Schedule

    Speakers

• K. Arashi (Nagoya University)
  On SO*(n) and its representations.
  The Lie group SO*(n) is an example of Hermitian Lie group. One of its definition is given by SO*(n) = SO(2n, C) \cap M(n, H), where M(n,H) is the set of quaternionic matrices realized as a subspace of M(2n,C). We show that the determinant of any matrix in M(n,H) is positive, which implies that SO*(n) coincides with O*(n) = O(2n, C) \cap M(n,H). Moreover, we discuss holomorphic discrete series representations of SO*(n).
• H. Ishi (Nagoya University/ JST PRESTO)
  New class of convex cones inspired by statistics
  Symmetric cones are fascinating research objects attracting mathematicians working on various areas including representation theory and mathematical statistics. Thanks to their rich symmetry, we have beautiful analytic formulas over the symmetric cones, and some of the formulas are generalized to non-self-adjoint homogeneous cones. On the other hand, statisticians found similar formulas for the cone of positive definite symmetric matrices with prescribed zero components, where the position of zeros are assigned by chordal graphs, despite the lack of transitive group actions. In this talk, we present a new class of convex cones consisting of real symmetric matrices with specific block decomposition. The class contains both homogeneous cones and the cones corresponding to chordal graphs, so that we get a unified machinery to investigate the two types of cones.
  
• Y. Isowaki (Nagoya University)
  Orbit method for some Lie groups.
  The theory of orbit method is founded by A.A. Kirillov in 1962. Kirillov's original orbit method claims the one-to-one correspondence between equivalence classes of irreducible unitary representations and coadjoint orbits of connected and simply connected nilpotent Lie group. Furthermore, it is known that this relation holds not only for nilpotent Lie groups but also for some other classes of Lie groups. In scheme of the orbit method, the notion of induced representation plays a fundamental role. Actually, induced representation is a universal method of constructing representations of a given group from representations of its (closed) subgroup. In this talk, we first introduce the notion of induced representation from the view point of vector bundles. Then, we observe orbit method for some simple examples.
• A. Keziou (Reims University)
  Duality of divergences and its use in semiparametric statistics
  We investigate the Fenchel duality theory, for the multivariate divergences between measures, viewed as convex functionals on well chosen topological vector spaces of signed finite measures. We give dual representations of these criteria, which we use to define new family of estimates and test statistics, with multiple samples under semiparametric density ratio models, extending the empirical likelihood method. Moreover, we present how to use the estimated ratios, obtained through the above approach, to define multi-group classifiers.
• G. Mendousse (Reims University)
  Some aspects of harmonic analysis on quaternionic homogeneous spaces
  We introduce specific representations of the Lie group G=Sp(n,C) (principal series) on the Hilbert space L^2(S^{4n-1}). We study them by restricting to its maximal compact subgroup K=Sp(n) and obtain the list of invariant irreducible components (K-types) that add up to L^2(S^{4n-1}). Elements of these components that satisfy some suitable invariance property can be associated to specific special functions. We explain how it works when one studies the orthogonal group O(4n) instead of the symplectic group Sp(n), we re-establish classical results and then generalise to Sp(n).
• T. Ohira (Nagoya University)
  Delayed stochastic systems
  Noise and time delay are two elements that are associated with many natural systems, and often they are sources of complex behaviors. Understanding of this complexity is yet to be explored, particularly when both elements are present. we investigate such delayed stochastic systems both in dynamical and probabilistic perspectives: Langevin equation with delay and a random-walk model whose transition probability depends on a fixed time-interval past, called Delayed Random Walks. A peculiar resonance behavior with noise and delay, called ?Delayed Stochastic Resonance? will be also discussed. We present some examples of applications of these delayed stochastic systems.
  
• Ph. Regnault (Reims University)
  Statistical inference based on divergence minimization and information geometry.
  Since the early developments of statistical mathematics, divergences between probability measures have played a prominent role, as they measure the dissimilarity from observations to some expected behavior. In the context of parametric statistics, classical divergences such as Kullback-Leibler or Cressi-Read divergences induce a Riemannian geometric structure on the model under study. Classical results -- and some others -- in statistics for independent samples can be derived through the study of this geometry, referred to as information geometry. In the first part of the talk, we will illustrate some of these informational geometric properties through its application to a large deviations principle for a sequence of estimators of Shannon entropy of a distribution, based on independent observations drawn according to this distribution. Very little attention has been devoted in the literature to informational geometric properties of divergence between Markov chains. As a first attempt to go further in that direction, we will establish in a second time, closed form expressions for divergences between Markov chains.