Benjamin Dozier
**Benjamin Dozier**
** Email:** bdozier@stanford.edu, benjamin.dozier@gmail.com
**Office:** 380-S, Building 380, 450 Serra Mall, Stanford CA, 94305
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ABOUT
ME

I'm currently working towards 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.

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PREPRINTS

*Equidistribution of saddle connections on translation surfaces,* arXiv: 1705.10847
(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,* arXiv:1701.00175.
**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.