We develop new techniques for understanding surfaces in
via bridge trisections. Trisections are a novel approach to smooth
–manifold
topology, introduced by Gay and Kirby, that provide an avenue to apply
–dimensional tools
to
–dimensional
problems. Meier and Zupan subsequently developed the theory
of bridge trisections for smoothly embedded surfaces in
–manifolds. The
main application of these techniques is a new proof of the Thom conjecture, which posits that
algebraic curves in
have minimal genus among all smoothly embedded, oriented surfaces in their
homology class. This new proof is notable as it completely avoids any gauge theory or
pseudoholomorphic curve techniques.
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