Mathematical Sciences: Problems in Complex Analysis

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The central focus of this mathematical research concerns problems in complex function theory and quasiconformal mappings. One of the primary objectives of the work is that of classifying certain groups of homeomorphisms of Euclidean space. Among the best understood groups are the so-called Mobius groups, those groups whose elements are represented by simplest linear (fractional) transformations. The classification problem is one of determining which other groups are topologically conjugate to a Mobius group and to understand what the obstruction is that might prevent such a group isomorphism. Among the target groups is one known as the convergence group which has many properties of groups of quasiconformal mappings. Many convergence groups are known to be conjugate to Mobius groups, but to date, no complete classification has been found. Work will be done in an effort to determine the extent to which these groups all enjoy the same properties relative to Mobius groups. A second line of investigation concerns injectivity criteria for locally univalent functions. This work seeks conditions which guarantee the extension of locally univalent meromorphic mappings from neighborhoods of a domain to a quasiconformal map of the entire complex plane. The extension is to be carried so that the complex dilatation remains bounded by a fixed amount. There is a sizeable literature concerning questions of this type. One specific objective of this work is to remove the condition that the meromorphic function be holomorphic in a neighborhood of a domain and only assume this property within a closed domain. If one drops the condition that the initial function be meromorphic, then the extension problem becomes too general. However, a natural condition on the cross-ratio of a complex function within a disc leads to a more restrictive class where quasiconformal extensions (the injectivity problem) have been shown to exist. Work will continue in determining precise conditions on cross-ratios which will guarantee injectivity. //