Problems in Function Theory

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Abstract: Garnett will work on several problems in classical one dimensionalcomplex analysis. The first problem is to approximate any Blaschke productuniformly on the open disc by Blaschke products whose zeros are sufficiently spread apart and thin that the corresponding Riesz mass isbounded in all holomorphic coordinate systems (i. e. is a Carleson measure)The approximation should be effected using explicit constructions.The second problem is to exhibit large compact sets whose complementsdo not support nonconstant bounded analytic functions. The third problemis a corona problem for infinitely connected plane domains whose boundarieslie on rectifiable Jordan curves. It too requires some new explicit constructions. The fourth problem is to show that a Jordan curve depends continuously on its welding map, which is the correspondence on theunit circle that connects the conformal mapping of the inside of thedisc to one side of the curve and the outside of the disc to the other side of the curve.The methods to be used on these problems will be constructive so that theycan be give explicit computer aided constructions of analytic functionsand conformal mappings. Analytic functions and conformal mappings havebroad applications in fluid dynamics, acoustics, and electrical engineering,and in these applications constructions are more useful than general existence theorems.