I am interested in differential topology, and in particular, in the global theory of singularities of differentiable mappings. It has been known that differentiable functions on a manifold can be well used to study its global geometric structures. In the 1950’s R. Thom began to try to generalize such a theory to that of differentiable mappings between manifolds. However, because of the difficulty in controlling local singularities, the theory has not been well developed until recently. I am thus studying differentiable mappings between manifolds with only mild singularities or those between low dimensional manifolds. This kind of global study of singularities is fairly new and my recent results have shown that the singularities of differentiable mappings play an essential role in the study of geometric structures of manifolds. In this way, it has been recognized that such a study is important in Topology.

Other than the above mentioned research, I am also interested in the following vast area of Topology and related fields: primary obstruction to topological embeddings, separation properties of codimension 1 maps, topology of complex isolated hypersurface singularities, fibered knots, 4-dimensional manifolds, codimension 1 embeddings, differential geometric invariants of space curves, unknotting numbers of knots, etc. I am also interested in the asymptotic behavior of generalized Fibonacci sequences. Furthermore, I am interested in the application of Topology to other areas in Science and Industry, such as DNA knots, visual data analysis for multivariate functions, analysis of materials from microscopic levels, etc.

Keywords | Topology, Singularity Theory, Differential Topology, DNA Knots |
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Division | Division of Fundamental Mathematics |

Links | Homepage |