Inscribing rectangles in Jordan curves and random rotations of surfaces in 4-manifolds

Inscribing rectangles in Jordan curves and random rotations of surfaces in 4-manifolds

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Cole Hugelmeyer, Princeton University
Fine Hall 314

The inscribed rectangle problem for Jordan curves has some interesting relationships with the geometry of surfaces in 4-manifolds. We will give a proof that every smooth Jordan curve has at least one third of all possible aspect ratios among its inscribed rectangles. This proof uses an unexpected total ordering which exists for certain collections of embedded Möbius strips in a 4-manifold. The following question will naturally arise: Given a 4-manifold with the structure of a principal S^1-bundle, and a surface embedded in that manifold, among all possible isotopies of that surface, what is the minimal probability that it intersects its rotation by a uniformly random element of S^1? We conjecture that this minimal probability is always either zero, one, or one third.