Modeling and understanding the structure of the universe is one the most interesting problems in modern cosmology. It is also an extremely difficult problem. Despite spectacular achievements in observations and computational capabilities at present there is no comprehensive theory of the structure formation. There are successful theoretical models however most of them rely on assumptions and various fits which are not fully physically justified. The Zeldovich Approximation is a rare exception. Suggested more than forty years ago it remains one of the most successful and profound analytic models of structure formation. Some of results stemmed from the Zeldovich Approximation will be briefly reviewed.

I discuss the formation of the modern cosmological paradigm. In more detail I describe the early study of dark matter and cosmic web and the role of Yakov Zeldovich in the formation of the present concepts on these subjects.

Yakov Borisovich Zeldovich was remarkably talented.† His active scientific career included major contributions in fields as diverse as chemical physics (adsorption & catalysis), the theory of shock waves, thermal explosions, the theory of flame propogation, the theory of combustion & detonation, nuclear & particle physics, and, during the latter part of his life: gravitation, astrophysics and cosmology (Zeldovich 1992).

In this brief paper, I cannot provide an overall review of the Planck results to date. Instead I will focus on a handful of results from both Planck and related cosmic microwave background (CMB) experiments that reflect Ya. B. Zel'dovich's legacy in cosmology. These include the Sunyaev-Zel'dovich effect in clusters of galaxies and in the cosmic web, and a map of the overall distribution of mass in the Universe derived from CMB maps.

Black holes are now commonplace, among the stars, in Galactic centers, and perhaps other places. But within living memory, their very existence was doubted by many, and few chose to look for them. Zeldovich and Guseinov were first, followed by Trimble and Thorne, using a method that would have identified HDE 226868 as a plausible candidate, if it had been in the 1968 catalogue of spectroscopic binaries. That it was not arose from an unhappy accident in the observing program of Daniel M. Popper long before the discovery of X-ray binaries and the identification of Cygnus X-1 with that hot, massive star and its collapsed companion.

We investigate the characteristics and the time evolution of the cosmic web from redshift, z=2, to present time, within the framework of the \nexus{} algorithm. This necessitates the introduction of new analysis tools optimally suited to describe the very intricate and hierarchical pattern that is the cosmic web. In particular, we characterising filaments (walls) in terms of their linear (surface) mass density, which is very good in capturing the evolution of these structures. At early times the cosmos is dominated by tenuous filaments and sheets, which, during subsequent evolution, merge together, such that the present day web is dominated by fewer, but much more massive, structures. We show also that voids are more naturally described in terms of their boundaries and not their centres. We illustrate this for void density profiles, which, when expressed as a function of the distance from void boundary, show a universal profile in good qualitative agreement with the theoretical shell-crossing framework of expanding underdense regions.

The initial shear field plays a central role in the formation of large-scale structures, and in shaping the geometry, morphology, and topology of the cosmic web. We discuss a recent theoretical framework for the shear tensor, termed the ‘peak/dip picture’, which accounts for the fact that halos/voids may form from local extrema of the density field – rather than from random spatial positions; the standard Doroshkevich's formalism is generalized, to include correlations between the density Hessian and shear field at special points in space around which halos/voids may form. We then present the ‘peak/dip excursion-set-based’ algorithm, along with its most recent applications – merging peaks theory with the standard excursion set approach.

In the conference presentation we have reviewed the theory of non-Gaussian geometrical measures for 3D Cosmic Web of the matter distribution in the Universe and 2D sky data, such as Cosmic Microwave Background (CMB) maps that was developed in a series of our papers. The theory leverages symmetry of isotropic statistics such as Minkowski functionals and extrema counts to develop post Gaussian expansion of the statistics in orthogonal polynomials of invariant descriptors of the field, its first and second derivatives. The application of the approach to 2D fields defined on a spherical sky was suggested, but never rigorously developed. In this paper we present such development treating the effects of the curvature and finiteness of the spherical space $S_2$ exactly, without relying on flat-sky approximation. We present Minkowski functionals, including Euler characteristic and extrema counts to the first non-Gaussian correction, suitable for weakly non-Gaussian fields on a sphere, of which CMB is the prime example.

