The aim of proposed research is to study various models of non-Markovian random walks modeling the dynamical properties of disordered media. Among these models, the focus is on random walks in random environments, random walks on random graphs like percolation clusters or random trees, reinforced random walks, and random walks in random sceneries. These models have different origins and motivations, but all have natural applications in physics, biology or engineering (see for instance [AH], [BDG], [BTDA], [CM], [dG], [MM], [lD], [St], ...) . All of these models exhibit strong dependence on the past, either by definition of the model itself, or as a consequence of averaging with respect to the random environment. This feature leads to a number of interesting questions, about recurrence/transience of these processes, limit theorems, or large deviations properties that we wish to address in the present project.

The objective of the ICLM project "Coherent Imaging of Multi-scattered Light" is to study light propagation through scattering media with different original techniques based on coherent interferometry, in order to apply it to biological media. This requires to work within the 650-1100nm spectral range in order to minimize absorption effects. One can extract information from the complex output field by interferometry with an external reference beam (two-wave mixing), and get a millimetric spatial localization by coupling light with an ultrasonic wave within the scattering medium (acousto-optic effect). One can generate by three-wave mixing a phase-conjugate beam, that once back-propagated within the medium, restores the input beam (plane wave) with the cumulated information of local absorption properties (phase conjugation process). But an efficient collection of these multi-scattered photons is puzzling in terms of flux, because the optical etendue of the output field (speckle) is huge, especially for thick media. Moreover, as it is the case for living tissues, this speckle field is not stationary versus time because of inner motions, and thus a coherent detection must be performed during a relatively short acquisition time (millisecond). The ambition of this project is multiple, since we wish to : 1 - Perform an efficient acousto-optic imagery and apply it to biological tissues, with a millimetric resolution. 2 - Understand and modelize back-propagation within the medium following the phase conjugation process. 3 - Control the spatial structure of the wavefront in order to optimize the propagation within the medium. We will use in two-wave mixing configuration a digital heterodyne off-axis holography in order to select photons that have crossed the ultrasonic field, and also a wavefront adaptive holography using original photosenstive materials at 800 and 1064nm (photorefractive crystals, gain media), which is well suited for biological media. We will also evaluate a new promising type of detection based on the spectral holeburning phenomenon. Phase-conjugation process will be generated in real time (picosecond source) by parametric amplification, making a three-wave mixing within a non-linear crystal, or considering the photosensitive materials used in the two-wave mixing configuration. The active control of the wavefront transmitted by the medium will be considered with spatial light modulators in order to study the properties of phase conjugation, and test the existence of Pendry's "Open Channels" within the medium.

Arakelov geometry and the Voevodsky-style theory of algebraic homology or homotopy (variants of which were discovered and studied by Suslin, Rost, Levine, Morel, Hanamura and Voevodsky, to quote just a few) provide entirely new methods of investigation for problems that have resisted the efforts of many researchers. In this project, our aim is twofold: 1> First of all, we intend to use the objects and the tools of the theory of scheme-theoretic homotopy to solve certain difficult problems coming from Arakelov theory. In particular, it seems to us that even to formulate, let alone prove the most general fixed point theorems (as well as their arakelov-theoretic extensions), this new framework is required. We thus hope to get closer to the proof of some very general conjectures that we made a few years ago. These conjectures have important consequences for the arithmetic geometry of Shimura varieties. 2> Reciprocally, we want to extend the constructions of Arakelov theory to the much more general context of scheme-theoretic homotopy. We expect that two types of results will be reached, if we carry out that plan: - New and very general constructions of non-trivial elements in motivic cohomology (please note that we have already obtained a few partial but original results in this direction), in the context of Beilinson's conjectures and its variants. - Results of finite generation, in some cases, for the motivic cohomology groups, thus generalising the classical approach of Dirichlet.