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Although theory of analytic differential equation is often viewed as being far from “Mathematics for Industry”, recent research has indicated that there are some unexpected connections between the two, which are described below. The theory of Schwarz mapping, which Schwarz began in the late 19th century, has provided geometrical meaning to Euler-Gauss hypergeometric differential equations. This theory has been further developed into complex higher dimensional ones throughout the 20th century, and research still continues today. Hence, this is a field where the geometry of symmetric space, automorphic form, algebraic geometry, and differential equations influence each other.

About four years ago, I proposed hyperbolic Schwarz mapping as an improved version of Schwarz mapping. In particular, I have tried to stress its advantage, that it is a good surface with its target in three-dimensional hyperbolic space. The produced image is a kind of surface which is called a flat front. Special differential equations represented by hypergeometric differential equations and special function of solutions can be “seen” as a surface.Software such as Mathematica and Maple has been used to visualize these surfaces, but obtaining a detailed view is difficult. Drawing hyperbolic Schwarz mapping using software is done with approximate values at grid points on the original plane. Therefore, the finer the grid, the more time consuming the process is, and achieving a better degree of approximation increases the process time. After all, there is no way to draw an image without discretization, and hence, a smart discretization is highly desired.

Software such as Mathematica and Maple has been used to visualize these surfaces, but obtaining a detailed view is difficult. Drawing hyperbolic Schwarz mapping using software is done with approximate values at grid points on the original plane. Therefore, the finer the grid, the more time consuming the process is, and achieving a better degree of approximation increases the process time. After all, there is no way to draw an image without discretization, and hence, a smart discretization is highly desired.

Thus, the question becomes, “What is a smart discretization?” Smart discretization is not an approximation of a smooth model to grid points or making a difference equation out of a given differential equation without policy, but it is a discretization that clarifies what properties (invariants) of a model (differential equation) to preserve. In our hyperbolic Schwarz mapping, we aim to preserve the shape of singularity at a finite plane and asymptotical behavior at infinity.

Having learned that there is a field of visualization and discrete differential geometry with the above consciousness, my group has recently begun a collaborative research project with Takeshi Sasaki (formerly at Kobe University) and Kotaro Yamada (Kyushu University), who specialize in differential geometry, and Wayne Rossman (Kobe University) and Tim Hofflamnn (Kyushu University), who specialize in discrete differential geometry.

This collaborative effort is an (unexpected) encounter of special differential equations (typical example: hypergeometric differential equation) and visualization. This work should provide a wealth of examples to discrete differential equations where there have been a plethora of general theories, but few interesting examples.

Hence, a smart discretization is nearing completion for hyperbolic Schwarz mapping of Airy equations, which is an extremely special confluent form of hypergeometric differential equations. Please see the images at the end of this article. One of our future projects will be discretization of hyperbolic Schwarz mapping of (general) hypergeometrical differential equations and their confluent equations.

Hyperbolic Schwarz mapping generally has singularities called cuspidal line and swallowtails. To date, research has not yet been conducted on the singularity of discrete surfaces. Although a general theory on discrete singularities will probably not be very fruitful, hyperbolic Schwarz mapping appears to have particularly good qualities, and so we continue to work hard to elucidate these qualities. Moreover, even though surfaces with singularities may appear to have few practical applications, understanding these singularities will provide insight on surfaces, because singularities are points with a larger amount of information (invariants) than ordinary points. Thus, with a certain transformation, a hyperbolic Schwarz surface can be transformed into a mean curvature constant surface, which makes it promising for industrial applications.