Old posts are being manually restored

During an update to the server in September and an issue with the backup system, the site was corrupted. Some posts have been totally lost. All previous comments have been totally lost.

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webcomic: re-search

Here begins the adventures of Massey into the unknown. Please enjoy my webcomic on what it feels like to run into a concept for the first time. I will be updating panel by panel, adding each one here as I make them.

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Fiber Bundles of Formal Disks

Here is an incomplete proof that varieties are fiber bundles of formal disks over their deRham Stacks. The fact makes intuitive sense, the deRham stack is the variety without infinitesimal data, and then by adding the infinitesimal data (formal disks) back in, you recover the result. However, the fact that you can build anything non infinitesimal out of formal disks fills me with confusion and awe.

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Ukulele Songs from the TamagaWHAT Seminar

This quarter (Spring 2018) we held a seminar on the Lurie-Gaitsgory proof of the function field case of the Tamagawa conjecture (Jora Belousov, Grigory Kondyrev, Yifeng Liu, and myself). I also started learning the uke, and I wrote a song for each talk (many use the tune of an existing song). I think the quality improved as I went on, so feel free to start from Lecture 8.

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Calculating the Period Matrix of a Shiga Curve, (y^3 = x^4-1).

Thanks to Dami Lee for patiently walking through how to compute the period matrix of this 12-fold cyclic cover of a thrice punctured sphere, and thanks Matthias Weber for showing me how to write the 3-fold cover of a 5-punctured sphere as a 12-fold cyclic cover of a thrice punctured sphere. Most of the figures are either hand-drawn or made using Geogebra. Note that I will sometimes use (\tau) to denote (2 \pi). All errors are mine and mine alone.

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Newspaper Ad: Looking for a Variety

Hello, my name is Catherine. I don’t want much, just looking for a nice Variety to spend my days with. If you apply, I’d like you to have a well understood group law that comes from some 3-fold symmetry, but I’m a simple girl, easy to please, and I don’t need your group law to be all fancy and closed — a group chunk (group law which closed at least locally to the origin) is fine by me. I’ll have to put you through an interview process to see if you’re group chunk gives me a formal group law which is height 3, but don’t worry, it’ll be painless. Please let me know if you have a friend that matches this description!

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How do I explicitly write an ODE as a linear ODE + a nonlinear ODE?

I was learning about linearized stability and was confused by where the magical linearized version of the equation was coming from. I finally understand it, and so stupidly simple so I want to tell you about it. First, I’ll motivate the question. If you don’t care about the motivation, just scroll ahead a bit.

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Spectrum of a Ring and Spectrum of a Linear Operator

A quick post before bed, an impressionist stroke on some nice things lurking in linear algebra. I love polynomials. They are the ultimate tools that make me feel like I’m touching something, calculating at the level of a polynomial is a good clean feeling. I want to show you that it is nice to think of a vector space over (F) as a (F[x])-module (thanks Emmy Noether). Thanks to Semon Rezchikov for helping me get over a few bumps in grasping some of the following.

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Classification of Conic Sections

Apollonius of Perga (262 BC) wrote an exhaustive treatise exploring conics. He presented a classification of conic sections by angle. I’ll show you a summary of what he did, and then a conceptually more pleasing and suggestive way to think about it.

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A question on primes

I recently encountered a result which seems to be analogous to the following result of Dirichlet, which I wrote in a few common forms to be more suggestive.

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Notes on One-Parameter Deformations of Cohomology Theories

Thanks to Dr. Lubin for graciously helping me derive and understand one-parameter families of formal group laws, thanks to Eric Peterson for introducing me to Morava’s Forms of K-theory, thanks to Agnes Beaudry for pointing out that there was a neater way to check for Landweber-exactness.

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Some comments on math communication

I read Bill Thurston’s On Proof and Progress this morning. This led me to consider a few things I’ve learned this summer about the sociology and psychology of being a part of the mathematical community, which I figured I’d share on the off chance that you might find it encouraging or helpful.

