This is an exploration of some interesting parallels.

One might say that the categories of geometry and algebra cover most of mathematics. I think most mathematicians see them not as categories but as different ways to see the same object. Time and again, geometric intuition uncovers hidden dimensions in algebraic objects: the algebraic geometric of elliptic curves, as well as the standard practice of studying a large set of functions by considering each function as a point in a "function space," serve as ready examples. On the other hand, geometric spaces that we're interested in, such as the fabric of space-time, or crystal lattices, often have underlying symmetries that can be encoded as algebraic structures which afford us a compact, high-level language to state and prove far-reaching claims on the properties of the geometry.

Why are these two viewpoints so effective, so illuminating, so incisive? If we see geometric intuition as an extension of our human capability for spatial visualization and reasoning, then I'd say that's a pretty fundamental source for geometric intuition—an internalization of our everyday visual and tactile interaction with out environment. What about algebra then? We could tie it to our faculties of language, which associate meanings to symbols or strings of them (literally called a word in group theory), communicate that meaning, and process the information in the symbols by manipulating them. We really do have a "language of algebra." So the first parallel goes from geometry and algebra to spatial reasoning and linguistic ability.

But I like to go even further, rise higher in abstraction: is there any more fundamental difference, or similarity, between our faculties of spatial reasoning and language? How about space vs. time? Obviously geometry is space. If we think of multiplication in algebra as composing transformations one after the other, such as the linear transformations in linear algebra (Cayley's theorem guarantees that we can view any group as a transformation group), then algebra is a form of "discrete time travel."

What about other fields of math? If geometry is space and algebra is time, then perhaps analysis is change, and topology is connection.

I'm honored to serve as the undergraduate representative on MIT's Task Force on Open Access. Some have asked me, "what is open access?" Among many things, Open Access (OA) aims to make research free to access, distribute and build on top of. One might call it a "Creative Commons or Free Software movement for research." This post gives a brief overview, but I'd be excited to answer your questions about anything OA!

Mainly because the majority of scholarly peer-reviewed articles are published in academic journals, ~75% of which with charge steep subscription fees (i.e. "toll access") [1]. In 2016 the MIT Libraries paid over $6 million to give MIT affiliates free access to some subset of journal articles [2].

Some facts: The research community submits articles, peer-reviews them and serves on editorial boards, all for no pay from publishers, with university salaries from mostly public funding. Toll-access journals do copy-editing, formatting, marketing etc, but make disproportionate profits: Publishing giant Elsevier reported a profit margin of 36%, more than Google, Amazon or ExxonMobil [1,3]. Most subscription revenue comes from publicly-funded university libraries, whose budgets grow much slower than subscription costs. Small universities and research institutions, especially those from developing countries, are most vulnerable.

No, it can cover research datasets, course materials (like MIT OCW), digitized print work, source code, images and much more. For example, the Open Science Framework aims to make transparent and provide OA to every aspect of the research cycle. Science and humanities face similar issues in OA. OA can include all forms of content like novels, movies, software etc, but it's focusing on research because researchers want to distribute their results for free.

Besides ensuring your current and future access to scholarly publishing, access to the fruits of research should be expanded beyond the elite institutions that can barely afford the steep journal subscription fees, to developing countries, to precollege classrooms, and to non-researchers such as journalists and policymakers. That would be more equitable, and would accelerate innovation in science and the humanities.

I welcome you to learn more about OA by reading or attending conferences, and contribute to OA projects like the aforementioned Open Science Framework and Right to Research, or even organize an open access hackathon!

Jean-Claude Guédon's article "Open Access: Toward an Internet of the Mind" introduces the history, landscape and nuances of OA quite well. If it seems long, just read the bullet-pointed history starting from page 8 to see how we got here.

My interest in art led me to take the Introduction to Art History course at MIT. I learned the most by thinking very hard about the essay assignments, one of which asked me to analyze the importance of authenticity in art--if it is important at all! We were to look at the case of the 17th-century Dutch painter Rembrandt van Rijn, because it had been notoriously hard to figure out which paintings were actually made by him. I spent way more time than necessary contemplating the issue and putting my thoughts together. Prof. Kristel Smentek, who taught the class, gave comments on drafts of this essay that guided how I developed it further. The essay recently won the Kelly Essay Prize for Excellence in Humanistic Scholarship. I hope you find it an intriguing read!

