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\textbf{SPEAKER}:
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Chloe Wawrzyniak, University of Kentucky
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{\bf TITLE}:
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On the Hull(s) of the $n$-sphere in $\mathbb{C}^n$
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{\bf ABSTRACT:}\\ 
This talk presents a result on the (global) Bishop problem for small perturbations of the unit sphere of $\mathbb{C}\times\mathbb{R}^{n-1}$ in $\mathbb{C}^n$. We show that if $S\subset\mathbb{C}^n$ is a sufficiently small perturbation of this sphere (in the $C^3$-norm), then $S$ bounds an $(n+1)$-dimensional ball $M\subset\mathbb{C}^n$ that is foliated by analytic disks attached to $S$. Furthermore, if $S$ is either smooth or real analytic, then so is $M$ (up to its boundary). Finally, if $S$ is real analytic (and satisfies a mild condition), then $M$ is both the envelope of holomorphy and the polynomially convex hull of $S$. This generalizes the previously known case of $n=2$ (CR singularities are isolated) to higher dimensions (CR singularities are nonisolated). This work is from a joint paper with Purvi Gupta, and the talk is intended to be accessible to graduate students.









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