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I will begin with a brief introduction to tropical geometry, and ex-
plain how algebraic curves give rise to tropical curves. I will then show that
every tropical plane quartic admits 7 families of bitangent lines. This is analo-
gous to the remarkable fact in classical geometry that a smooth plane quartic
has exactly 28 bitangent lines. While the proof is purely combinatorial, I will
discuss recent developments which suggest that the classical and tropical results
are closely related. This is joint work with Matt Baker, Ralph Morrison, Nathan
Pueger, and Qingchun Ren.
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