I am going to tell about one of important applications of ideas of soliton theory to complex analysis of Riemann surfaces. It has been used, since several years ago, for the construction of operator fields in string theory by Krichever and myself. Since then other people have been working here. The idea is originated from the study of discrete systems in soliton theory. The most well-known disc...

An edge detection scheme is developed robust enough to perform well over a wide range of signal-to-noise ratios. It is based upon the detection of zero crossings in the output image of a nonlinear Laplace filter. Specific characterizations of the nonlinear Laplacian are its adaptive orientation to the direction of the gradient and its inherent masks which permit the develop ment of approximatel...

The Laplace–Beltrami operator (LBO) is the fundamental geometric object associated with manifold surfaces and has been widely used in various tasks in geometric processing. By understanding that the LBO can be computed by differential quantities, we propose an approach for discretizing the LBO on manifolds by estimating differential quantities. For a point on the manifold, we first fit a quadra...

In the context of synthetic differential geometry, we study the Laplace operator an a Riemannian manifold. The main new aspect is a neighbourhood of the diagonal, smaller than the second neighbourhood usually required as support for second order differential operators. The new neighbourhood has the property that a function is affine on it if and only if it is harmonic.