The analysis and explanation of the cosmic web is one of the challenges
in modern cosmology. The cosmic web has formed at the nonlinear stage
of the growth of tiny inhomogeneities in the density field
probably originated at the end of the inflationary stage.
These inhomogeneities are responsible for spectacular anisotropies
of the cosmic microwave background mapped by the Wilkinson Microwave
Anisotropy Probe as well as by other experiments.
A quantitative analysis of the cosmic web poses a number
of challenges: 1) a sensible identification of the constituents of the web,
2) practical and robust numerical techniques able to operate on real
data sets, 3) selection of adequate quantitative characteristics and
statistics. Recently new ideas based on the concepts of differential and
integral geometry as well as achievements in computational geometry have
been applied to the morphological analysis of the cosmic web.
In the talk I will describe these problems and outline some
of the approaches used in modern cosmological studies.