Thesis Defense | Jiaqi Zang (Northwestern University)

Title: Reifenberg Theorem for Locally Finitely Almost Splitting Sets

Abstract: The well-known Reifenberg theorem states that if a subset of R^n can be well approximated by k-planes at every point and every scale, then it is biHölder homeomorphic to a k-disk. This article concerns a subset S of R^n which can be approximated by at most N parallel k planes at each point and scale.  As a subset of R^n such an S may be quite degenerate; S may clearly not be homeomorphic to a disk, and indeed we will see may not be homeomorphic to a union of disks.  However, we prove that S is still the image of a multivalued map on R^k, which is itself a biHölder homeomorphism of the disk into the set of subsets of R^n.

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