Hi. I'm Hannah Cairns.
I'm a postdoc at McGill University. I have a PhD in mathematics from Cornell University, where I worked on connections between probability theory and analysis. I'm also into algorithms.
Here is my CV, and here is my page at Cornell. My LinkedIn page is here.
My email is hannah dot abigail dot cairns at the domain gmail dot com.
I'm looking for a postdoctoral position or job in industry, ideally a job as a programmer or researcher.
Papers and preprints:
The smash sum is the unique sum of open sets satisfying a natural list of axioms. To appear in the Journal d'Analyse Mathematique. arXiv link. 224 KB PDF.
A proof that the Diaconis-Fulton smash sum of open sets is the only one that satisfies a short list of natural axioms. It comes with an expository note that shows the existence and uniqueness of quadrature sets for a large class of positive weight functions: 295 KB PDF.
[Draft] The critical sleep rate for activated random walk on the cycle with one chip per vertex: 240 KB PDF.
Activated random walk is a process with chips on a graph which move around and wake each other up, and fall asleep at a rate. In the extreme case where the walk starts with one chip on every vertex of a cycle, there is a sharp boundary between fast and slow sleep.
Perron's Theorem in an Hour.
A short proof of the Perron-Frobenius theorem, intended to fit in a single class period. Published in 2021 in The American Mathematical Monthly, 128 (8), 748-752.
Hipster random walks.
A paper about a process on a tree that doesn't like popular things. Published in 2020 in Probability Theory and Related Fields, 178, 437-473. Arxiv link: 1909.07367.
Some halting problems for abelian sandpiles are undecidable in dimension three.
This was published in 2018, SIAM Journal on Discrete Mathematics, 32(4), 2636–2666. An older version is on arxiv: arXiv link.
Round sequence Ramachandran-Schild algorithm. Explanation (PDF) and implementation.
This is my improvement and simplification of the Ramachandran-Schild algorithm for stabilizing sandpiles on trees. It introduces the concept of "round sequences."
Miscellaneous exposition:
A short proof of Perron's theorem, 2014, 189 KB PDF.
The ancestor of "Perron's Theorem in an Hour." This is a proof of the popular Perron theorem, written for a presentation in a graduate class. (Note: if you have a stochastic matrix, this isn't the best way. There is a more intuitive probabilistic proof which works not only for finite matrices but countable ones also.)

As of October 2021, the pdf (on the class web page) is still the second result on Google when you search for proof of perron frobenius.

Watson's third integral, 2007, 238 KB PDF.
The probability that three-dimensional simple random walk returns to the origin can be written in terms of elliptic integrals; this was proven by G. N. Watson in 1930-something. But the reasons are complicated and there are many moving parts, and so, in an effort to understand them, I wrote them down in this note. That was in my second year of undergrad, I think.

It used to be on the UBC Math Club website, but that's gone now. I recovered this from some kind person's file locker on baidu.com.

Programming projects:
I have a couple github pages.
cairnsh tends to have more mathematical programs, while...
colaprograms has less serious programs, like an AR game and an informative kiosk.
Most of the programs are written in Python. The AR game is written in Unity with C#.
Some other things:
I wrote a single-page app, Solidplotter, for visualizing three-dimensional shapes defined by equations. It's designed to be usable on a phone.
I participated in a team at the 2019 AI4Good hackathon at Mila. We wrote a system for parsing handwritten strings of digits , for Charitable Analytics International's challenge that asked teams to improve their Meza product.

Our team's program came in second out of five in its category, with 85% accuracy. Here is my later version that works a little better, with 92% accuracy.

In the fall of 2018, I participated in a class competition at Universite de Montreal (https://www.kaggle.com/c/ift3395-6390-f2018). The goal was to classify noisy images created from Google's Quick, Draw! dataset into one of 31 categories. I placed second among the 57 teams in the class, using PyTorch and an SE-ResNet-like architecture with a lot of inter-block concatenation. My submission is here: https://www.kaggle.com/cairnsh/ift6390-submission
koreader is an open-source ebook reader written in Lua, which can be used on e-ink readers like Kindles or Kobos in place of the stock software. I contributed by submitting a patch that made the touchscreen on the Kobo Aura H2O work. This took a little bit of reverse engineering: I had to figure out how the touch event protocol had changed.
I made a portable wide-band radio scanner which can be used from a cellphone, using a Raspberry Pi Zero W and an RTL-2832 TV tuner dongle. The Pi Zero W runs a wifi access point, a simple web server, and a separate backend server that runs the dongle. The user carries this all around in a box, and uses it by connecting their cellphone to the Zero's access point and opening the web page provided by the server. They can command the scanner to look at different 10-MHz bands between 24 and 1766 MHz, and see the power at each frequency at 2 KHz resolution. The backend and web server are written in Python, using scipy to do the computations, and the web page uses ajax and jQuery.
Master's thesis:
Ellipse packing, 2013, 2.51 MB PDF.
My master's thesis was on the use of ellipse packing (via the very general blunt packing theorem of Schramm) to get approximate solutions to the "bar-Beltrami equation."
Translation:
The Ticket, 2015. Translation of Edogawa Ranpo's story 一枚の切符 (Ichimai no Kippu), 1923. 79 KB PDF.
This is a translation into English of the second published story of Edogawa Ranpo, an influential mystery writer who worked from 1923 to 1960. To my knowledge it is the first translation of this story. It was done in a couple weeks for a translation competition at Cornell, and, not so surprisingly, it didn't win.
Updated April 2023.