Division algebras (E. Dieterich)

The notion of a division algebra makes sense over any ground field. It requires only bilinearity of the algebra multiplication and unique solvability of the equations ax = b and ya = b for given algebra elements a and b, a being nonzero. Classical examples of real division algebras are the real number field, the complex number field, the quaternion algebra and the octonion algebra. They play a crucial role in the algebraic theory of spinors, which in turn forms an important part of the general theory of quantum fields and strings. Real division algebras in general are closely related to vector fields on spheres, a classical subject in topology. Division algebras over arithmetic fields are related to the theory of quadratic forms over these fields, a classical subject in algebraic number theory. Our main objective is to classify, up to isomorphism, all division algebras over a given ground field, and to study their deformation theory

Representations of associative algebras (E.Dieterich)

The notion of a representation of an associative algebra with identity element generalizes and unifies various classical algebraic objects such as, for example, matrix representations of a group, several linear operators acting simultaneously on a vector space, or configurations in a vector space given by linear subspaces satisfying a prescribed inclusion diagram. During the last 30 years three originally independent influences (Auslander's and Reiten's almost split sequences, Gabriel's notion of quivers and their representations, Nazarova's and Roiter's matrix techniques) have merged to a synthesis which for representation theory meant a period of revival by opening a surprisingly rich and unexpected computational potential. The philosophy is to study a given associative algebra by studying its category of representations which in turn is expected to reveal internal properties of the algebra. More precisely, the main objective is to classify, up to isomorphism, all indecomposable representations, to describe the irreducible morphisms between them and to understand the combinatorial structure given by this information.

Toric Varieties and Intersection Cohomology (K.-H.Fieseler)

Toric varieties are, roughly speaking, (normal) equivariant compactifications of an algebraic torus (C^*)^n acting on itself via group multiplication. They can be completely described in combinatorial terms, i.e. by finite fans in R^n. Such a fan is a collection of strictly convex polyhedral cones, such that any two of them meet in a common face, and all the faces of a cone in the fan also belong to the fan. The aim of the research in that area is to provide a combinatorial interpretation of the intersection cohomology of a toric variety and to study its algebraic properties. Intersection Cohomology is a cohomology theory particularly adapted to the study of singular varieties: It agrees with usual cohomology for smooth spaces, but satisfies Poincar´e duality in every case.

Infinite-dimensional Lie algebras and quasi-crystals (V.Mazorchuk)

Although there is no uniquely defined notion of a quasi-crystal, in the physical literature one usually uses this notion to describe some aperiodic analogs of crystals. For example, there is a way to construct quasi-crystals using non-crystallographic Coxeter groups. Very recently there appeared a procedure how one can associate infinite-dimensional Lie algebra to certain one-dimensional quasi-crystals. The properties of this Lie algebras and its representations gives some information about the properties of the physical system, described by the quasi-crystal. It is now a very challenging problem to extend this construction to quasi-crystals of higher dimensions. In particular, this should include the famous non-periodic Penrose tiling of the plane.

Representations of Lie algebras, quantum algebras and related associative algebras (V.Mazorchuk)

Lie algebras are algebras with an anti-commutative binary operation, which satisfies the Jacoby identity. Although they were defined to study properties of continuous groups, it was quite soon understood that they have numerous applications to many branches of mathematics, including functional analysis, algebra, number theory, theoretical physics, quantum physics etc. Most of the physical applications of Lie algebras rely on the study of their representations, that is, Lie subalgebras of matrix algebras. Approximately 30 years ago it was understood that many natural classes of these representations can be described in terms of usual associative algebras. This connection was established for classical representations. Recently there have appeared a lot of new examples of representations, important for applications and it is now a very popular problem to connect them to associative algebras and to find out what kind of information can be derived from this connection. The same problems appear also in study of quantum groups.

Transformation semigroups (V.Mazorchuk)

A semigroup is a set equipped with a binary associative operation. There are many examples of finite semigroups, naturally appearing in different branches of mathematics. For instance, the full transformation semigroup, the full inverse symmetric semigroup, the semigroups of all doubly-stochastic matrices and so on. The study of properties of (finite) semigroups naturally leads to beautiful and difficult combinatorial problems. The principal question is usually to find the number of elements, satisfying some natural conditions (for example, the number of nilpotent elements). Even for classical semigroups a lot of questions of this kind still are not answered.