This volume presents the proceedings from the conference on low dimensional topology held at the University of Madeira (Portugal). The event was attended by leading scientists in the field from the U.S., Asia, and Europe. The book has two main parts. The first is devoted to the Poincare conjecture, characterizations of $PL$-manifolds, covering quadratic forms of links and to categories in low dimensional topology that appear in connection with conformal and quantum field theory. The second part of the volume covers topological quantum field theory and polynomial invariants for rational homology 3-spheres, derived from the quantum $SU(2)$-invariants associated with the first cohomology class modulo two, knot theory, and braid groups. This collection reflects development and progress in the field and presents interesting and new results.