Quartic curves and their bitangents

Plane quartic curves appear in many different areas and over 150 years of mathematics. This theory began with the beautiful combinatorics of the 28 bitangents of a quartic curve. A classical theorem of Hilbert states that every positive ternary quartic is a sum of squares. Using their bitangents, we will represent positive quartics as the sum of three squares in 8 different ways and start to understand the 6-dimensional spectrahedron of all sums of squares representations. This is joint work with Daniel Plaumann and Bernd Sturmfels appearing in arXiv:1008.4104.