I am a mathematician generally interested in topology, geometry, and dynamics. More precisely, I am interested in problems that arise in studying translation surfaces and their moduli spaces. These are surfaces equipped with a special type of geometry that can be encoded as a piece of complex analytic (or algebro-geometric) information attached to the surface. This fascinating area has its genesis in a very simple-sounding dynamical problem related to billiard balls bouncing around a billiard table, and combines tools from low-dimensional topology, complex analysis, algebraic geometry, and ergodic theory, and so my interests overlap with many other areas of mathematics. For a little more information about this area and my research projects, see the research section below.

Before developing an interest in mathematics in graduate school, I was interested in computer programming and completed a B.S. in computer science from Western Carolina University. That knowledge has been beneficial as a mathematician, as it has allowed me to quickly write programs to perform simulations, gather data for research problems, and generate images for papers and animations for presentations.

Currently I am a Visiting Assistant Professor at Bucknell University in Lewisburg, PA. Before coming to Bucknell I held visiting positions at Indiana University and Wake Forest University. I received my Ph.D. from Clemson University under the direction of Martin Schmoll.

Teaching


My teaching philosophy boils down to the following principles: all students are capable of success, and all students should be held to a high standard, though some students require more guidance and encouragement than others. Besides relaying mathematical information, I strive to help my students become independent learners, critical thinkers, and problem solvers; these are skills that students can transfer to other areas of their life outside of the classroom.

More detailed information, including specifics of what I do in the classroom, can be found in my statement of teaching philosophy.

Fall 2019
Spring 2020
Previous Courses

Research


My research broadly breaks down into two categories: geometry and dynamics. Most recently I've been interested in the isoperiodic foliation of the Hodge bundle, and the ergodic theory of affine interval exchanges. I have also collaborated with Robert Niemeyer on projects related to billiards in fractals. A more thorough overview of my research interests is available in my research statement.

Because of the nature of the topics I am interested in, my research uses tools from a variety of different mathematical disciplines, which is one of the things I really like about this area. Algebraic topology, complex algebraic geometry, complex analysis, differential geometry, dynamical systems, ergodic theory, and hyperbolic geometry are all relevant to my interests.

Some of my interests also have a more combinatorial flavor and would be suitable for collaborations with undergraduate students. In particular, projects related to counting the number of square-tiled surfaces with some special properties can be tackled without requiring a lot of advanced machinery, and I am very open to working on such projects with any interested students.

Works in progress
The Haupt condition for strata
(Joint with Matt Bainbridge, Chris Judge, and Insung Park.)
Applying a recent result of work Calsamiglia, Deroin, and Francaviglia, we extend the results of Haupt and Kapovich (done independently) that classifies which periods of 1-forms on a surface can occur as the periods of a holomorphic 1-form for some choice of complex structure. In particular, once a collection of allowable periods is selected, we describe which connected components of strata of the Hodge bundle contain Abelian differentials with those periods.
Non-singular ergodic theory of affine interval exchanges
Interval exchanges are tools for studying the dynamics of flows on translation surfaces that have been well-studied over the last 40 years. A simple generalization allows for the sizes of exchanged intervals to change. While the Lebesgue measure is no longer preserved, ergodic theoretic questions about these maps can still be considered. Right now I am studying a particular family of affine interval exchanges to which an infinite interval exchange can be associated. Taking a suspension of the infinite interval exchange gives rise to a translation surface of infinite area where ergodic theoretic properties of the vertical flow on that surface can be deduced from the dynamics of the initial affine interval exchange.
Preprints & Publications
The wild, elusive singularities of the T-fractal
(with R. Niemeyer)
Submitted for publication
Pre-print available on the arXiv.
Cutting sequences on square-tiled surfaces
Geometriae Dedicata, 190 (2017), 53 -- 80.
Full text available for online reading; pre-print available on the arXiv
Hyperelliptic translation surfaces and folded tori
(with M. Schmoll)
Topology and its Applications, 161 (2014), 73 -- 94.
Preprint
Pseudo-Anosov eigenfoliations on Panov planes
(with M. Schmoll)
Electronic Research Announcements in Mathematical Sciences, 21 (2014), 89 -- 108.

Travel and Presentations


June 5 - 9, 2020
AIMS Conference Series on Dynamical Systems and Differential Equations
Atlanta, GA
January 15-18, 2020
AMS/MAA Joint Mathematics Meetings
Denver, CO
November 21, 2019
Colloquium
Bucknell University
Lewisburg, PA
November 15, 2019
Analysis seminar talk
Clemson University
Clemson, SC
March 20 - 24, 2019
AMS Spring Central and Western Joint Sectional Meeting
University of Hawai'i at Mānoa
Honolulu, Hawaii
January 16 - 20, 2019
AMS/MAA Joint Mathematics Meetings
Baltimore, MD
November 9, 2018
Seminar talk in Teichmüller theory seminar
Indiana University
Bloomington, IN
October 21 - 27, 2018
Workshop on Dynamics and Moduli Spaces of Translation Surfaces
Fields Institute
Toronto, Ontario, Canada
April 22, 2017
Carolina Dynamics Symposium
University of North Carolina at Charlotte
Charlotte, NC
February 13 - 17, 2017
Teichmüller space, polygonal billiards, and interval exchange transformations
Centre International de Rencontres Mathématiques
Marseille, France
November 29, 2016
Seminar Talk in the Ergodic Theory and Dynamical Systems Seminar
University of North Carolina
Chapel Hill, NC
November 4 - 6, 2016
Midwest Dynamical Systems Seminar 2016
IUPUI
Indianapolis, IN
November 1, 2016
Analysis seminar talk
Clemson University
Clemson, SC
October 12 - 15, 2016
Colloquium Presentation & research meeting with Rob Niemeyer
University of Maine
Orono, ME
August 1 - 5, 2016
Cycles on Moduli Spaces, Geometric Invariant Theory, and Dynamics
ICERM
Providence, RI
May 8 - 13, 2016
Flat Surfaces and Dynamics in Moduli Space
Casa Matemática Oaxaca
Oaxaca, Mexico
April 1 - 3, 2016
Carolina Dynamics Symposium
Furman University
Greenville, SC
March 5 - 6, 2016
AMS South-Eastern Sectional
University of Georgia
Athens, GA
October 30 - November 1, 2015
Midwest Dynamical Systems Seminar
The Ohio State University
Columbus, OH
September 28 - October 2, 2015
Clay Research Conference and Workshops
Oxford University
Oxford, UK
June 27 - July 5, 2015
Research collaboration with Robert Niemeyer
University of New Mexico
Albequerque, NM