Contact type hypersurfaces in R^4

Contact type hypersurfaces in R^4

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Thomas Mark, University of Virginia

Zoom link:  https://princeton.zoom.us/j/453512481?pwd=OHZ5TUJvK2trVVlUVmJLZkhIRHFDUT09

Given a closed, orientable 3-manifold M, it is a subtle and often difficult problem to determine whether M may be smoothly embedded in R^4. Even among integer homology spheres, and restricting to special classes such as Seifert manifolds, the problem is open in general, with positive answers for some such manifolds and negative answers in other cases. However, recent work shows that if suitable geometric conditions are imposed then there is a uniform answer for an important class of 3-manifolds called Brieskorn homology spheres: no such 3-manifold admits an embedding as a hypersurface "of contact type" in R^4, which is to say as the boundary of a region that is convex from the point of view of symplectic geometry. I'll describe further context and background for this result, which is joint work with Bülent Tosun, and give some indication of the proof.