Abstract:

One of the features of undergraduate and even graduate education is that everything is divided into compartments -- courses and those annoying qualifying exams, too, are on specific subjects , and this division comes to seem, perhaps, an aspect of mathematics itself, rather than a curricular convenience. But in fact some of the most interesting and important mathematics of modern times has come from interaction between these nominally separated subjects. In this talk, I am going to try to explain how ideas from geometry have been used in complex analysis and how in turn these applications of geometric ideas have led to interesting new areas of investigation in geometry. This is a large subject ,to say the least. To cut it down to a size negotiable in a one- hour lecture , I shall be discussing primarily the case of complex dimension one, that is, complex analysis of one variable. There the connections between geometry and the purely analytic concepts can be described very explicitly. In particular, the complex structure of the unit disc in the complex plane is (relatively) easily seen to be associated to a geometry which is different from but not all that much different from ordinary plane geometry. This alternative geometry is what one obtains by modifying Euclid’s axioms in a simple way. Namely, one gives up the uniqueness part of the "parallel postulate" [that there is exactly one parallel to a given line through a given point not on the line]. In the new geometry, called "hyperbolic geometry", there are in fact infinitely many such parallels. The real point here ,however, is not so much the specific form of the geometry as just that there is some geometry that arises from the analytical situation. This connection between complex function theory and geometry holds in a wide context about which I shall try to give at least an outline at the end of the lecture.