Siegel-Veech Constants of Cyclic Covers of Generic Translation Surfaces

Date: 
Wednesday, 15 January, 2025 - 11:00 - 12:00
Venue: 
AB1 502A
Seminar Type: 
Seminar
Speaker Name: 
Prof. David AULICINO
Affiliation: 
Brooklyn College
Abstract: 

We consider generic translation surfaces of genus g>0 with marked points and take covers branched over the marked points such that the monodromy of every element in the fundamental group lies in a cyclic group of order d. Given a translation surface, the number of cylinders with waist curve of length at most L grows like L^2. By work of Veech and Eskin-Masur, when normalizing the number of cylinders by L^2, the limit as L goes to infinity exists and the resulting number is called a Siegel-Veech constant. The same holds true if we weight the cylinders by their area. Remarkably, the Siegel-Veech constant resulting from counting cylinders weighted by area is independent of the number of branch points n. All necessary background will be given and a connection to combinatorics will be presented. This is joint work with Aaron Calderon, Carlos Matheus, Nick Salter, and Martin Schmoll.