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Group Contractions for Elliptic Curves

When you construct a particular sheaf over an elliptic curve and then continuously vary the elliptic curve, what happens to the sheaf? I’m not sure, so I am first trying to understand what group contractions mean for elliptic curves.

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Complex Analysis: Poles, Residues, and Child’s Drawings

Thanks to Laurens Gunnarsen for his superb pedagogy and for this amazing explanation on the incredible depth of connections springing from the Sperner lemma. All errors are mine not his. This started with a chain of events, sitting in on number theory seminars and encountering Abel’s differentials of the first and second kind, interest in the dessin, and led up to asking Laurens:

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What is the difference between homotopy and coherent homotopy?

This question had been bugging me for a while, and I have been unable to find a source that is suited to the beginning topologist. Eric Peterson kindly answered this for me, and I found his explanation so astoundingly beautiful that I wish to share on the off chance that you, dear reader, will similarly appreciate this visually rich narrative. All errors are mine and not his.

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A First Look at an Equivariant Elliptic Cohomology

Usually, besides the information preserved by the formal group law of the elliptic curve, we can’t see any information about the elliptic curve when looking at the output of its associated cohomology theory. The formal group law only* remembers if the curve was singular/supersingular, and the characteristic of its field.

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Landweber-Ravenel-Stong Construction Flowchart

Here’s a flowchart I made while preparing for an upcoming talk. I fear that it may be hard to follow without being already familiar with the story, but there’s little harm in posting it. Maybe it’ll help someone navigate the literature.

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The Landweber exact-functor theorem

This post assumes familiarity with formal group laws, the definition of exact sequences, the motivation of the Landweber-Ravenel-Stong construction, that the exactness axioms are one of the generalized Eilenberg-Steenrod axioms, and the fact that formal group laws over (R) are represented by maps from the Lazard ring to (R).

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(Pic(X)) vs. (CP^infty)

There are likely inaccuracies in this post, as I wrote it quickly and am just beginning to learn the basics of algebraic geometry. Constructive criticism is strongly encouraged.

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Oriented Cobordism Cohomology

Edit: When I say cobordism, I mean oriented cobordism unless stated otherwise. Also note that I accidentally flip-flopped \(\Omega^*\) and \(\Omega*\) — \(\Omega^n\) should be cobordism classes of maps from manifolds of codimension \(n\) to \(X\), and \(\Omega_n\) is cobordism classes of maps from manifolds of dimension \(n\) to \(X\)._

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A Recipe for Constructions ((R_F(G)), (A(G)), (K_0(X)), …)

I noticed an informal “recipe” for taking a type of object and constructing invariants (of the object). It’s been useful for removing the feeling of “what, why? where did that come from?” when learning new constructions that fit this recipe. Hopefully it will help you!

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Staring.

And I sit here, undeserving,
in a small flat
staring out the window, dear.
Thinking of patterns, and sometimes of regrets.
Staring at symbols, trying to see what the author saw.
Trying to be them, just for a second,
to glimpse the beauty they’d uncovered
by sitting in a small flat,
staring out of their window
thinking of patterns, and sometimes of regrets.

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Spin Manifolds

Disclaimer: This exposition is profane. It is the result of me trying to be productive during a raging headache, and ending up with comic relief.

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Introduction to Bundles

Treating spaces as fiber bundles allows us to tame twisted beasts. Most of spin geometry is phrased in the language of fiber bundles, and this post will begin to introduce that language — extremely powerful in its simplicity.

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Studying Symmetry

Group(oid) theory is the study of symmetry. When we are dealing with objects that appear symmetric, group theory assists with analysis of these objects. The label of “symmetric” is applied to anything which stays invariant under some transformations.

