What's New in Mathematics?

What's New in Mathematics?

It's coming toward the end of the year, and that means there may be some "Best of 2023" articles coming about all kinds of things.

Why not have something similar for math? What are some of the latest newsworthy pieces in mathematics for this year?

I was going to write this kind of article, but then I realized that mathematics is a discipline that is (conservatively) thousands of years old and more likely tens of thousands of years old. A year's worth of mathematical progress is a drop in a lake. A small drop.

Instead, let's take a look at some of the biggest achievements of the past 60 years of mathematics, a somewhat bigger piece but closer to a glass of water in a lake.

Andrew Wiles finally solves Fermat's Last Theorem

One of the most prominent theorems of European mathematics is Fermat's Last Theorem. This was famously defined by Pierre de Fermat, a lawyer and amateur mathematician around 1637 who stated "I have discovered a truly marvellous demonstration of this proposition that this margin is too narrow to contain." (source).

The proof was elusive, however, and took over 350 years to be found. In 1994, the mathematician Andrew Wiles produced a proof that was validated in 1995, proving Fermat was correct. Since then, Wiles was knighted by Queen Elizabeth II in 2000.

The Four Colour Theorem is - basically - proven

Another famous mathematical theorem from recent times is the Four Colour Theorem. This is a theorem from graph theory that seems to be fairly simple at first glance. A loose statement of this theorem is that every two-dimensional map (such as a map of the United States depicting each state's boundaries) can be coloured using four colours such that each region of the map does not share a line with an adjacent region with the same colour. This was proven in 1976 by Kenneth Appel and Wolfgang Haken, but with this proof came considerable controversy.

First, there have been many false proofs of the four colour theorem, going back to the 1800s. As well, there were many false disproofs, where folks showed "counterexamples" of maps that were coloured with five or more colours. These supposed counterexamples could always be shown to be recoloured using four colours instead. But the biggest controversy came with the proof by Appel and Haken themselves, when they used a new and unfamiliar tool for their work: a computer.

The Four Colour theorem was the first major proof to be assisted by a computer. Appel and Haken took an approach of breaking down the theorem into smaller cases and proceeding by a sort of proof by cases approach. The twist is that there were many cases and these cases were quite computational in nature, so the pair invoked using a computer. Mathematicians were hesitant at the time to accept this proof since there were many cases and it was possible there was a mistake made in one of them. Eventually, the proof was confirmed and accepted. Four colours suffice.

The aperiodic monotile

In the mathematical equivalent of breaking news, a group of mathematicians, David Smith, Joseph Samuel Myers, Craig S. Kaplan, and Chaim Goodman-Strauss, have found the first example of an aperiodic monotile, also known as an einstein. The idea of such a tile is simple: a monotile is a single rigid tile that has a shape such that it can "tile" or cover the entire plane without repeating a pattern. There are other examples of aperiodic tilings involving distinct multiple tiles such as Penrose tilings but the tile found by the Smith et al is a single tile that will not repeat itself.

The aperiodic monotile could open up new avenues in mathematics, not just in planar geometry but elsewhere. This monotile also has been displayed artistically in several places.

The Invention of the Rubik's Cube

A beloved toy the world over, this colourful cube was formally invented by Ernő Rubik, a professor of applied arts in Hungary. First starting as an art project, he found the prototype helpful to illustrate group theoretic concepts, hence making the Rubik's cube the first non-mathematical application of group theory.

The key to the Rubik's cube is that it is solvable from any initial configuration. In the language of group theory, a Rubik's cube is a finite symmetric group where the group operation is a permutation of the faces of the cube. Since this is the case, every element of the group can be "translated" into another element by disjoint permutations. What this really means is that you can always solve a Rubik's cube, no matter the starting point.

That's it for now, please subscribe to this newsletter if you want to get more mathematical news and thoughts.