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.