The question of geometric realization is whether or not a particular

algebraic object has a corresponding geometric analogue.

This time the resulting

algebraic object is called a local role algebra.

We say that some

algebraic objects defined over an infinite global field k locally have a property P if for all places v [member of] [V.sub.k], considered as object over k(v), it has property P.

He takes a quantum groups approach (rather than Chern-Simons field theory or 2-dimensional conformal field theory) in order to derive invariants of knots and 3-manifolds from

algebraic objects which formalize the properties of modules over quantum groups at roots of unity.

Algebraic insight can become a focus of classroom conversations which help students get an overall feeling of

algebraic objects. For example, a teacher might routinely start discussion of a question by looking at the overall structure of an expression or equation, pointing out what are the main structural groups, what is varying and what is staying the same.

To inspire researchers at all levels to investigate algebraic questions posed by design theory, they provide a large selection of the

algebraic objects and applications to be found in design theory.

His topics include quantum angular momentum, including combinatorial features, composite systems, including binary coupling theory, graphs and adjacency diagrams, including nonisomorphic trivalent trees and cubic graphs, generating functions, the form of certain polynomials, operator actions in Hilbert space, structure of certain polynomials, the general linear and unitary groups, tensor operator theory, basic

algebraic objects, and combinatorial objects.