I am interested in Geometric Topology, Geometric Group Theory, Ergodic Theory and Dynamics. In particular; 

• Mapping class groups, Teichmüller spaces, Moduli spaces, Geodesic currents on surfaces

• Out(FN), Culler-Vogtmann's Outer Space, Geometry of free factor and free splitting complexes, Geodesic currents on free groups

• Veech Surfaces, Quadratic and Abelian Differentials, Dynamics on translation surfaces

• Substitutions and symbolic dynamics 

• Random walks on groups and spaces, hyperbolic groups

• Kleinian groups, representations of free and surface groups

 Papers:

• Random walk speed is a proper function on the Teichmüller space (with A. Azemar, V.Gadre, S. Gouëzel, T. Haettel, P. Lessa) submitted, 2023. [pdf] [arxiv]

• Slope gap distribution of saddle connections for the 2n-gon (with J. Berman, T. McAdam, A. Miller-Murthy, H.Wan)
  Discrete and Contin. Dyn. Syst. 43 (2023), no.1, 1-56.  [pdf] [arxiv] [journal]

• Exceptional directions for the Teichmüller geodesic flow and Hausdorff dimension (with H. Al-Saqban, P. Apisa, A. Erchenko, O. Khalil, S. Mirzadeh)
  J. Eur. Math. Soc. 23 (2021), 1423-1476. [pdf] [arxiv] [journal]

• Atoroidal dynamics of subgroups of Out(FN) (with M. Clay)
  J. London Math. Soc., 102 (2020), 818-845. [pdf] [arxiv] [journal]

• Length functions on currents and applications to dynamics and counting (with V. Erlandsson)
  In the Tradition of Thurston, Springer (2020), 423-458 [pdf] [arxiv] [book]

• Hyperbolic extensions of free groups from atoroidal ping-pong
  Algebr. Geom. Topol. 19 (2019), no.3, 1385-1411. [pdf] [arxiv] [journal]

• Simultaneous construction of hyperbolic isometries (with M. Clay)
  Pacific J. Math. 294 (2018), no. 1, 71-88. [pdf] [arxiv] [journal]

• Perron-Frobenius theory and frequency convergence for reducible substitutions (with M. Lustig)
  Discrete Contin. Dyn. Syst. 37 (2017), no. 1, 355-385. [pdf] [arxiv] [journal]

• North-South dynamics of hyperbolic free group automorphisms on the space of currents (with M. Lustig)
  J. Topol. Anal. 11 (2019), no. 2, 427-466. [pdf] [arxiv] [journal]

• The distribution of gaps for saddle connections on the octagon (with G. Work)
  Int. Math. Res. Notices, 18 (2016), 5569-5602.  [pdf] [arxiv] [journal]

• Dynamics of hyperbolic iwips
  Conform. Geom. Dyn. 18 (2014), 192-216. [pdf] [arxiv] [journal]

• Generalized north-south dynamics on the space of geodesic currents
  Geom. Dedicata, 177 (2015), 129-148. [pdf] [arxiv] [journal]

Collaborators / Mentors

• Hamid Al-Saqban (Paderborn, Germany)

• Paul Apisa (Wisconsin)

• Aitor Azemar (Glasgow)

• Jayadev Athreya (Washington)

• Matt Clay (Arkansas)

• Spencer Dowdall (Vanderbilt)

• Vaibhav Gadre (Glasgow) 

• Alena Erchenko (Dartmouth )

• Viveka Erlandsson (Bristol)

• Sébastian Gouëzel (Rennes)

• Thomas Haettel (Montpellier)

• Ilya Kapovich (CUNY)

• Osama Khalil (UIC)

• Chris Leininger (Rice) 

• Pablo Lessa (IMERL)

• Martin Lustig (Aix- Marseille)

• Joseph Maher (CUNY)

• Taylor McAdam (Yale) 

• Yair Minsky (Yale) 

• Shahriar Mirzadeh (Brandeis)

• Catherine Pfaff (Queens)

• Grace Work (Wisconsin)

Students 

• Yandi Wu (PhD student at UW Madison, co-advisor)

• Catherine Lee (Senior thesis advisee at Yale, currently at UC Berkeley)

• Jonah Berman (REU student at Yale)

• Ananth Miller-Murthy (REU student at Yale)

• Hamilton Wan (REU student at Yale)

I also supervise several other undergraduates through the Madison Experimental Mathematics Lab.