The proposer plans to investigate purely algebraic analogs of the theory of the characteristic submanifold of a Haken 3-manifold. Some results in this direction have already been obtained by several authors but their uniqueness results are much weaker than that in the 3-manifold context. It is expected that such results should also yield new understanding of the purely topological theory of 3-manifolds.
The universe appears to be 3-dimensional and it is a very interesting question to understand its shape. There is an abstract theory of spaces called 3-manifolds which attempts to describe all the possibilities for the shape of such a space. While this theory is not complete, there is a great deal known. In particular, many (and perhaps all) such spaces can be decomposed into simple geometric pieces in a natural way. In this proposal, it is planned to investigate further the properties of this decomposition into geometric pieces.
