Complex approximation on Riemann surfaces and in infinite dimensions

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The topic of this research is approximation theory. It is theoretical in nature, which means that it is potentially applicable to a wide variety of situations, but in order to apply to a particular field, say energy, collaboration will be required between applied mathematicians, physcists and engineers in order to make explicit how such mathematical entities as a, b and c can be interpreted as say force, potential energy and kinetic energy. More specifically, my research involves trying to approximate given functions by nicer ones. The functions we encounter in the world are often too complicated to deal with, so we seek simpler mathematical functions which approximate the real functions of the world. If these simpler functions approximate the real functions well enough, we can make the desired predictions within an acceptable degree of accuracy. For example, if we can make a watch which loses only one second in one hundred years, most people would be satisfied. Such a watch only approximates real time, but extremely well. The nice functions I wish to approximate with are the so-called analytic functions. These are functions f(x), which have the virtue that they can be written in the form of infinite series* f(x) = a + bx + cx2 + . . . ***where x is the variable and the coefficients a, b, c, . . . are constants. Analytic functions have extremely nice properties. For example, their global behaviour is uniquely determined by their local behavior. This means, for example, that if we know that some universal phenomenon behaves analytically, then, in principle, its behavior behind some distant star is completely determined by its behaviour on earth. Thus, (again in principle) we can know the behavior behind the star without going there, provided we can know it sufficiently well right here on earth. *** ***Our approach to approximating a given function g by an analytic function f is a localization process. We first approximate the function g in various local regions by analytic functions in these local regions. Then, we try to piece together these local analytic functions in order to get a global analytic function f which approximates our initial function g. This is hard to do, because the aforementioned uniqueness property of analytic functions makes it very difficult to patch analytic functions together and still retain analyticity. **