Computers and Mathematics: Applications of computers to integrable systems methods for constructing constant mean curvature surfaces

This article is about using computer algorithms and graphics in the study of constant mean curvature surfaces. The advantages of using computers in surface theory is clear, as it helps one to visually see the geometric behavior of the objects of study in a way that is not possible in most fields of mathematics. But on a more fundamental level it is about turning an infinite dimensional problem (that a computer can never truly solve) into a finite dimensional one (that is easily solved with a computer), an idea that applies equally well in a great variety of mathematical fields, and is in no way unique to surface theory. Think of a simple first order smooth ordinary differential equation with a given initial condition. If you cannot write down its solution explicitly, you might think about finding a discrete approximate solution by using the Euler algorithm or Runga-Kutta algorithm, just to have some initial idea how the smooth solution behaves. In this case, your interest in the approximate solution is only as a stepping stone for understanding the smooth true solution. We can think of the equation (i.e. the algorithm) for the discrete approximate solution as a ”finite dimensional” problem because the full space of objects (a vector space of discrete functions) that can be inserted to test for validity in the equation is finite dimensional. Likewise, we can call the smooth differential equation an ”infinite dimensional” problem, because the objects insertable into the equation form an infinite dimensional vector space. This is a somewhat unconventional way to use the expressions ”finite dimensional” and ”infinite dimensional”, but some geometers do use these expressions in this way in conversations, although generally not in papers they write. Or you might instead look at a related ordinary difference equation, with little concern that the resulting discrete solution approximates the smooth solution, and rather be more concerned that the difference equation maintains some property found in the smooth differential equation that you deem important. In this case, as your primary interest is the ”finite dimensional” difference equation situation itself, you might discard the smooth equation altogether, or you might acknowledge the existence of the smooth equation but regard it only as an incidental limiting case of the difference equation you care much more about. Both approaches are of interest, but for clearly different reasons, and are philosophically quite separate, although both clearly benefit from the existence of computers. Both are now common in surface theory, though usually involving partial differential equations, not ordinary ones. Regarding the second approach, finding discrete analogs of smooth objects has recently become an important theme in mathematics, appearing in a variety of places in analysis and geometry. So it