¾ÅÉ«ÊÓƵ

I was an undergraduate math major at the University of Maryland, where I did some work in the Experimental Geometry Lab, where I was interested in geometric structures and in dynamical systems arising from constructions in classical geometry. I was a graduate student at Stony Brook, where I learned low dimensional topology and geometry, including hyperbolic geometry and Teichmüller theory. I spent two years visiting Yale and completed my dissertation under the direction of Yair Minsky. My dissertation was interested in the dynamical behavior of billiards in polygons and connections to Teichmüller theory. I received my PhD in 2006 from Stony Brook. I spent a little over three years as a Boas Assistant Professor at Northwestern, where I learned Dynamical systems and Ergodic theory. I came to City College in the spring of 2010. City College and CUNY have offered me the opportunity to continue to develop my mathematical background, to teach interesting classes, and to interact with interesting students.

PHD, Mathematics, Stony Brook University, 2006.

M.A., Mathematics, University of Maryland, College Park, 2001.

B.S., Mathematics, University of Maryland, College Park, 2001.

I study dynamical systems defined by piecewise continuous maps which preserve some nice structure (such as a metric) away from their discontinuities on the phase space. This subject is frequently motivated by connections to geometry. Indeed the simplest such systems, interval exchange maps, are closely related to Teichmüller theory. For many nice spaces, the group of isometries of a space is quite rigid making a dynamical analysis of the action of an isometry uninteresting. By considering piecewise continuous isometries, we obtain a richer class of dynamical systems which give rise to new dynamical phenomena. The goal of his research is to better understand these systems from a topological or ergodic theoretic point of view.

• Professor, Department of Mathematics, City College of New York, 2017-present
• Associate Professor, Department of Mathematics, City College of New York, 2014-2017
• Member of the Doctoral Faculty, Department of Mathematics, CUNY Graduate Center, 2013-present
• Assistant Professor, Department of Mathematics, City College of New York, 2010-2014
• Ralph Boas Assistant Professor, Department of Mathematics, Northwestern University, 2006-2009
• Visiting Assistant in Research, Department of Mathematics, Yale University, 2004-2006

• Platonic solids and high genus covers of lattice surfaces with and and with an appendix by .
  Experimental Mathematics, (2020), doi: .
  Preprint:
  This paper also has .
• The Extrinsic Primitive Torsion Problem with
  Algebraic & Geometric Topology, 20 (2020) 3329--3376, doi: .
  Preprint:
• Renormalizing an infinite rational IET with and ,
  Discrete & Continuous Dynamical Systems - A, 40 (2020), no. 9, 5105--5116, doi: .
  Preprint:
• Indiscriminate covers of infinite translation surfaces are innocent, not devious with
  Ergodic Theory and Dynamical Systems, 39 (2019) no. 8, 2071--2127, doi: .
  .
• An infinite surface with the lattice property â…¡: Dynamics of pseudo-Anosovs
  Journal of Modern Dynamics, 14 (2019) no. 1, 243--276, doi: .
  Preprint:
• Periodicity and ergodicity in the trihexagonal tiling with
  Commentarii Mathematici Helvetici, 93 (2018) no. 4, 661--707, doi: .
  .
• Immersions and translation structures I: The space of structures on the pointed disk,
  Conformal Geometry and Dynamics, 22 (2018), 235-270, doi: .
  Preprint: .
• Rel leaves of the Arnoux-Yoccoz surfaces with and with an appendix by Lior Bary-Soroker, Mark Shusterman, and Umberto Zannier,
  Selecta Mathematica New Series (2018) 24, no. 2, 875-934, doi: .
  .
• The Invariant Measures of some Infinite Interval Exchange Maps
  Geometry & Topology 19 (2015) 1895-2038, doi: .
  Abbreviated abstract: We study classify the locally finite ergodic invariant measures of some skew rotations and other infinite interval exchange maps.
• An infinite surface with the lattice property I: Veech groups and coding geodesics
  , 366 (2014), no. 5, 2625-2649, doi: .
  
• Grid graphs and lattice surfaces
  , 12 (2013), no. 1, 2657-2698. doi: .
• Dynamics on the infinite staircase with and ,
  , 33 (2013), no. 9, 4341 - 4347, doi: .
  
• Renormalization of Polygon Exchange Maps arising from Corner Percolation
  191 (2013), Volume 191, no. 2, 255-320, doi: .
  Published version: []
• Another Veech Triangle
  , 141 (2013), no. 3, 857-865, doi: .
  Abstract: The triangle with angles (Pi/12, Pi/3, 7 Pi/12) has the lattice property.
• Generalized Staircases: Recurrence and Symmetry with
  , no. (), p. 1581-1600, doi: 
  Journal's article page: []
  Abbreviated abstract: We study properties of Z-covers of translation surfaces.
• Billiards in nearly isosceles triangles with
  Journal of Modern Dynamics 3 (2009), no. 2, 159--231.
  Abbreviated abstract: We prove that nearly isosceles triangles have periodic billiard paths.
  View original: from []
• Lower bounds on growth rates of periodic billiard trajectories in some irrational polygons
  Journal of Modern Dynamics 1 (2007), no. 4, 649--663.
  View original: from []
  Abbreviated abstract: Examples of irrational polygons where the growth rate of periodic trajectories is slightly super-linear are provided.
• Periodic Billiard Paths in Right Triangles are Unstable
  Geom. Dedicata 125 (2007), no. 1, 39--46.
  The original publication is available at . []
• From Pappus' theorem to the twisted cubic
  Geometriae Dedicata 110 (2005), 103--134.
  The original publication is available at . []