these geometric measures in concrete physical systems has attracted a lot of attention in the physics community. The aim of the conference is to try to gather together the various attempts proposed so far within a single general geometric framework, and to discuss its link with the more traditional entropic approach.

  • Geo-Sci-Info

    Organisers: Carlo Rovelli

    Speakers :

    Livine Etera, Laboratoire de Physique, ENS Lyon, Lyon, France. Antonino Marcianò, Fudan University, China. Carlo Rovelli, Centre de Physique Theorique de Luminy, Marseille, France.

    Abstract:
    The objective of the session is to make the point on the way geometry and quantum correlations are related in quantum gravity. The focus would be on the recent developments in the possibility of describing quantum states of the geometry with long distance correlations.


  • Stéphanie Allasonnière, Xavier Pennec

    Geo-Sci-Info

    Organisers:

    Stéphanie Allasonnière (School of medicine, Paris Descartes university, France) Xavier Pennec (INRIA sophia, France)

    Mini-course:

    Xavier Pennec (INRIA sophia, France) Alain Trouvé (ENS Cachan, France)

    Speakers:

    Marc Arnaudon (IMB, France) Aasa Feragen (DIKU, Denmark) Stanley Durrleman (ARAMIS lab, France) Ian Dryden (University of Nottingham, UK) Alice Le Brigant (IMB, Université de Bordeaux, France)

    Abstract:
    This session presents recent progresses in geometric statistics. In many applications domains such as computational anatomy, computer vision, structural biology, computational phylogenetics, etc one models data as elements of a manifold which is quotiented by a proper and isometric Lie group action (a shape space). For instance, Kendall shape spaces encode the invariants of the configuration of a fixed number of points under the action of Euclidean or similarity transformations. One can also construct (infinite dimensional) spaces of curves or surfaces by quotienting out the parametrization. Shape spaces are stratified spaces whose manifold part is Riemannian. Looking for summary, explanatory or discriminative statistics on such measurements is first complicated by the presence of the curvature. The theory of statistics on Riemannian manifolds now begins to be powerful enough to provide simple notions (mean values, moments) and some (limited) central limit theorems, although the role of the curvature is not fully elucidated. However, subspace estimation methods going beyond the mean value have only recently emerged. A second complication arise in quotient spaces when data are scattered over several regular strata or close to singular points of the stratification. Other differential geometric structures such as affine connection spaces or fiber spaces raise new questions for the generalization of statistical notions and estimation methods.

  • Geo-Sci-Info

    Organisers: Nihat Ay, František Matúš

    Speakers :

    Nihat Ay
    Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany. Tobias Fritz
    Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada. Luigi Malago
    Romanian Institute of Science and Technology - RIST, Romania. František Matúš
    Institute of Information Theory and Automation, Academy of Sciences,Prague, Czech Republic. Guido Montúfar
    Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany. Johannes Rauh
    Leibniz Universität Hannover, Institut für Algebraische Geometrie, Hannover, Germany. Milan Studený
    Institute of Information Theory and Automation, Academy of Sciences, Prague, Czech Republic.

    Abstract: This session will review the role in network analysis within the fields of artificial intelligence and machine learning, of geometric objects defined in terms of information equalities as well as information inequalities.



TGSI2017

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