Hannah Markwig (Tübingen) 
Counting bitangents of plane quartics  tropical, real and arithmetic
A smooth plane quartic defined over the complex numbers has precisely 28 bitangents. This result goes back to Plücker. In the tropical world, the situation is different. One can define equivalence classes of tropical bitangents of which there are seven, and each has 4 lifts over the complex numbers. Over the reals, we can have 4, 8, 16 or 28 bitangents. The avoidance locus of a real quartic is the set in the dual plane consisting of all lines which do not meet the quartic. Every connected component of the avoidance locus has precisely 4 bitangents in its closure. For any field k of characteristic not equal to 2 and with a nonArchimedean valuation which allows us to tropicalize, we show that a tropical bitangent class of a quartic either has 0 or 4 lifts over k. This way of grouping into sets of 4 which exists tropically and over the reals is intimately connected: roughly, tropical bitangent classes can be viewed as tropicalizations of closures of connected components of the avoidance locus. Arithmetic counts offer a bridge connecting real and complex counts, and we investigate how tropical geometry can be used to study this bridge.

09:30  10:45 
Hannah
Markwig (Tübingen)
Counting bitangents of plane quartics  tropical, real, and arithmetic 

10:45  11:15  Break  
11:15  12:15 
Hannah Larson (Stanford)
BrillNoether theory over the Hurwitz space 

12:15  12:45  Break 
Organizers:
Dan Abramovich (Brown),
Kenny Ascher (MIT),
Asher Auel (Dartmouth),
Dori Bejleri
(Harvard),
Madeline Brandt (Brown),
Melody Chan (Brown),
Yuchen Liu
(Yale),
Kalina Mincheva (Tulane),
Nathan Pflueger (Amherst),
and
Jeremy Usatine (Brown).