Research:

Low-dimensional geometry and topology.

Curriculum Vitae

Teaching:

Math 2202   Multivariable Calculus

Publications:

• Immersed essential surfaces in hyperbolic 3-manifolds.  Comm. Anal. Geom. 10 (2002), 275--290.

• Boundary curves of surfaces with 4-plane property.      Geometry & Topology, 6 (2002), 609--647.

• Laminar branched surfaces in 3-manifolds.               Geometry & Topology, 6 (2002), 153--194.

• Commutator groups and foliations without holonomy.    Proc. Amer. Math. Soc. 130 (2002), 2471--2477.

• Boundary train tracks of laminar branched surfaces    Proc. of Symposia in Pure Math., 71, AMS (2003), 269--285.

• An algorithm to detect laminar 3-manifolds. (with Ian Agol)  Geometry & Topology, 7 (2003), 287--309.  

• Property P for knots admitting certain Gabai disks.     (with Oliver Dasbach)  Topology and its Applications, 142, (2004) 113--129.

• The Waldhausen conjecture (a 4-page outline of the proof of the Waldhausen conjecture) Oberwolfach Report 2005.

• An algorithm to find vertical tori in small Seifert fiber spaces.         Commertarii Mathematici Helvetici, 81 (2006) 727--753.

• Heegaard surfaces and measured laminations, I: the Waldhausen conjecture  Invent. Math. 167 (2007) 135 – 177   (arXiv:math.GT/0408198)

• Heegaard surfaces and measured laminations, II: non-Haken 3-manifolds    J. Amer. Math. Soc., 19 (2006) 625--657 (arXiv:math.GT/0408199)

• On the Heegaard splittings of amalgamated 3-manifolds. Geometry & Topology Monographs , 12 (2007) 157 – 190

• Saddle tangencies and the distance of Heegaard splittings. Albegraic & Geometric Topology , 7 (2007) 1119 – 1134

• Thin position and planar surfaces for graphs in the 3-sphere. Proc. Amer. Math. Soc.138 (2010)  333 – 340

• A quadratic bound on the number of boundary slopes of essential surfaces with bounded genus. (with Ruifeng Qiu and Shicheng Wang). Geom. Dedicata. 147  (2010)   131 – 138

• Heegaard surfaces and the distance of amalgamation. Geometry & Topology 14  (2010), 1871 – 1919

• An algorithm to determine the Heegaard genus of a 3-manifold. Geometry & Topology 15  (2011), 1029 – 1106 (arXiv:1002.1958)

• Images of the disk complex. Geom. Dedicata. 158 (2012) 121 – 136

• Rank and genus of 3-manifolds. J. Amer. Math. Soc. 26 (2013) 777--829. (arXiv:1106.6302)

• Small 3-manifolds with large Heegaard distance. Math. Proc. Cambridge Phil. Soc. 155 (2013), 431 – 441 (arXiv:1211.3064)

• Taut foliations in knot complements. (with Rachel Roberts) Pacific J. of Math. 269 (2014), 149 – 168    (arXiv:1211.3066)

• On the degeneration of tunnel numbers under connected sum. (Joint with Ruifeng Qiu) Trans. Amer. Math. Soc. 368 (2016), 2793 – 2807

• Tunnel number one knots satisfy the Berge Conjecture (Joint with Yoav Moriah and Tali Pinsky)

• Heegaard genus, degree-one maps, and amalgamation of 3-manifolds.  arXiv:2007.14534

• Taut foliations of 3-manifolds with Heegaard genus two.  arXiv:2202.00737