9224464 Viro The first part of this project is devoted to quantum invariants of low dimensional topological objects. It seeks to give a purely combinatorial construction of known quantum invariants and to look for non-trivial topological quantum field theories not related to quantum groups; to study polynomial link invariants defined by a construction similar to Seifert's construction for the Alexander module, but with homology replaced by any topological quantum field theory; to develop link theory in special 3-manifolds like projective space; to study Vassiliev invariants and their generalizations; and to investigate the possibility of combinatorial construction of 4-manifold invariants. The second part is devoted to realizing bistellar transformations of smooth triangulations. The third part is related to the topology of real algebraic varieties. It involves investigation of the relationship between special classes of real algebraic varieties and their topological models and an attempt to study the topology of real algebraic knots in 3-dimensional projective space. The main idea of this project is to develop the topological side of quantum topology, a theory which has flowered recently as a result of the introduction of modern physics into low-dimensional topology. The effect of this progress on topology needs to be better understood. The new tools deserve more detailed study and systematic application to geometric problems. They also provide a new opportunity to study topological properties of real algebraic varieties. Of course, traditional methods are to be used together with the new ones. ***
