Approximately differentiable a.e. homeomorphisms might have surprising properties. For example, Goldstein and Hajłasz constructed a homeomorphism of the unit cube, which coincides with identity on the boundary and whose approximate derivative equals the reflection matrix a.e. In this talk, I will show that under some mild assumptions on a map $T: [0,1]^n \to GL(n)^+$, it is possible to find a homeomorphism of the unit cube which is approximately differentiable a.e. on an arbitrarily large set and whose derivative equals $T$. I will also present the connections between this problem and the Oxtoby-Ulam theorem about homeomorphic measures and the Dacorogna-Moser theory of mappings with prescribed Jacobian.

This is work in progress with Paweł Goldstein and Piotr Hajłasz.
Meeting ID: 850 0118 0234 Passcode: 017052