Invited Speaker



Charles Michel Marle (Professeur honoraire à l'Université Pierre et Marie Curie, Institut Mathématique de Jussieu, Correspondant de l’Académie des Sciences, Paris, France)
Biography:
CharlesMichel Marle was born in 1934; He studied at Ecole Polytechnique (19531955), Ecole Nationale Supérieure des Mines de Paris (19571958) and Ecole Nationale Supérieure du Pétrole et des Moteurs (19571958).
He obtained a doctor's degree in Mathematics at the University of Paris in 1968. From 1959 to 1969 he worked as a research engineer at the Institut Français du Pétrole. He joined the Université de Besançon as Associate Professor in 1969, and the Université Pierre et Marie Curie, first as Associate Professor (1975) and then as full Professor (1981). His resarch works were first about fluid flows through porous media, then about Differential Geometry, Hamiltonian systems and applications in Mechanics and Mathematical Physics.

Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Hamiltonian systems
Abstract: I will present some tools in Symplectic and Poisson Geometry in view of their applications in Geometric Mechanics and Mathematical Physics. Lie group and Lie algebra actions on symplectic and Poisson manifolds, momentum maps and their equivariance properties, first integrals associated to symmetries of Hamiltonian systems will be discussed. Reduction methods taking advantage of symmetries will be discussed.
Bibliography :
 Abraham, R., Marsden, J.E.: Foundations of Mechanics, 2nd edn. AddisonWesley, Reading (1978).
 Arnold, V.I.: Mathematical methods of Classical Mechanics, 2nd edn. Springer, New York (1978).
 Ganghoffer, J.F., Maldenov, E. (editors): Similarity and Symmetry Methods; Applications in Elasticity and mechanics of Materials. Lecture Notes in Applied and Computational Mechanics 73, Springer, Heidelberg (2014).
 Guillemin, V., Sternberg, S.: Symplectic Techniques in Physics. Cambridge University Press, Cambridge (1984).
 LaurentGengoux, C., Pichereau, A., Vanhaecke, P.: Poisson structures. Springer, Berlin (2013).
 Libermann, P., Marle, C.M.: Symplectic Geometry and Analytical Mechanics. Symplectic Geometry and Analytical Mechanics. D. Reidel Publishing Company, Dordrecht (1987)




Keynote Speakers



Marc Arnaudon (Université de Bordeaux, France)
Biography:
Marc Arnaudon was born in France in 1965. He graduated from Ecole Normale Supérieure de Paris, France, in 1991. He received the PhD degree in mathematics and the Habilitation à diriger des Recherches degree from Strasbourg University, France, in January 1994 and January 1998 respectively. After postdoctoral research and teaching at Strasbourg, he began in September 1999 a full professor position in the Department of Mathematics at Poitiers University, France, where he was the head of the Probability Research Group. In January 2013 he left Poitiers and joined the Department of Mathematics of Bordeaux University, France, where he is a full professor in mathematics.
Prof. Arnaudon is an expert in stochastic differential geometry and stochastic calculus in manifolds, he has published over 50 articles in mathematical and physical journals.

Stochastic EulerPoincaré reduction
Abstract: We will prove a EulerPoincaré 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. We will also show examples of its application to the rigid body and to the group of diffeomorphisms, which includes the NavierStokes equation on a bounded domain and the CamassaHolm equation.
References:
 M. Arnaudon, A.B. Cruzeiro and X. Chen, "Stochastic EulerPoincaré Reduction", Journal of Mathematical Physics, to appear
 V. I. Arnold and B. Khesin, "Topological methods in hydrodynamics", Applied Math. Series 125, Springer (1998).
 J. M. Bismut, "Mécanique aléatoire", Lecture Notes in Mathematics, 866, Springer (1981).
 D.G. Ebin and J.E. Marsden, "Groups of diffeomorphisms and the motion of an incompressible fluid", Ann of Math. 92 (1970), 102163.
 J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry: a basic exposition of classical mechanical systems", Springer, Texts in Applied Math. (2003).


Biography:
 BA in Mathematics, University of Timisoara, Romania, 1973
 MA in Applied Mathematics, University of Timisoara, Romania, 1974
 Ph.D. in Mathematics, University of California, Berkeley, 1980
 T.H. Hildebrandt Research Assistant Professor, University of Michigan, Ann Arbor, USA 19801983
 Associate Professor of Mathematics, University of Arizona, Tuscon, USA 19831988
 Professor of Mathematics, University of California, Santa Cruz, USA, 19882001
 Chaired Professor of Mathematics, Ecole Polytechnique Federale de Lausanne, Switzerland, 1998  present
 Professor of Mathematics, Skolkovo Institute of Science and Technonology, Moscow, Russia, 2014  present

