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Contact geometry, in dimension three, lies at the
intersection of many fields: symplectic geometry, 3- and 4-dimensional
topology, CR-geometry, dynamics, etc. Contact structures come in two
flavors: tight contact structures which tend to reflect the ambient
geometry of the 3-manifold, and overtwisted contact structures which are
flabby (homotopic in nature) and are relatively well-understood. In
this talk, I will explain the dichotomy of 3-manifolds that carry
finitely many tight contact structures vs. those that carry infinitely
many. This is joint work with Vincent Colin and Emmanuel Giroux.
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