Title: Improvements to the Brill-Noether Theorem

Abstract: In 1980 Griffiths and Harris proved what is known as the "Brill-Noether Theorem," which essentially says that for a general curve C of genus g that the dimension of a variety of special linear series on C is precisely equal to the Brill-Noether number of that variety. However, it is also known that a general curve C of genus g must have a particular gonality k, so the next natural question to ask is, "Can we compute the dimension of a variety of special linear series on a general curve C of genus g with specified gonality k?" This week we will see a result from Nathan Pfleuger that says we can, at least, bound the dimension above via a modification to the Brill-Noether number using recent results from Tropical Geometry.