Check out the other amazing art pieces in the Jaffa Museum's abums below! Vincent Floderer's haunting "sea creatures" fashioned out of crumpled paper are some of my favourites.

(This post is for prospective freshmen enrolling in the Massachusetts Institute Technology. I wish I knew all this early enough, so now I'm telling it to you!)

Origami? Folding bits of paper? You mean that children's pastime? It can't possibly be of any "real" use, right?
What if I told you that origami has been applied in space technology?

The 6th International Meeting on Origami in Science, Mathematics and Education (6OSME) was held in Tokyo University in August to help origami researchers, artists and educators share their ideas on the connections between origami and a wide range of other fields, including applications in theory and industry.

Back in 2010 I tinkered around with 3D graphics, and I learned how to use OpenGL, a 3D graphics engine, with the only language I knew well, Haskell. Eager to try it out, I embarked on a project to build from scratch a molecular graphics engine, because cool images of molecules always impressed me. Below are some snapshots from the program Hmol that I wrote:

Hmol is publicly available via Github under the GNU GPLv3 License (details in the Licensing section at the bottom).

I distinctly remember finishing much of the development during my exam period—no classes and no homework, so lots of spare time!

Grab a milk or juice carton and there's a fair chance you'd find that it's from Tetra Pak, a Swedish company that focuses on food packaging and processing solutions. A few months ago Tetra Pak invited ten people from around the world to a package design workshop aimed at brainstorming an improved design to one of their commercial packages. I qualified through their "Can You Fold It?" competition, in which contestants had to dream up a simple but creative way to close the end of a cardboard tube—which is basically what package manufacturers have to do.

Image from "Can You Fold It?" qualification phase

Tetra Pak's pioneering product was the Tetra Classic®, but probably more familiar is the Tetra Rex®.

One of the talks I usually give is on geometric dissections, which I begin with the following puzzle from (Lindgren 1964):

I thought it might have been more fun to give the puzzle to the prospective audience one week before I actually presented the solution, so I did that for an upcoming presentation on geometric dissections for students from Raffles Girls' School. I hope they got very frustrated by the puzzle. I also promised to give a hint to the puzzle, but I forgot to post it! Highlight the hint if you want to read it...

Lindgren, Harry (1964). Geometric Dissections.  D. Van Nostrand.

There was this thing going on Facebook:

I was given Stefan Banach (Poland). I will not discuss the Banach-Tarski Paradox.

Stefan Banach was one of the 20th century's greatest mathematicians. His self-taught prowess impressed the renowned Prof. Hugo Steinhaus, who supported Banach's career. His doctoral thesis sparked and grew into the field of "Functional Analysis", which has proved fundamental to Quantum Mechanics and Differential Equations (used widely in engineering). During World War II he and fellow academics were sheltered from Nazi persecution by employment as "lice feeders" at a Typhus Research Institute. (from Wikipedia)

He proved that:

1. Throw a world map on the floor. There's always a way to hammer a nail through the map and the floor so that the nail's position on the map exactly corresponds to its actual location.
2. Take a piece of paper, fold a paper airplane and press it flat within the original boundary of the paper. One of the points in the paper has been returned to its original position.

These are consequences of the Banach fixed-point theorem, which actually says that if we "shrink" the world (e.g. into a "map") so every pair of points can only get closer (this also happens when folding), there will be a "fixed point". The world might be (1) Earth, (2) a piece of paper or some other strange, convoluted, theoretical world.

This is one of many "fixed-point theorems", all of which say that if you take a "world", transform it so-and-so, then some point in the world must have remained immobile. The more famous Brouwer fixed-point theorem has been applied in Economics, to show that markets (the "world") have "fixed points" (e.g. balances between supply and demand).

I must thank a friend for telling me a basic form of this layman explanation many years ago; it has stuck with me all these years.