We consider the problem of finding all linear algebraic groupsubgroup pairs such that the rational invariants of the group and of the subgroup coincide. In this paper the solution will be given for the case where both the group and the subgroup are connected complex irreducible linear groups.

Let k be an algebraically closed field and let G be a reductive linear algebraic group over k. Let P be a parabolic subgroup of G, Pu its unipotent radical and pu the Lie algebra of Pu. A fundamental result of R. Richardson says that P acts on pu with a dense orbit (see [9]). The analogous result for the coadjoint action of P on pu is already known for char k = 0 (see [5]). In this note we prov...

We introduce two algebraic completely integrable analogues of the Mumford systems which we call hyperelliptic Prym systems, because every hyperelliptic Prym variety appears as a fiber of their momentum map. As an application we show that the general fiber of the momentum map of the periodic Volterra lattice ȧi = ai(ai−1 − ai+1), i = 1, . . . , n, an+1 = a1, is an affine part of a hyperelliptic ...

Let G be a unipotent algebraic subgroup of some GLm(C) defined over Q. We describe an algorithm for finding a finite set of generators of the subgroup G(Z) = G ∩ GLm(Z). This is based on a new proof of the result (in more general form due to Borel and Harish-Chandra) that such a finite generating set exists.

If the ring Z of integers is the union of finitely many residue classes a1(mod n1), . . . , ak(mod nk), then the system A = {as(mod ns)}s=1 is called a cover of Z or a covering system. There are many problems and results on this topic initiated by Paul Erdös. In this talk we introduce progress on the main open problems in this field, as well as the algebraic theory of periodic arithmetical maps...

One way to view Theorem 1.1 is as a statement that the algebraic structure of a finitely generated profinite group somehow also encodes the topological structure. That is, if one wishes to know the open subgroups of a profinite group G, a topological property, one must only consider the subgroups of G of finite index, an algebraic property. As profinite groups are compact topological spaces, an...

If we are given a smooth differential operator in the variable x ∈ R/2πZ, its normal form, as is well known, is the simplest form obtainable by means of the Diff(S)-group action on the space of all such operators. A versal deformation of this operator is a normal form for some parametric infinitesimal family including the operator. Our study is devoted to analysis of versal deformations of a Di...