## Research

My papers are available on my arXiv page.

Generating
mapping class groups with elements of fixed finite order

*Journal of Algebra *, 511 (2018) — (pdf) (arXiv) (journal)

*Abstract:* We show that for *k* ≥ 6 and *g* sufficiently large,
the mapping class group of a surface of genus *g* can be generated by three
elements of order *k*. We also show that this can be done with four
elements of order 5. We additionally prove similar results for some permutation
groups, linear groups, and automorphism groups of free groups.

The slides I used
for a talk about this research at the 2016 Spring Topology and Dynamics
Conference.

Normal generators for mapping class groups are abundant

with Dan Margalit

submitted —
(pdf) (arXiv)

*Abstract:* We provide a simple criterion for an element of the mapping
class group of a closed surface to have normal closure equal to the
whole mapping class group. We apply this to show that every nontrivial
periodic mapping class that is not a hyperelliptic involution is a normal
generator for the mapping class group when the genus is at least 3. We
also show every pseudo-Anosov mapping class with stretch factor less
than √2 is a normal generator. This answers in the affirmative a question
of D. D. Long. We also give pseudo-Anosov normal generators with
arbitrarily large stretch factors and arbitrarily large translation lengths
on the curve graph, disproving a conjecture of Ivanov.

A video of a five-minute talk about this
research I gave at No Boundaries.

The slides I
used
for a talk about this research at Oberwolfach in September 2018.

How to hear the shape of a billiard table

with Aaron Calderon,
Solly Coles,
Diana Davis, &
Andre Oliveira

submitted —
(pdf) (arXiv)

*Abstract:* The bounce spectrum of a polygonal billiard table is the
collection of all bi-infinite sequences of edge labels corresponding to billiard
trajectories on the table. We give methods for reconstructing from the bounce
spectrum of a polygonal billiard table both the cyclic ordering of its edge
labels and the sizes of its angles. We also show that it is impossible to
reconstruct the exact shape of a polygonal billiard table from any finite
collection of finite words from its bounce spectrum.

This work began at a research cluster on
polygonal billiards, organized in summer 2017 by Moon
Duchin.

Generalizing Brouwer: adding points to configurations in closed balls

with Lei Chen and
Nir Gadish

—
(pdf) (arXiv)

*Abstract:* We answer the question of when a new point can be added in a continuous way to a given configuration
of *n* distinct points in a closed ball of arbitrary dimension. We show that this is possible given an ordered configuration
of *n* points if and only if *n* ≠ 1. On the other hand, when the points are not ordered and the dimension of the ball is at least 2,
a point can be added continuously if and only if *n* = 2.

In summer 2017 I led a research team at an REU at Georgia
Tech that was
organized by Dan Margalit. I
worked with Santana Afton, Sam Freedman, and Liping Yin. We investigated
generators, relations, and homomorphisms of big mapping class groups.
Research from the REU is in preparation.

The slides that Santana and Sam used for a talk about our research at MathFest
2017.

A poster about
our research from a Georgia Tech REU poster session.

I am working with Jim Belk, Dan Margalit, and Becca Winarski on a project where we give a new approach to studying Thurston maps.

A video of an fifty-minute talk about this
research Dan gave at a conference at Warwick.

In summer 2018 I led a research team at an REU at Georgia
Tech that was
organized by Dan Margalit. I
worked with Santana Afton, Xian Li, and Abby Saladin. We looked for and analyzed subgroups of a certain mapping class group that fix the equivalence class of the rabbit polynomial as a Thurston map.
Research from the REU is in preparation.

The poster that Abby used at the poster session at YMC 2018.