Table of contents

Nonlocal boundary value problems are considered for elliptic equations of the following form. A nonperiodic mapping q of the boundary to itself is given, and the boundary condition connects the values of the unknown function and its derivatives at the points x, g(x), , ... . The author obtains necessary and sufficient conditions for the problem to be Noetherian (i.e. for the operator to be Fredholm) in terms of the invertibility of an auxiliary functional operator (the symbol of the problem), acting in a function space on the bundle of unit cotangent vectors to the boundary. Explicit necessary and sufficient conditions for the Noether property are presented for a number of examples. The main constructions and proofs are based on the theory of C*-algebras generated by dynamical systems.

Bibliography: 36 titles.

It is proved that a triangulated category generated by a strong exceptional collection is equivalent to the derived category of modules over an algebra of homomorphisms of this collection. For the category of coherent sheaves on a Fano variety, the functor of tightening to a canonical class is described by means of mutations of an exceptional collection generating the category. The connection between mutability of strong exceptional collections and the Koszul property is studied. It is proved that in the geometric situation mutations of exceptional sheaves consist of present sheaves and the corresponding algebra of homomorphisms is Koszul and selfconsistent.

Bibliography: 18 titles.

The operator is considered in ; here , ,  is a periodic potential, and  is a perturbation parameter, . An analytic perturbation theory with respect to  is constructed for Block eigenfunctions and the corresponding eigenvalues of . It is proved that, for large energies, when the quasimomentum belongs to a sufficiently rich set they admit expansion in a Taylor series in the disk , and these series are asymptotic in the energy and infinitely differentiable with respect to the quasimomentum.

Bibliography: 14 titles.

The author gives conditions on the behavior of a controlled diffusion process within and on the boundary of a domain that are sufficient for the value function to have two bounded generalized derivatives and to satisfy the Bellman equation. These conditions are almost necessary even for uncontrolled diffusion processes, and at the same time they encompass, for example, the heat equation in a disc and the Monge–Ampère equation in a convex domain.

Bibliography: 24 titles.

The author defines a new construction of free product in the variety  of profinite groups of the family of groups in , continuously indexed by points of the profinite space . In the case where  is closed relative to extensions with Abelian kernels, a number of assertions about the homology groups of  are obtained. Using homological methods, a theorem of Kurosh type on decomposition of an arbitrary pro--subgroup in  into a free pro--product is proved, under a certain separability condition on .

Bibliography: 19 titles.

For rings with a large number of units the authors prove a strengthened theorem on homological stabilization: the homomorphism H_k(GL_n(A)) → H_k(GL(A)) is surjective for n ≥ k + sr A – 1 and bijective for n ≥ k + sr A. If A is a local ring with an infinite residue field, then this result admits further refinement: the homomorphism H_n(GL_n(A)) → H_n(GL(A)) is bijective and the factor group H_n(GL(A)) / H_n(GL_{n -1}(A)) is canonically isomorphic to Milnor's nth K-group of the ring A. The results are applied to compute the Chow groups of algebraic varieties.

Bibliography: 16 titles.

The authors study the spectrum of a C*-algebra of pseudodifferential operators whose symbols have discontinuities along submanifolds (the supports of the discontinuities) intersecting at nonzero angles. All (up to equivalence) irreducible representations of such algebras are enumerated, a realization of these representations is indicated, and the spectral topology is described.

Bibliography: 7 titles.

For infinite-particle systems of classical statistical mechanics, a method of describing the dynamics of a charged one-dimensional Brownian particle is obtained.

Bibliography: 7 titles.

An asymptotic formula is obtained for the number of solutions of the congruence

where  is a sufficiently large prime.

Bibliography: 7 titles.