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The stochastic dynamics preserving an equilibrium distribution of quantum gravity is considered. This is the first detailed theoretical investigation of this dynamics (earlier it was used for Monte-Carlo simulation). The main result is related to the existence and certain properties of local correlation functions in the thermodynamic limit. At the same time, the paper can serve as a mathematical introduction to quantum gravity because we give a rigorous exposition of quantum gravity in the case of planar pure gravity. We mainly use the combinatorial approach instead of matrix models, which are more popular in physics, and the central point is the famous exponent

In the paper, the relationship between the theory of holomorphic functions on two-dimensional complex manifolds and their differential topology is described. The basic fact, established by using the Seiberg-Witten invariants, is that the topological characteristics of embedded real surfaces in Stein surfaces satisfy adjunction-type inequalities. A version of Gromov's h-principle for totally real embeddings shows that these topological inequalities are sharp. In some cases, these results can be used to describe the envelopes of holomorphy of embedded real surfaces in a given complex surface. Our examples include real surfaces in  and  and in products of  with non-compact Riemann surfaces. A similar technique can be applied to the study of geometric properties of strictly pseudoconvex domains in dimension two.

We study analogues of analytic capacity for classes of analytic functions representable via some special analytic machinery, which we refer to as "Golubev sums". A Golubev sum contains derivatives of various (given) orders of Cauchy potentials (in particular, the Cauchy potentials themselves can occur in a Golubev sum). Furthermore, the measures determining distinct terms of a Golubev sum are in general defined on distinct compact sets. We consider Golubev sums with various types of measures: complex, real, and positive. We present an abstract scheme for studying extremal problems like the analytic capacity problem. The dual problems turn out to be approximation problems in which the size of the approximants is taken into account. In the case of positive measures, the approximation problem is transformed into a problem in which one has to move a given element of a space into a given cone in that space by adding linear combinations of elements of a given subspace with coefficients as small as possible. As a preliminary, we state criteria for the representability of an analytic function by Golubev sums of various kinds. These criteria generalize known criteria for representability by Cauchy potentials.