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Graphical Supersymmetry Algebras

Thanks to Dr. James Hughes, Matthew Lynn, Chris Walker, Chas Leichner, Alex Ray, Alex Zhu, Dr. Cornelia Van Cott, Paul Sebexen, Colin Aitken, Chuck Moulton, Sebastien Zany, and Nick for joining me in playing with this problem.

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Respect Our Work.

Already, criticism has be thrown at me for deciding to focus on pure mathematics. I am often told (with the best intentions) that I should go back to doing applied math/engineering. I’m guessing that other people who’ve decided to study theoretical subjects suffer from a similar lack of respect.

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Pressure Ulcer Prevention

A pressure ulcer is an ulcerated area of skin caused by irritation and continuous pressure on part of the body. It starts as an area of skin damage which spreads to the tissues underlying the skin. In severe cases (Stage IV), there can be permanent damage to muscle or bone underneath the skin. Pressure ulcers are most common over bony prominences. Leading contributors to pressure ulcers: pressure, moisture, friction/shear and nutrition. Tissue is stressed between the bony mass and a rigid surface (such as a chair seat or bed mattress).

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Reframing the Gluten Scanner

A lesson that every scientist (or any person in a fast-paced creative field) learns: be glad when we find out that our research idea has already been done.

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Finding Routine in Freedom

I’ve just recently taken on a contract project. Adding that project onto my wheelchair and protein detector projects is a fun exercise in multi-threading that contributed to a recent realization.

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Old Posts Are Being Manually Restored

During an update to the server in September and an issue with the backup system, the site was corrupted. Some posts have been totally lost. All previous comments have been totally lost.

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Controlling Fear

Having a reliable method to control your fear and achieve focus quickly is indispensable. During one sleepless night reading Frank Herbert, I discovered my solution.

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Force Quit in Ubuntu

A surprising amount of Linux users are unfamiliar with how to force quit programs via the command-line. It is often that your only option to escape a process gone awry is using the term window.

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Penrose Triangle

I wanted wall art for my office (\rightarrow) I made myself wall art with painter’s tape during a 30min break from working on my simulation. The triangle is fairly large, (h \approx) 130 cm (\approx) 4’4”

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Introductory Science Kit

Katriona Guthrie-Honea, Austin Russell, and I recently brainstormed to come up with an introductory science kit. The goal of the kit is to create wonder and light the flame of curiosity; revealing that science is a series of fun puzzles and that magic is more beautiful when you understand how it is done.

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Megan’s Present

I visited my friend, Megan Brusnahan, before I left for CA. She had a picture I’d drawn her in middle school (2009) mounted on her wall in a frame. I’d forgotten that I’d drawn this picture! It’s done entirely in black sharpie.

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The Engineer-Musician

There is a connection between engineering and music. For the sake of simplicity, I will shorten “math, physics, computer science, mechanical & electrical engineering” to “engineering.”

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Braille System Font

I mentioned that I set my system font to Braille in a previous post. This tutorial generalizes to any font you can find in a .ttf format!

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Simple Cubic Lattice

Today, let’s have some fun playing with perspective rendering in Python! My graphics package of choice is VPython:

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Unschärferelation

Konzepte der Quantenmechanik durch den dunklen Nebel der Unwissenheit aufgefressen. Lass mich den Nebel eines verbreiteten Missverständnisses aufheben. Lass mich deine gegenwärtige Meinung ändern.

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Blame Bias & Project Unbreakable

Project Unbreakable is a project run by Grace Brown, Kaelyn Siversky, Christina Dunlop, and Kerri Pang. It was created to raise awareness of the common nature of sexual assault and serves to alter the general sociocultural perception of rape. Disproving the regular assumption that rape is an uncommon, unfortunate occurrence that happens only to those who deserve it. The project is a composed of a collection of art submitted by survivors. These submissions are photographs of a survivors holding posters decorated with quotes from their attackers. Submissions also include quotes from others in reaction to survivors seeking help (e.g., “You deserved it,” “I don’t believe you”). This project features male rape survivors and showcases the reality that men can get raped. There is no discrimination as to who can participate in Project Unbreakable (“anyone who has experienced any form of sexual abuse, whether physical or emotional”). However, they do not accept admissions from children.

