Table of contents

Finite groups generated by Euclidean reflections have been commonplace in various problems of singularity theory since their relationship with the classification of critical points of functions was discovered by Arnol'd [1], [2]. We show that a number of finite groups generated by unitary reflections are also naturally related to singularities of functions, namely, those invariant under a unitary reflection of finite order. To this end, we consider germs of functions on a manifold with boundary and lift them to a cyclic covering of the manifold, ramified over the boundary. This construction provides a new notion of roots for the groups under consideration and provides skew-Hermitian analogues of these groups.

This survey consists of three parts: partial geometries, diagram geometries, and extensions of partial geometries. In the first part we present the main definitions, all known examples of partial geometries, some characterization theorems, and also certain results on semipartial geometries. In the second part we give definitions related to geometries belonging to diagrams, a series of general results concerning these geometries, examples of rank 2 geometries that play the role of building blocks for geometries of higher rank, and also theorems on the structure of some flag-transitive rank 2 geometries. In the third part we give a survey of results characterizing extensions of partial geometries, namely, 2-designs and dual 2-designs, nets and dual nets, and also generalized quadrangles. Moreover, in the concluding subsection of this part we present some generalizations of the notion of extension and the corresponding characterization results.

We describe a wide class of boundary-value problems for which the application of elliptic theory can be reduced to elementary algebraic operations and which is characterized by the following polynomial property: the sesquilinear form corresponding to the problem degenerates only on some finite-dimensional linear space  of vector polynomials. Under this condition the boundary-value problem is elliptic, and its kernel and cokernel can be expressed in terms of . For domains with piecewise-smooth boundary or infinite ends (conic, cylindrical, or periodic), we also present fragments of asymptotic formulae for the solutions, give specific versions of general conditional theorems on the Fredholm property (in particular, by modifying the ordinary weighted norms), and compute the index of the operator corresponding to the boundary-value problem. The polynomial property is also helpful for asymptotic analysis of boundary-value problems in thin domains and junctions of such domains. Namely, simple manipulations with  permit one to find the size of the system obtained by dimension reduction as well as the orders of the differential operators occurring in that system and provide complete information on the boundary layer structure. The results are illustrated by examples from elasticity and hydromechanics.