Benjamin Dozier
Benjamin Dozier
Email: benjamin.dozier@gmail.com
Office: 426, Fields Institute, Toronto, Ontario
About Me
I've recently completed my PhD in mathematics at Stanford University.
From July-December 2018, I will be postdoctoral fellow at the Fields Institute in Toronto for the thematic program on Teichmuller theory.
In January 2019, I will start a postdoc as a Simons Instructor at Stony Brook University.
My research is in Teichmuller dynamics, and I have been advised by Maryam Mirzakhani and Alex Wright.
Papers
- Equidistribution of saddle connections on translation surfaces, Accepted to Journal of Modern Dynamics [pdf | arxiv]
(Images illustrating the theorem)
Abstract:
Fix a translation surface X, and consider the measures on X coming from averaging the uniform measures on all the saddle connections of length at most R. Then as R goes to infinity, the weak limit of these measures exists and is equal to the Lebesgue measure on X. We also show that any weak limit of a subsequence of the counting measures on S^1 given by the angles of all saddle connections of length at most R_n, as R_n goes to infinity, is in the Lebesgue measure class. The proof of the first result uses the second result, together with the result of Kerckhoff-Masur-Smillie that the directional flow on a surface is uniquely ergodic in almost every direction.
- Convergence of Siegel-Veech constants, Geometriae Dedicata 2018 [pdf | arxiv]
Abstract:
We show that for any weakly convergent sequence of ergodic SL2(R)-invariant probability measures on a stratum of unit-area translation surfaces, the corresponding Siegel-Veech constants converge to the Siegel-Veech constant of the limit measure. Together with a measure equidistribution result due to Eskin-Mirzakhani-Mohammadi, this yields the (previously conjectured) convergence of sequences of Siegel-Veech constants associated to Teichmuller curves in genus two.
The proof uses a recurrence result closely related to techniques developed by Eskin-Masur. We also use this recurrence result to get an asymptotic quadratic upper bound, with a uniform constant depending only on the stratum, for the number of saddle connections of length at most R on a unit-area translation surface.
Videos of Talks