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We develop a structure theory of connected solvable spherical subgroups in semisimple algebraic groups. Based on this theory, we obtain an explicit classification of all such subgroups up to conjugation.

This is a survey paper of the theory of crystal bases, global bases and the cluster algebra structure on the quantum coordinate rings.

This is a report on our long term project to find an algorithm to decide if a finitely presented group has a non-trivial action on a tree.

We survey some recent constructions of cluster algebra structures on coordinate rings of unipotent subgroups and unipotent cells of Kac-Moody groups. We also review a quantized version of these results.

We classify real 6-dimensional nilpotent Lie algebras for which the corresponding Lie group has a left-invariant complex structure, and estimate the dimensions of moduli spaces of such structures.

In this article we try to explore the relation between real conjugacy classes and real characters of finite groups at more refined level. This refinement is in terms of properties of groups such as strong reality and total orthogonality. In…

The article surveys published and not yet published results about moduli spaces of algebraic surfaces.

The purpose of this paper is to survey the structure of closed and transitive transformation groups acting on a closed surface. In particular, we prove a number of relations between groups acting on the sphere that contain the rotation…

In structural rigidity, one studies frameworks of bars and joints in Euclidean space. Such a framework is an articulated structure consisting of rigid bars, joined together at joints around which the bars may rotate. In this paper, we will…

We present a comprehensive survey of constructions of the real numbers (from either the rationals or the integers) in a unified fashion, thus providing an overview of most (if not all) known constructions ranging from the earliest attempts…

We study the local Killing Lie algebra of meromorphic almost rigid geometric structures on complex manifolds. This leads to classification results for compact complex manifolds bearing holomorphic rigid geometric structures.

We give a purely algebraic treatment of reduction theory for connections over the formal punctured disc. Our proofs apply to arbitrary connected linear algebraic groups over an algebraically closed field of characteristic 0. We also state…

We survey different topologizations of the set $\mathcal{S}(G)$ of all closed subgroups of a topological group $G$ and demonstrate some applications in Topological Grous, Model Theory, Geometric Group Theory, Topological Dynamics.

In this paper we explore a new method of analysis of associative algebras.

By analytic deformations of complex structures, we mean perturbations of the Dolbeault operator. By algebraic deformations of complex structures, we mean deformations of holomorphic glueing data. For complex manifolds there is,…

We study actions of linear algebraic groups on finite-dimensional central simple algebras. We describe the fixed algebra for a broad class of such actions.

We study groups, exponential groups and ordered groups equipped with valuations. We investigate algebraic and topological features of such valued structures, and apply our findings in order to solve regular equations over groups using…

In this work we deal with left invariant complex and symplectic structures on simply connected four dimensional solvable real Lie groups. We search the general form of such structures, when they exist and we make use of this information to…

In this paper we look into the structure of finite-dimensional graded superalgebras of various types such as associative, Lie and Jordan over an algebraically closed field of characteristic zero.

In this work, the complex Lie affgebra structures on three-dimensional solvable Lie algebras are completely determined.