Abstract:
The aim of this review article is to give a comprehensive description of the scaling properties detected for the distribution of cosmic structures. Due to the great variety of statistical methods to describe the large-scale structure of the Universe, I will mainly concentrate on those methods which reveal remarkable regularities and scaling in the structure of the Universe. Although in most cases I prefer not to enter into the technical aspects of how implementing such methods, more details will be furnishes about the description of galaxy clustering in terms of fractal concepts. Despite recent claims for a Universe, which behaves like a fractal at arbitrarily large scales, I will show that the fractal language can be usefully employed to disprove this picture. The emerging scenario is that of a Universe, which behaves like a self-similar structure at small scales, where fractality is dynamically generated by non-linear gravitational clustering, while preserving large-scale homogeneity. Nevertheless, even at scales $\magcir 10\hm$, where gravity still acts linearly, the distribution of galaxy clusters shows remarkable scale-invariant features, which could give precise hints about the initial conditions for the evolution of the large-scale structure of the Universe.

Abstract:
These Lecture Notes are devoted to an introductory description of some of the most widely applied statistical methods for the analysis of the Large-Scale Structure (LSS) of the Universe. Rather than providing technical details about the implementation of such methods, I concentrate more on their significance and on the statistical information they provide in the comparison between data and LSS formation models. I introduce the concept of correlation functions and their connection with the count-in-cell statistics of a point distribution. Also some geometrical descriptions of the LSS, like the void probability function and the topological genus characteristics, are briefly reviewed.

Abstract:
This Meeting featured the recent advancements in our understanding of galaxy clusters and the distant Universe, achieved by the past and new generation of X-ray satellites. I summarize here the main themes that have been discussed: (a) Clusters of galaxies as probes of cosmological models; (b) The physics of cosmic baryons trapped within the potential wells of galaxy clusters; (c) The origin of the cosmic X-ray background and the nature of the contributing sources.

Abstract:
We review the prospects for dark matter (DM) in two classes of supersymmetric (SUSY) models which are characterized by the different mechanism of SUSY breaking, namely the more common supergravity models with very large scale of SUSY breaking and the recent schemes where SUSY is broken at a relatively low scale and gravitinos are likely to be the lightest SUSY particle. We point out that the former scheme is in general associated with the cold dark matter (CDM) scenario, while the latter predicts a warm dark matter (WDM) dominated Universe.

Abstract:
We review recent progress in the description of the formation and evolution of galaxy clusters in a cosmological context by using numerical simulations. We focus our presentation on the comparison between simulated and observed X-ray properties, while we will also discuss numerical predictions on properties of the galaxy population in clusters. Many of the salient observed properties of clusters, such as X-ray scaling relations, radial profiles of entropy and density of the intracluster gas, and radial distribution of galaxies are reproduced quite well. In particular, the outer regions of cluster at radii beyond about 10 per cent of the virial radius are quite regular and exhibit scaling with mass remarkably close to that expected in the simplest case in which only the action of gravity determines the evolution of the intra-cluster gas. However, simulations generally fail at reproducing the observed cool-core structure of clusters: simulated clusters generally exhibit a significant excess of gas cooling in their central regions, which causes an overestimate of the star formation and incorrect temperature and entropy profiles. The total baryon fraction in clusters is below the mean universal value, by an amount which depends on the cluster-centric distance and the physics included in the simulations, with interesting tensions between observed stellar and gas fractions in clusters and predictions of simulations. Besides their important implications for the cosmological application of clusters, these puzzles also point towards the important role played by additional physical processes, beyond those already included in the simulations. We review the role played by these processes, along with the difficulty for their implementation, and discuss the outlook for the future progress in numerical modeling of clusters.

