Title: Whitney’s extension theorem for curves in the Heisenberg group
Speaker: Scott Zimmerman (The Ohio State University)
Speaker's URL: https://www.zimmermanmath.com/
Abstract: Given a compact set $K$ in $\mathbb{R}^n$ and a continuous, real valued function $f$ on $K$, when is there a $C^{m,\omega}$ function $F$ defined on $\mathbb{R}^n$ such that $F|_K = f$? ($C^{m,\omega}$ is the space of $C^m$ functions whose $m$th order derivatives are uniformly continuous with modulus of continuity $\omega$.) Whitney famously answered this question in 1934 when extra data is provided on the derivatives of the extension on $K$, and less famously answered the original question for subsets of $\mathbb{R}$. The question was answered in full by Charles Fefferman in 2009.
The Heisenberg group $H$ is $\mathbb{R}^3$ with a sub-Riemannian and metric structure generated by a class of admissible curves. We will consider Whitney's original question for curves in $H$: given a compact set $K$ in $\mathbb{R}$ and a continuous, map $f:K \to H$, when is there a $C^{m,\omega}$ admissible curve $F$ such that $F|_K = f$? I will present a project with Andrea Pinamonti and Gareth Speight in which we address this question.
URL associated with Seminar: https://u.osu.edu/aots/
