Abstract: Classifying proper holomorphic maps between unit balls in higher dimensions has been an active area of research in Complex Analysis. In this talk, we introduce several homotopy equivalence relations for proper holomorphic mappings between balls. We show that for proper holomorphic maps from balls of dimension at least 2 to higher dimensions, the set of homotopy classes is finite. By contrast, when the target dimension is at least twice the domain dimension
there are uncountably many spherical equivalence classes. A generalization of this result is obtained by showing arbitrary homotopy of rational maps whose endpoints are spherically inequivalent must contain uncountably many spherically inequivalent maps. Time permitting we introduce the notion of ‘Whitney Sequences’ in higher dimensions which is an analogue of finite Blaschke products
in one dimension. The talk is largely based on a paper by D’Angelo and Lebl.
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