Table of contents

Discrete differential geometry aims to develop discrete equivalents of the geometric notions and methods of classical differential geometry. This survey contains a discussion of the following two fundamental discretization principles: the transformation group principle (smooth geometric objects and their discretizations are invariant with respect to the same transformation group) and the consistency principle (discretizations of smooth parametrized geometries can be extended to multidimensional consistent nets). The main concrete geometric problem treated here is discretization of curvature-line parametrized surfaces in Lie geometry. Systematic use of the discretization principles leads to a discretization of curvature-line parametrization which unifies circular and conical nets.

The main results in the area of set-theoretic non-standard analysis (non-standard set and class theories) obtained over the last few years are presented. It is demonstrated how a universe of a comparably simple theory (beginning with the usual Zermelo-Fraenkel set theory ZFC) can be extended to a universe of a more complicated non-standard set or class theory. The last section develops the foundations of Boolean-valued analysis as a part of set-theoretic non-standard analysis.

The theory of maps close to representations (almost representations, approximate representations, quasi-representations, pseudorepresentations, and so on) has accumulated a great amount of material during the last 25-30 years, and has been enriched with technical tools having non-trivial applications in algebra and topology, from bounded cohomology to Finsler metrics and the Calabi invariant in symplectic geometry. In this survey the main notions and facts of the theory are presented in connection with the proof of the 'triviality theorem', presented here for finite-dimensional quasi-representations of compact Lie groups: every (not necessarily continuous) finite-dimensional unitary quasi-representation with small defect of a semisimple compact Lie group is close to an ordinary (continuous) representation of the group. This theorem, which gives a complete answer to the 1982 question of Kazhdan and Mil'man, is also a partial answer to Gromov's 1995 question, namely, although a semisimple compact group is not amenable in the discrete topology, all finite-dimensional unitary quasi-representations of it are still perturbations of ordinary representations. Moreover, necessary and sufficient conditions for the validity of an analogue of the van der Waerden theorem (that is, conditions for the automatic continuity of all locally bounded finite-dimensional representations) for a given connected Lie group are indicated, and a description of the structure of all finite-dimensional locally bounded quasi-representations of arbitrary connected semisimple Lie groups is given. Results related to some other directions of investigation concerning the theory of maps close to representations of groups and algebras and their applications to geometry and group theory are also discussed.