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Do You Even Search, Bro?

I recently came across a software engineer I respect greatly who is unfamiliar with the basics of grep (I know, right? Blew my mind). This is for him, hopefully it will help others. If you’re already familiar with this black magic   want to see a cool implementation, check out Grep is a Magical Beast.
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A Simple Look at Nanowire Assemblies in Optics

Below is a quick attempt to summarize and extrapolate from an article on Optical Routing And Sensing With Nanowire Assemblies in a simple manner, without assuming the reader is deeply familiar with the esoteric lexicon of photonics and optoelectronics. I can’t freely distribute the article which inspired this post; check with your local university to find a copy. More information on the relevant article is post script.

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English to Morse Translator

Today, I wanted to code an efficient letter to Morse code translator in Python and whipped this up. I’ve found that a familiarity with many of Python’s lesser-known built-in functions is quite useful in situations such as this!

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Visual Grade 2 Braille Dictionary Introduction

I learned braille for 3 reasons. The first is my hobby of picking something random and learning it. The second is because I wanted to learn touch typing. The third is because I often fell asleep while reading and left the light on. This way, I can read myself to sleep without a light on!

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My Weekly Planner Template

Sometimes, when juggling a particularly busy lifestyle, a stand-alone Todo list isn’t enough. I find it useful to supplement my Todo system with a weekly schedule in order to quickly allocate Todos to each day.

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Why Young Innovators Should Answer the Call

Disclaimer: The following is only my opinion. I am not directly affiliated with the Thiel Fellowship or Thiel Foundation, but I did have the opportunity this summer to interact with and work with many of the fellows and other people involved in the community.

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Temporarily Mute: An Overview of Communication Methods

If you’ve run into me in the past 2 days, I’ve squeaked at you and scribbled on my notebook “lost my voice! How are you?” As of this post, I am still mute. However, I will write it in the past tense to create a false sense of encouragement that my voice will return soon.

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Procrastinating? Fix it.

I am often asked how to stop procrastinating or asked how I get so much done. While I think I don’t get enough done, I will share some of my methods of crushing procrastination with mindfulness.

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Popular Weekdays

The code on my blog will range in quality from “I’m waiting in line and have 10 minutes to code” to “I’ve been working on this all day.”

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Duct-Tape Decoration

Let’s say you want to decorate an all-black surface, without damaging it deeply. If engraving and sharpies aren’t within your acceptable option set, I suggest duct-tape and an exact-o knife. Begin covering your surface with a duct tape canvas. Next, sketch your desired design on some paper (I suggest graph paper) and secure each piece of paper with scotch tape.

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Installing CoffeeScript on Ubuntu 13.04

Unfortunately, the current CoffeeScript docs do not support installation on the latest Ubuntu distro. To get around this, we must manually install the dependencies. Don’t worry, I’ve done most of the work for you.

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Small Math Puzzles Make My Day

I was recently hanging out with some friends, and one of them brought out an old math problem sheet. This problem sheet was briefly passed around and then put away again. One of the problems was a cute math puzzle. This problem was…

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mySQL: LIKE vs. REGEXP

Using LIKE in a query is an order of magnitude faster than using REGEXP. The downside is that LIKE doesn’t offer the control and generality that REGEXP does.

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Todo system

I used to drown in a sea of post-it notes and repetitive todo lists, my thoughts spread across unorganized Dropbox files, repetitive Evernote entries and various iPhone apps (Checklist, Notes, GoogleTasks).

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Setting Up pygeoip Environment on Ubuntu

GeoIP uses a large database to find information about a given IP address or website. I went through a couple different package installations before it integrated with my programs successfully. The fairly simple process I settled upon to set up the functional package, pygeoip (python API for GeoIP), is as follows:

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The Divine Computer

I have been drawing mainly hands for a while, so I figured I’d use my free time to sketch something different.

