Francisco Arana-Herrera

How to reach me

• E-mail: farana at umd dot edu.
• Office: 4311 Kirwan Hall, 4176 Campus Drive, College Park, MD 20742, USA.

About me

I am a Brin Postdoctoral Fellow at the Department of Mathematics of the University of Maryland, College Park.
I was a member of the School of Mathematics at the Institute for Advanced Study.
I was a PhD student at Stanford University advised by Alex Wright (University of Michigan) and Steve Kerckhoff (Stanford University).
I am particularly interested in Teichmüller theory, hyperbolic geometry, and dynamics on moduli spaces of Riemann surfaces.

Preprints and publications

1. The boundary of complex totally geodesic submanifolds of Teichmüller space.
   (with Alex Wright). In preparation.
   
2. Closed geodesics on surfaces: topology, geometry, arithmetic. (pdf)
   (with Pouya Honaryar). Preprint.
   
3. Weak mixing rational billiards. (pdf, arXiv)
   (with Jon Chaika and Giovanni Forni). Preprint.
   
4. From limit theorems to mixing limit theorems. (pdf, arXiv)
   (with Giovanni Forni). Preprint.
   
5. The distribution of critical graphs of Jenkins-Strebel differentials. (pdf, arXiv)
   (with Aaron Calderon). Preprint.
   
6. The shapes of complementary subsurfaces to simple closed hyperbolic multi-geodesics. (pdf, arXiv)
   (with Aaron Calderon). Preprint.
   
7. Effective count of square-tiled surfaces with prescribed real and imaginary foliations in connected components of strata. (pdf, arXiv)
   Ergodic Theory and Dynamical Systems, to appear.
   
8. Counting problems from the viewpoint of ergodic theory: from primitive integer points to simple closed curves. (pdf, arXiv)
   Ergodic Theory and Dynamical Systems, to appear.
   
9. The asymmetry of Thurston's earthquake flow. (pdf, arXiv, journal)
   (with Alex Wright). Geometry & Topology, to appear.
   
10. Bounds on the Hausdorff dimension of Teichmüller horocycle flow orbit closures. (pdf, arXiv, journal)
    International Mathematics Research Notices, 2022.
    
11. Effective mapping class group dynamics III: Counting filling closed curves on surfaces. (pdf, arXiv)
    Cambridge Journal of Mathematics, to appear.
    
12. Effective mapping class group dynamics II: Geometric intersection numbers. (pdf, arXiv)
    Preprint.
    
13. Effective mapping class group dynamics I: Counting lattice points in Teichmüller space. (pdf, arXiv, journal)
    Duke Mathematical Journal, 2023.
    
14. Counting hyperbolic multi-geodesics with respect to the lengths of individual components and asymptotics of Weil-Petersson volumes. (pdf, arXiv, journal)
    Geometry & Topology, 2022.
    (Part 1 and Part 2 of a talk I gave on this paper at the Nearly Carbon Neutral Geometric Topology Conference, Online, June 2020).
15. Equidistribution of families of expanding horospheres on moduli spaces of hyperbolic surfaces. (pdf, journal)
    Geometriae Dedicata, 2021.
    
16. Square-integrability of the Mirzakhani function and statistics of simple closed geodesics on hyperbolic surfaces. (pdf, arXiv, journal)
    (with Jayadev Athreya). Forum of Mathematics, Sigma, 2020.
    
17. Counting square-tiled surfaces with prescribed real and imaginary foliations and connections to Mirzakhani’s asymptotics for simple closed hyperbolic geodesics. (pdf, arXiv, journal)
    Journal of Modern Dynamics, 2020.
    (Video of a talk I gave on this paper at the Flat Surfaces and Dynamics on Moduli Space II Workshop, Oaxaca (BIRS), May 2019).

Notes

1. Normalization of Thurston measures on the space of measured geodesic laminations. (pdf)
   Preprint.