Relying on a separable modal expansion of the bispectrum, the implementation of a fast estimator for the full bispectrum of a 3d particle distribution is presented. The computational cost of accurate bispectrum estimation is negligible relative to simulation evolution, so the bispectrum can be used as a standard diagnostic whenever the power spectrum is evaluated. As an application, the time evolution of gravitational and primordial dark matter bispectra was measured in a large suite of N-body simulations. The bispectrum shape changes characteristically when the cosmic web becomes dominated by filaments and halos, therefore providing a quantitative probe of 3d structure formation. Our measured bispectra are determined by ∼ 50 coefficients, which can be used as fitting formulae in the nonlinear regime and for non-Gaussian initial conditions. We also compare the measured bispectra with predictions from the Effective Field Theory of Large Scale Structures (EFTofLSS).

The Zeldovich approximation (ZA) predicts the formation of a web of singularities. While these singularities may only exist in the most formal interpretation of the ZA, they provide a powerful tool for the analysis of initial conditions. We present a novel method to find the skeleton of the resulting cosmic web based on singularities in the primordial deformation tensor and its higher order derivatives. We show that the A3 lines predict the formation of filaments in a two-dimensional model. We continue with applications of the adhesion model to visualise structures in the local (z < 0.03) universe.

The phase space distribution of matter out to ∼ 100 \rm Mpc is probed by two types of observational data: galaxy redshift surveys and peculiar motions of galaxies. Important information on the process of structure formation and deviations from standard gravity have been extracted from the accumulating data. The remarkably simple Zel'dovich approximation is the basis for much of our insight into the dynamics of structure formation and the development of data analyses methods. Progress in the methodology and some recent results is reviewed.

I review the nature of three-dimensional collapse in the Zeldovich approximation, how it relates to the underlying nature of the three-dimensional Lagrangian manifold and naturally gives rise to a hierarchical structure formation scenario that progresses through collapse from voids to pancakes, filaments and then halos. I then discuss how variations of the Zeldovich approximation (based on the gravitational or the velocity potential) have been used to define classifications of the cosmic large-scale structure into dynamically distinct parts. Finally, I turn to recent efforts to devise new approaches relying on tessellations of the Lagrangian manifold to follow the fine-grained dynamics of the dark matter fluid into the highly non-linear regime and both extract the maximum amount of information from existing simulations as well as devise new simulation techniques for cold collisionless dynamics.

The powerful Lagrangian view of structure formation was essentially introduced to cosmology by Zel'dovich. In the current cosmological paradigm, a dark-matter-sheet 3D manifold, inhabiting 6D position-velocity phase space, was flat (with vanishing velocity) at the big bang. Afterward, gravity stretched and bunched the sheet together in different places, forming a cosmic web when projected to the position coordinates.

Here, I explain some properties of an origami approximation, in which the sheet does not stretch or contract (an assumption that is false in general), but is allowed to fold. Even without stretching, the sheet can form an idealized cosmic web, with convex polyhedral voids separated by straight walls and filaments, joined by convex polyhedral nodes. The nodes form in ‘polygonal’ or ‘polyhedral’ collapse, somewhat like spherical/ellipsoidal collapse, except incorporating simultaneous filament and wall formation. The origami approximation allows phase-space geometries of nodes, filaments, and walls to be more easily understood, and may aid in understanding spin correlations between nearby galaxies. This contribution explores kinematic origami-approximation models giving velocity fields for the first time.

The Cosmic Web is a complicated highly-entangled geometrical object. Remarkably it has formed from practically Gaussian initial conditions, which may be regarded as the simplest departure from exactly uniform universe in purely deterministic mapping. The full complexity of the web is revealed neither in configuration no velocity spaces considered separately. It can be fully appreciated only in six-dimensional (6D) phase space. However, studies of the phase space is complicated by the fact that every projection of it on a three-dimensional (3D) space is multivalued and contained caustics. In addition phase space is not a metric space that complicates studies of geometry. We suggest to use Lagrangian submanifold i.e., x = x(q), where both x and q are 3D vectors instead of the phase space for studies the complexity of cosmic web in cosmological N-body dark matter simulations. Being fully equivalent in dynamical sense to the phase space it has an advantage of being a single valued and also metric space.