Symmetry methods in geometric mechanics
Abstract: The goal of these lectures is to show the influence of symmetry in various aspects of theoretical mechanics. Canonical actions of Lie groups on Poisson manifolds often give rise to conservation laws, encoded in modern language by the concept of momentum maps. Reduction methods lead to a deeper understanding of the dynamics of mechanical systems. Basic results in singular Hamiltonian reduction will be presented. The Lagrangian version of reduction and its associated variational principles will also be discussed. The understanding of symmetric bifurcation phenomena in for Hamiltonian systems are based on these reduction techniques. Time permitting, discrete versions of these geometric methods will also be discussed in the context of examples from elasticity.
References
 Demoures, F., GayBalmaz, F., Ratiu, T.S.: Multisymplectic variational integrators and space/time symplecticity, Communications in Analysis and Applications (2015), to appear
 GayBalmaz, F., Ratiu, T.S.: The geometric structure of complex fluids, Advances in Applied Mathematics, 42 (2009), 176275
 GayBalmaz, F., Marsden, J.E., Ratiu, T.S.: Reduced variational formulations in free boundary continuum mechanics, Journ. Nonlinear Sci., 22 (2012), 463497
 Libermann, P., Marle, C.M.: Symplectic Geometry and Analytical Mechanics. Symplectic Geometry and Analytical Mechanics. D. Reidel Publishing Company, Dordrecht (1987)
 Marsden, J.E, Ratiu, T.S.: Introduction to Mechanics and Symmetry, Texts in Applied Mathematics 17, second edition, Springer Verlag, (1998)
 Marsden, J.E, Misiolek, G., Ortega, J.P., Perlmutter, M., Ratiu, T.S.: Hamiltonian Reduction by Stages, Springer Lecture Notes in Mathematics, 1913, SpringerVerlag, New York (2007)
 Marsden J.E, West M.: Discrete mechanics and variational integrators, Acta Numer., 10 (2001), 357–514.
 Ortega, J.P., Ratiu, T.S.: Momentum Maps and Hamiltonian Reduction, Progress in Mathematics 222, Birkh\"auser, Boston (2004)



Mathilde Marcolli (Caltech, USA)
Biography:
 Laurea in Physics, University of Milano, 1993
 Master of Science, Mathematics, University of Chicago, 1994
 PhD, Mathematics, University of Chicago, 1997
 Moore Instructor, Massachusetts Institute of Technology, 19972000
 Associate Professor (C3), Max Planck Institute for Mathematics, 20002008
 Professor, California Institute of Technology, 2008present
 Distinguished Visiting Research Chair, Perimeter Institute for Theoretical Physics, 2013present

From Geometry and Physics to Computational Linguistics
Abstract: I will show how techniques from geometry (algebraic geometry and topology) and physics (statistical physics) can be applied to Linguistics, in order to provide a computational approach to questions of syntactic structure and language evolution, within the context of Chomsky's Principles and Parameters framework.
References:
 N. Chomsky, Lectures on Government and Binding, Foris, Dordrecht, 1981.
 H. Edelsbrunner, J.L. Harer, Computational Topology: An Introduction, American Mathematical Society, 2010.
 M. Drton, B. Sturmfels, S. Sullivant, Lectures on Algebraic Statistics, Birkhauser, 2009
 K. H. Fischer, J. A. Hertz, Spin Glasses, Cambridge University Press, 1993.



Dominique Spehner (université Grenoble Alpes, France)
Biography:
 Diplôme d'Études Approfondies (DEA) in Theoretical Physics at the École Normale Supérieure de Lyon, 1994
 Civil Service (Service National de la Coopération), Technion Institute of Technology, Haifa, Israel, 19951996
 PhD in Theoretical Physics, Université Paul Sabatier, Toulouse, France, 19962000.
 Postdoctoral fellow, Pontificia Universidad Católica, Santiago, Chile, 20002001
 Research Associate, University of DuisburgEssen, Germany, 20012005
 Maître de Conférences, Université Joseph Fourier, Grenoble, France, 2005present
 Habilitation à diriger des Recherches (HDR), Université Grenoble Alpes, 2015
 Member of the Institut Fourier (since 2005) and the Laboratoire de Physique et Modélisation des Milieux Condensés (since 2013) of the university Grenoble Alpes, France

Geometry on the set of quantum states and quantum correlations Short Course (Chaired by Roger Balian)
Abstract:
I will show that the set of states of a quantum system with a finitedimensional Hilbert space can be equipped with various Riemannian distances having nice properties from a quantum information viewpoint, namely they are contractive under all physically allowed operations on the system. The corresponding metrics are quantum analogs of the Fisher metric and have been classified by D. Petz. Two distances are particularly relevant physically: the BogoliubovKuboMori distance studied by R. Balian, Y. Alhassid and H. Reinhardt, and the Bures distance studied by A. Uhlmann and by S.L. Braunstein and C.M. Caves. The latter gives the quantum Fisher information playing an important role in quantum metrology. A way to measure the amount of quantum correlations (entanglement or quantum discord) in bipartite systems (that is, systems composed of two parties) with the help of these distances will be also discussed.
Bibliography:
 D. Petz, Monotone Metrics on Matrix Spaces, Lin. Alg. and its Appl. 244, 8196 (1996)
 R. Balian, Y. Alhassid, and H. Reinhardt, Dissipation in manybody systems: a geometric approach based on information theory, Phys. Rep. 131, 1 (1986)
 R. Balian, The entropybased quantum metric, Entropy 2014 16(7), 38783888 (2014)
 A. Uhlmann, The ``transition probability'' in the state space of a *algebra, Rep. Math. Phys. 9, 273279 (1976)
 S.L. Braunstein and C.M. Caves, Statistical Distance and the Geometry of Quantum States, Phys. Rev. Lett. 72, 34393443 (1994)
 D. Spehner, Quantum correlations and Distinguishability of quantum states, J. Math. Phys. 55 (2014), 075211
