—By singularity theory, we treat singularities of various objects and study their mathematical structures. In particular we investigate the structures on the subspace consisting of singular objects, so called the bifurcation set.
—In the 16th problem of Hilbert, we study the topological structure of the zero-set of a real polynomial. Naturally we apply complex algebraic geometry, manifold theory, algebraic topology and singularity theory etc for the solution of the problem.
—One of my themes is to develop the singularity theory in symplectic geometry which is related to classical mechanics and quantum mechanics. Moreover I am interested in unknown area, such as quantum topology, quantum singularity theory etc.
—I was involved in various mathematical works and then recently I have noticed that my interests tend to be gravitated more strongly towards “topology on solution spaces of differential systems” and now I take great delight in working on them.
- G. Ishikawa, Generic bifurcations of framed curves in a space form and their envelopes, Topology and its Applications, 159 (2012), 492–500.
- G. Ishikawa, Infinitesimal deformations and stability of singular Legendre submanifolds. Asian Journal of Mathematics. 9-1 (2005), 133–166.
- G. Ishikawa, Symplectic and Lagrange stabilities of open Whitney umbrellas, Invent. math., 126-2 (1996), 215–234.
—Advise to the students who proceed to the graduate school on mathematics:
When you try to do good mathematical works, you must find good mathematical problems and solve them in a wise manner.
To do so, of course, you need deep and wide knowledge on mathematics
and related areas. However I believe that it is enough to get such knowledge after you feel its real necessity. In a large time scale, I could say that most important thing is the “intellectual curiosity”.
Singularity theory of mappings:
Lagrange and Legendre singularities, Singularity theory of exterior differential systems, Differentiable structures of mapping space quotients, Catastrophe theory, Thom-Mather theory. Symplectic and contact geometry: Characteristic Cauchy problem for Hamilton-Jacobi equations, Sub-Riemannian geometry, Projective and Grassmann duality, Fronts with degenerate Gauss mappings. Real algebraic geometry: Topology of real algebraic varieties, The16th problem of Hilbert, Topology of real rational curves and Fourier (trigonometric) curves, Real algebraic homogeneous spaces, Euler characteristics and Euler obstructions.