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.
I am an Assistant Professor at my alma mater, Western Carolina University. Before coming back to WCU I held visiting positions Bucknell University, 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 2024
- Math 153: Calculus I
- Math 362: Linear Algebra I
Previous Courses
- Complex Analysis (Western Carolina University)
- Introduction to Logic & Proof (Western Carolina University)
- Real Analysis (Western Carolina University)
- Geometry (Bucknell University)
- Introduction to Numerical Analysis (Indiana University)
- Ordinary Differential Equations (Western Carolina University, Indiana University)
- Introduction to Probability & Statistics (Indiana University)
- A Brief Survey of Calculus, pt. 2 (Indiana University)
- Linear Algebra (Indiana University, Wake Forest University)
- Intro. to Number Theory (Clemson University)
- Calculus I (Bucknell University, Indiana University, Wake Forest University, Clemson University)
- Calculus II (Western Carolina University, Bucknell University, Clemson University)
- Calculus III (Western Carolina University, Clemson University)
- Mathematics for Liberal Arts Majors (Clemson University, Western Carolina University)
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.
Preprints & Publications
- (Almost) Everything is Differentiable
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To appear in Mathematics Magazine
Preprint available - Haupt's theorem for strata of abelian differentials
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(with M. Bainbridge, C. Judge, and I. Park)
Israel Journal of Mathematics, 252 (2022), 429 -- 459.
Pre-print available on the arXiv. - The wild, elusive singularities of the T-fractal
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(with R. Niemeyer)
Submitted for publication
Pre-print available on the arXiv. (Revised April 2020.) - Cutting sequences on square-tiled surfaces
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Geometriae Dedicata, 190 (2017), 53 -- 80.
Full text available for online reading; pre-print available on the arXiv - Hyperelliptic translation surfaces and folded tori
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(with M. Schmoll)
Topology and its Applications, 161 (2014), 73 -- 94.
Preprint - Pseudo-Anosov eigenfoliations on Panov planes
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(with M. Schmoll)
Electronic Research Announcements in Mathematical Sciences, 21 (2014), 89 -- 108.
Notes
Lecture Notes
These are notes I've developed for some of the classes I've taught. There are likely typos and errors (though hopefully they are all minor); if you notice any mistakes, please feel free to contact me so that I can correct them.
Some courses I only have handwritten notes for, but those notes will likely be typed the next time I teach the course and will then appear here.
- Calculus I (updated Fall 2024)
- Calculus II (updated Spring 2024)
- Calculus III (updated Spring 2022)
- Elementary Linear Algebra (updated Fall 2024)
- Introduction to Logic & Proof (updated Fall 2023)
- Introduction to Probability & Statistics (updated Fall 2018)
- Ordinary Differential Equations (updated Fall 2022)
- Introduction to Numerical Analysis (updated Spring 2019)
Travel and Presentations
A solid star, ★, indicates I gave an invited talk at the conference; an empty star, ☆, indicates I gave a contributed talk.
- ★September 14 - 15, 2024
- AMS Fall Central Sectional Meeting
University of Texas at San Antonio
San Antonio, TX - April 12 - 14, 2024
- Carolina Dynamics Symposium 2024
University of North Carolina at Charlotte
Charlotte, NC - ★May 31 - June 4, 2023
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The 13th AIMS Conference on Dynamical Systems, Differential Equations, and Applications
University of North Carolina at Wilmington
Wilmington, NC - ☆April 14 - 16, 2023
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Carolina Dynamics Symposium 2023
Davidson College
Davidson, NC - ★January 6 - 8, 2023
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AMS Joint Mathematics Meeting
Boston, MA - January 1 - 5, 2023
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Salt Flats MRC-style meeting on Flat Surfaces and Teichmüller Dynamics
University of Utah
Salt Lake City, UT - October 15 - 16, 2022
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AMS Fall Southeastern Sectional Meeting
University of Tennessee at Chattanooga
Chattanooga, TN - April 22 - 24, 2022
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Carolina Dynamics Symposium
Furman University
Greenville, SC - ☆March 16, 2021
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Teichmüller theory seminar talk (Zoom)
Indiana University
Bloomington, IN - ☆November 05, 2020
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Colloquium (Zoom)
Western Carolina University
Cullowhee, NC June 5 - 9, 2020Rescheduled due to Coronavirus-
AIMS Conference Series on Dynamical Systems and Differential Equations
Atlanta, GA
- ★February 26 - 27, 2020
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Colloquium
Western Carolina University
Cullowhee, NC - ★January 15-18, 2020
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AMS/MAA Joint Mathematics Meetings
Denver, CO - ★November 21, 2019
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Colloquium
Bucknell University
Lewisburg, PA - ★November 15, 2019
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Analysis seminar talk
Clemson University
Clemson, SC - ★March 20 - 24, 2019
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AMS Spring Central and Western Joint Sectional Meeting
University of Hawai'i at Mānoa
Honolulu, Hawaii - ★January 16 - 20, 2019
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AMS/MAA Joint Mathematics Meetings
Baltimore, MD - ☆November 9, 2018
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Seminar talk in Teichmüller theory seminar
Indiana University
Bloomington, IN - October 21 - 27, 2018
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Workshop on Dynamics and Moduli Spaces of Translation Surfaces
Fields Institute
Toronto, Ontario, Canada - April 22, 2017
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Carolina Dynamics Symposium
University of North Carolina at Charlotte
Charlotte, NC - February 13 - 17, 2017
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Teichmüller space, polygonal billiards, and interval exchange transformations
Centre International de Rencontres Mathématiques
Marseille, France - ★November 29, 2016
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Seminar Talk in the Ergodic Theory and Dynamical Systems Seminar
University of North Carolina
Chapel Hill, NC - November 4 - 6, 2016
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Midwest Dynamical Systems Seminar 2016
IUPUI
Indianapolis, IN - ★November 1, 2016
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Analysis seminar talk
Clemson University
Clemson, SC - ★October 12 - 15, 2016
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Colloquium Presentation
University of Maine
Orono, ME - August 1 - 5, 2016
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Cycles on Moduli Spaces, Geometric Invariant Theory, and Dynamics
ICERM
Providence, RI - May 8 - 13, 2016
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Flat Surfaces and Dynamics in Moduli Space
Casa Matemática Oaxaca
Oaxaca, Mexico - ☆April 1 - 3, 2016
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Carolina Dynamics Symposium (Talk canceled due to illness.)
Furman University
Greenville, SC - March 5 - 6, 2016
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AMS South-Eastern Sectional
University of Georgia
Athens, GA - October 30 - November 1, 2015
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Midwest Dynamical Systems Seminar
The Ohio State University
Columbus, OH - September 28 - October 2, 2015
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Clay Research Conference and Workshops
Oxford University
Oxford, UK - June 27 - July 5, 2015
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Research collaboration with Robert Niemeyer
University of New Mexico
Albequerque, NM