Résumé des résultats de ma première année de thèse qui se trouvent dans l'article joint.
Plus une petite explication sur la fragmentation (extension) qui est le sujet de ma deuxième année.

One of our research goal is the design of algorithms capable of detecting, with some robustness guaranteed by a suitable performance criterion, random signals with unknown distributions and occurrences in additive and stationary noise. This research is motivated by numerous practical applications where the lack of prior information about the signals or most of their describing parameters is an obstacle to the use of standard likelihood ratio theory. Several results established and assessed in various signal processing applications suggest that our goal could actually be attainable in a nearby future. Part of these results concern robust and non-parametric statistical inference based on assumptions of weak sparseness for the signal. The notion of weak sparseness slightly differs from that introduced by Donoho and Johnstone. Another set of results address statistical properties of the wavelet transform of wide-sense stationary random processes and fractional brownian motions. Some of the aforementioned results have been published separately, whereas others have been submitted only recently. Our purpose is then to present these results altogether so as to examine the links between them and pinpoint possible extensions. In particular, a perspective is the design of algorithms that perform more than the detection of random sigals with unknown distributions, but actually acquire statistical knowledge about these signals to optimize their detection. Beyond the description of standard possible applications in image processing and non-parametric statistics, the presentation will also give the opportunity to initiate a discussion on the application of these results to the design of new types of cell automata.

A piecewise expanding unimodal map on the unit interval admits a unique absolutely continuous invariant measure. By Birkhoff's Ergodic Theorem Lebesgue almost every point in the unit interval is typical for this measure, i.e., for each continuous function its time average along the forward orbit of a typical point is equal to its space average over the absolutely continuous invariant measure. In this talk we look at one-parameter families of piecewise expanding unimodal maps and we show that in the generic case there exists a set of full Lebesgue measure in the parameter space such that for each map corresponding to a parameter in this set the turning point is typical for the absolutely continuous invariant measure. This almost sure typicality result in the parameter space also applies to points different from the turning point.

Until the eighties, it was a general though that hyperbolic surfacesplayed a marginal role in the theory of minimal surfaces in Euclidean $3$-space. However the constructions of Jorge-Meeks in 1980 and Nadirashvili in 1996 have inspired the proof of general existence theorems for complete hyperbolic minimal surfaces. In this talk, we will discuss some techniques of construction of such surfaces and show several applications.