Masato Tanabe

I am interested in singularity theory of (differentiable) maps, particularly from a topological viewpoint.

Singularities of maps are concepts corresponding to critical points of functions or to apparent contours of objects under projection onto the retina.
By focusing on such local data associated with these singularities, one can extract global information such as the shape of the graph of a function or an object.
A guiding principle in singularity theory is based on these facts and stated as follows:
"to recover global information (topological invariants) of a given space from the singularities of maps defined on it."

Recently, I have been studying a relative version of the above idea:
"to recover invariants of a given map from the singularities of its extensions."
In fact, it is known that invariants of nonsingular maps, such as immersions and embeddings, can often be described in terms of the singularities of their extensions.
My aim is to develop a theoretical framework underlying these phenomena and also to apply it to broader classes of maps and to geometric (or physical) objects, thereby clarifying what singularities detect.