Abstract:
We analyze the statistical properties of bubble models for the large-scale distribution of galaxies. To this aim, we realize static simulations, in which galaxies are mostly randomly arranged in the regions surrounding bubbles. As a first test, we realize simulations of the Lick map, by suitably projecting the three-dimensional simulations. In this way, we are able to safely compare the angular correlation function implied by a bubbly geometry to that of the APM sample. We find that several bubble models provide an adequate amount of large-scale correlation, which nicely fits that of APM galaxies. Further, we apply the statistics of the count-in-cell moments to the three-dimensional distribution and compare them with available observational data on variance, skewness and kurtosis. Based on our purely geometrical constructions, we find that a well defined hierarchical scaling of higher order moments up to scales $\sim 70\hm$. The overall emerging picture is that the bubbly geometry is well suited to reproduce several aspects of large-scale clustering.

Abstract:
In this review, we describe our current understanding of cluster formation: from the general picture of collapse from initial density fluctuations in an expanding Universe to detailed simulations of cluster formation including the effects of galaxy formation. We outline both the areas in which highly accurate predictions of theoretical models can be obtained and areas where predictions are uncertain due to uncertain physics of galaxy formation and feedback. The former includes the description of the structural properties of the dark matter halos hosting cluster, their mass function and clustering properties. Their study provides a foundation for cosmological applications of clusters and for testing the fundamental assumptions of the standard model of structure formation. The latter includes the description of the total gas and stellar fractions, the thermodynamical and non-thermal processes in the intracluster plasma. Their study serves as a testing ground for galaxy formation models and plasma physics. In this context, we identify a suitable radial range where the observed thermal properties of the intra-cluster plasma exhibit the most regular behavior and thus can be used to define robust observational proxies for the total cluster mass. We put particular emphasis on examining assumptions and limitations of the widely used self-similar model of clusters. Finally, we discuss the formation of clusters in non-standard cosmological models, such as non-Gaussian models for the initial density field and models with modified gravity, along with prospects for testing these alternative scenarios with large cluster surveys in the near future.

Abstract:
In this Letter, we analyse the predicted physical properties of massive galaxies, in the framework of recent semi-analytic models of galaxy formation. All models considered account for winds driven by supernovae explosions and suppression of gas condensation at the centre of relatively massive haloes by active galactic nuclei (AGN). We show that, while these models successfully reproduce the old stellar populations observed for massive galaxies, they fail in reproducing their observed chemical abundances. This problem is alleviate but still present if AGN feedback is completely switched off. Moreover, in this case, model predictions fail in accounting for the old stellar ages of massive galaxies. We argue that the difficulty of semi-analytical models in simultaneously reproducing the observed ages and metallicities of massive galaxies, signals a fundamental problem with the schemes that are currently adopted to model star formation, feedback, and related recycling of gas and metals.

Abstract:
In theories with a gauge-mediated mechanism of supersymmetry breaking the gravitino is likely to be the lightest superparticle and, hence, a candidate for dark matter. We show that the decay of the next-to-lightest superparticle into a gravitino can yield a non-thermal population of gravitinos which behave as a hot dark matter component. Together with the warm component, which is provided by the population of gravitinos of thermal origin, they can give rise to viable schemes of mixed dark matter. This realization has some specific and testable features both in particle physics and astrophysics. We outline under which conditions the mechanism remains viable even when R parity is broken.

Abstract:
We apply the ZTRACE algorithm to the optical NOG and infra-red PSCz galaxy catalogues to reconstruct the pattern of primordial fluctuations that have generated our local Universe. We check that the density fields traced by the two catalogues are well correlated, and consistent with a linear relation (either in $\delta$ or in log(1+$\delta$)) with relative bias (of NOG with respect to PSCz) b_rel = 1.1 $\pm$ 0.1. The relative bias relation is used to fill the optical zone of avoidance at |b| < 20$^\circ$ using the PSCz galaxy density field. We perform extensive testing on simulated galaxy catalogues to optimize the reconstruction. The quality of the reconstruction is predicted to be good at large scales, up to a limiting wavenumber k_lim $\simeq$ 0.4 h/Mpc beyond which all information is lost. We find that the improvement due to the denser sampling of the optical catalogue is compensated by the uncertainties connected to the larger zone of avoidance. The initial conditions reconstructed from the NOG catalogue are found (analogously to those from the PSCz) to be consistent with Gaussian paradigm. We use the reconstructions to produce sets of initial conditions ready to be used for constrained simulations of our local Universe.