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:
Bounded projections to the Z-factor graph (with M.Clay), preprint, 2023 [pdf][arxiv]
Random walk speed is a proper function on the Teichmüller space (with A. Azemar, V. Gadre, S. Gouëzel, T. Haettel, P. Lessa)
J. Mod. Dyn. 19 (2023), 815-832. [pdf] [arxiv][journal]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]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]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]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]
PhD Students
Yandi Wu (PhD, 2024) First position: Lovett Instructor at Rice
Rongsong (Ron) Yang (PhD student at UW Madison)
Collaborators
Yassine Guerch (ENS Lyon)
Radhika Gupta (TATA)
Sébastian Gouëzel (Rennes)
Thomas Haettel (Montpellier)
Sanghoon Kwak (KIAS)
Pablo Lessa (IMERL)
Marissa Loving (Wisconsin)
Joseph Maher (CUNY)
Jean Pierre Mutanguha (Princeton)
Yandi Wu (Rice)
Undergraduate/ Master's Students
Hart Easley (MA at UW Madison)
Katerina Stuopis (Senior thesis advisee and Hilldale fellow at UW Madison)
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, currently at MIT)
Michael Beers (Summer undergraduate research at UW Madison, 2023)
I also supervise several other undergraduates through the Madison Experimental Mathematics Lab.