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Hadoop’s 5 Daemons

At work the other day, I was reading about Hadoop’s 5 daemons. The information wasn’t quite clicking, so I drew a picture to cement the concepts into my mind. (I’ve checked that all information regarding Hadoop in this blogpost is publicly available.)

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Python: Planning Lunches and Groceries

I bring my lunch to work to avoid food poisoning (I have food allergies, not paranoia). To be able to make these lunches for work when exhausted, I’d like a list of what to pack for a set number of days.

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Matlab: Smooth Rotating Animation for Line Plots

I recently became stuck trying to create an animation which consists of a smooth rotation of a viewpoint around the Lorenz attractor. Methods I use for changing viewpoints with respect to surface objects were creating jerky, lagging animations when applied to my line plot.

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Matlab: Lorenz Attractor

I’m a big fan of the Lorenz Attractor, which, when plotted, resembles the half open wings of a butterfly. This attractor was derived from a simplified model of convection in the earth’s atmosphere. One simple version of the Lorenz attractor is pictured below:

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An Essay: Why Doctor Who is Awesome

Doctor Who depicts the adventures of a humanoid alien, the Doctor, who roams the universe on a sentient spaceship. This spaceship is referred to as the TARDIS which is an acronym for Time and Relative Dimension(s) in Space. As a Time Lord, the Doctor is able to regenerate his body when he is near death. Each of his incarnations has their own quirks but otherwise share the memories and basic personality of the previous incarnations. The Doctor often brings human companions to accompany him on his quests through parallel universes and different dimensions. On these adventures he saves civilizations and rights wrongs. The Doctor regularly gains new companions and loses old ones. The companions provide a surrogate with whom the audience can identify, and they serve to further the story by manufacturing peril for the Doctor to resolve.

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Some Trombone Basics

In which I play a bit, debunk some misconceptions, teach you the slide positions and embouchure (i.e., mouth shape), and show you how to safely bike with a trombone.

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Matlab: Rotating Sphere Animation

Let’s say you want to make a simple simulation of a sphere spinning in Matlab. First, you set the pop-up window to have the title ‘Spheres,’ the window to have black background, and specify said window’s position and size.

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DC Josephson Effect

I created a video for my Modern Physics class explaining the DC Josephson Effect in less than 5 minutes. You might enjoy it, I sure enjoyed making it!

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Chess Tips for Beginners

Disclaimer: I’m a mediocre chess player, I do not consider myself a chess expert. These simple tips are for beginners looking to surprise intermediate players during games.

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Doctor Who Poems

I was writing a letter to my grandma. In the letter, I included a poem. My Uncle asked if I would send him more poems.

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CAMEL paper

I used Braille as a test language, but this is a framework to automate the decoding of any partially understood (ancient) language by creating probabilistic dictionaries.

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Pythonic Method Calling

I have an aversion to hard-coding. Hard coding is when you write out a long, elaborate code that could also be written with a dynamic loop. This usually limits the ability to easily adjust your own code in case you want to change something later (or re-use it). “Hard coding” refers to “rigidly” writing out things instead of keeping them dynamic.

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Listing Programming Languages on Your Resume

I love programming and know a few programming languages. Every programmer I’ve met has a favorite language, however, most of us can code in more than one language. As I was revising the ‘Language’ portion of my resume, I became confused.

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Introduction to Braille

I’m currently researching machine learning using Braille as my language platform. There is a Research Symposium for Mason’s College of Science at the end of April.

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Introduction to FractalCows

I am creating this blog in order to keep my family and friends updated on my adventures. I often neglect to update those close to me on my daily life and changing interests, because of my tendency to get caught up in my schoolwork and research. I expect this blog an eclectic patchwork of explanations, project updates, doodles, and code.

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