---
title: Grad School Personal Statement
author: Colton Grainger
revised: 2018-03-21
status: finished
belief: likely 
math: true
---

The robustness of a mathematical method determines its utility. Just
imagine designing a communication network that fails to account for
topological perturbations, or modeling an epidemic with strictly
deterministic differential equations. My goal, then, is to research
robust methods in algebraic topology that can be applied in both
abstract and computational settings.

For example, consider training an AI to distinguish the tone signature of
different musical instruments. Applying persistent homology, we associate
“holes” in an audio recording’s time-delay phase-space with a sample statistic:
the persistent rank function (PRF). Corresponding informally with 
[Nikki Sanderson](https://arxiv.org/abs/1708.09359), 
I learned a computer trained on PRFs more accurately classifies tones than one
trained on Fast Fourier Transforms. Here a topological invariant answers the
question “which instrument?” with higher fidelity than a recording’s frequency
transform.

My research interest stems from my exposure to topology and its applications as
an undergraduate, a base of computational skill, and insight from two years of
service work since graduating.

Advised by Dr. Jonny Comes, my senior independent study examined how
Galois theory constrains the solution space of Fuchsian-type
differential equations. Following Michio Kuga, we developed a
correspondence between the fundamental group of
$D = \mathbf{C}\setminus\\{x_1,\ldots,x_n\\}$ and this surface’s universal
covering space $\tilde{D}$. Exploiting the representation of the group
of covering transformations $\Gamma(\tilde{D} \to \tilde{D})$ as a group
of linear automorphisms, I parameterized solutions to the hypergeometric
DE. For interesting cases, I found the monodromy representation at
singular points, and generated plots. My [conference poster](http://coltongrainger.com/docs/galois.pdf)
summarizes our method, its history, and discusses applications to
mathematical physics. At the same time, I studied point-set topology
under Dr. Dave Rosoff, who led a seminar in the Moore method. I reasoned
from counter examples, occasionally contributing a stronger hypothesis
to our notes to account for non-intuitive spaces. We had enough time to
abstract spaces to objects and maps to morphisms as a means of
analogously introducing category theory from topology. I am enthusiastic
to build from this ground to higher results, one of which I reached in
my senior study, another of which Nikki Sanderson demonstrated ahead of
me.

After graduating, I took two years to perform stipended service work. In
Houston, under Shaoli Bhadra, I developed scalable
[resources](https://github.com/coltongrainger/ymca-resources) for
refugee case management, which included crowd-sourcing a 
[map of clinics and languages spoken](https://drive.google.com/open?id=1kk9yn6-4nifHLIf2tGYbW_7PiYo&usp=sharing)
through the Google Maps API. I wrote bug reports for the implementation
three databases, and, when Texas cut funding for Refugee Medical
Assistance, I contributed to a data management plan for the small
refugee population transitioning from state to federal medical care. In
Olympia, I am a community organizer at a 24/7 homeless shelter. I rely
on distributed version control, and have therefore become a staunch
advocate for “deploying early and often”. For example, I built 
[volunteer.fscss.org](https://web.archive.org/web/20171223032857/http://volunteer.fscss.org:80/) 
to maintain a schedule of events, but it now doubles as a wiki. As I have begun
mentoring interns and work-study students who are preparing to embark on
careers in social work, I am noticing skills I think will transfer neatly to a
teaching assistantship. In all, I have cultivated working methods I believe to
be of value in the scientific disciplines: collaboration allows me to focus my
effort on tasks where I have a comparative advantage, cumulation allows me to
work where others have left off, and transparency allows others to work off of
me.

I am serious, however, to pursue a career in mathematics. To prepare for
graduate study, I have enrolled in an affordable selection of
correspondence courses: Probability, Differential Equations, and
Numerical Analysis. I am also reading from Hatcher’s *Algebraic
Topology*, surveying topics in Gower’s *Companion to Mathematics*, and
building a base of computing skills in the UNIX philosophy.

(CU Boulder) Though I am open to a variety of research at CU Boulder,
my experience with messy data-sets and my appetite for topology leads
naturally into topological data analysis. I would be enthusiastic to
collaborate with Profs Meiss (APPM), Bradley (CSCI) or Beaudry 
(MATH). I see fruitful work to be done with TDA in signal processing and
network analysis; I am also curious to study higher-dimensional data in
material science.

(University of Rochester) While I am open to the breadth of mathematical
inquiry at the University of Rochester, I would like to “push hard on a clear
signal” and study algebraic topology — specifically homotopy theory and
homology groups — with computational topology in mind for applications. I would
be glad to collaborate with Profs Pakianathan, Iosevich, or Ravenel.