The cosmic web is a complex spatial pattern of walls, filaments, cluster nodes and underdense void regions. It emerged through gravitational amplification from the Gaussian primordial density field. Here we infer analytical expressions for the spatial statistics of caustics in the evolving large-scale mass distribution. In our analysis, following the quasi-linear Zel'dovich formalism and confined to the 1D and 2D situation, we compute number density and correlation properties of caustics in cosmic density fields that evolve from Gaussian primordial conditions. The analysis can be straightforwardly extended to the 3D situation. We moreover, are currently extending the approach to the non-linear regime of structure formation by including higher order Lagrangian approximations and Lagrangian effective field theory.

We investigate large scale structure formation of collisionless dark matter in the phase space description based on the Vlasov-Poisson equation. We present the Schrödinger method, originally proposed by \cite{WK93} as numerical technique based on the Schrödinger Poisson equation, as an analytical tool which is superior to the common standard pressureless fluid model. Whereas the dust model fails and develops singularities at shell crossing the Schrödinger method encompasses multi-streaming and even virialization.

Zel'dovich proposed Lagrangian perturbation theory (LPT) for structure formation in the Universe. After this, higher-order perturbative equations have been derived. Recently fourth-order LPT (4LPT) have been derived by two group. We have shown fifth-order LPT (5LPT) In this conference, we notice fourth- and more higher-order perturbative equations. In fourth-order perturbation, because of the difference in handling of spatial derivative, there are two groups of equations. Then we consider the initial conditions for cosmological N-body simulations. Crocce, Pueblas, and Scoccimarro (2007) noticed that second-order perturbation theory (2LPT) is required for accuracy of several percents. We verify the effect of 3LPT initial condition for the simulations. Finally we discuss the way of further improving approach and future applications of LPTs.

There is now no doubt that neutrinos are massive particles fully involved in the non-linear growth of the large-scale structure of the universe. A problem is that they are particularly difficult to include in cosmological models because the equations describing their behavior in the non-linear regime are cumbersome and difficult to handle. In this manuscript I present a new method allowing to deal with massive neutrinos in a very simple way, based on basic conservation laws. This method is still valid in the non-linear regime. The key idea is to describe neutrinos as a collection of single-flow fluids instead of seeing them as a single hot multi-flow fluid. In this framework, the time evolution of neutrinos is encoded in fluid equations describing macroscopic fields, just as what is done for cold dark matter. Although valid up to shell-crossing only, this approach is a further step towards a fully non-linear treatment of the dynamical evolution of neutrinos in the framework of large-scale structure growth.

In the Century since Slipher's first observations, roughly three million galaxy redshifts have been measured. The resulting maps of large-scale structure have taught us much of central importance in cosmology, ranging from the matter content of the universe to the study of the primordial density fluctuations. This talk aims to review some of the key observational and theoretical milestones on this journey, and to speculate about what the future may bring.

Our view of the low-redshift Cosmic Web has been revolutionized by galaxy redshift surveys such as 6dFGS, SDSS and 2MRS. However, the trade-off between depth and angular coverage limits a systematic three-dimensional account of the entire sky beyond the Local Volume (z < 0.05). In order to reliably map the Universe to cosmologically significant depths over the full celestial sphere, one must draw on multiwavelength datasets and state-of-the-art photometric redshift techniques. We have undertaken a dedicated program of cross-matching the largest photometric all-sky surveys – 2MASS, WISE and SuperCOSMOS – to obtain accurate redshift estimates of millions of galaxies. The first outcome of these efforts – the 2MASS Photometric Redshift catalog (2MPZ, Bilicki et al. 2014a) – has been publicly released and includes almost 1 million galaxies with a mean redshift of z=0.08. Here we summarize how this catalog was constructed and how using the WISE mid-infrared sample together with SuperCOSMOS optical data allows us to push to redshift shells of z∼ 0.2 –0.3 on unprecedented angular scales. Our catalogs, with ∼ 20 million sources in total, provide access to cosmological volumes crucial for studies of local galaxy flows (clustering dipole, bulk flow) and cross-correlations with the cosmic microwave background such as the integrated Sachs-Wolfe effect or lensing studies.