Hausdorff Dimension and Quasisymmetric Mappings

Refreshments will be served at 4:00 PM.

ABSTRACT

Hausdorff dimension is a standard notion of dimension for arbitrary metric spaces. Quasisymmetric maps are a natural generalization of conformal mappings in complex analysis to the setting of metric spaces. In this talk, we examine the interaction of these two concepts. A problem of wide interest is to determine when the Hausdorff dimension of a metric space can be lowered by applying a quasisymmetric map. For some spaces (e.g., Sierpinski carpets and gaskets) this can be done; for others (e.g., many product spaces) this is not possible. As our main result, we show how to construct quasisymmetric maps that lower dimensions for positive-measure subsets of R^n and give applications of this construction

BIOGRAPHY

Matthew Romney is a Lecturer in the Department of Mathematical Sciences at Stevens. He received his Ph.D. in Mathematics from the University of Illinois under the direction of Jeremy Tyson. Dr. Romney works in the areas of geometry and analysis in metric spaces and complex analysis. His research is supported by the National Science Foundation.

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