# Composing and decomposing surfaces in $\mathbb{R}^n$ and $\mathbb{H}_n$

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Robert Young, New York University

How do you build a complicated surface?  How can you decompose a surface into simple pieces?  Understanding how to construct an object can help you understand how to break it down.  In this talk, we will present some constructions and decompositions of surfaces based on uniform rectifiability.  We will use these decompositions to study problems in geometric measure theory and metric geometry, such as how to measure the nonorientability of a surface and how to optimize an embedding of the Heisenberg group into $L_1$ (joint with Assaf Naor).