# Algebraic Topology via Analogy

“So, what kind of math do you study?”

Whenever someone finds out that I am a graduate student studying mathematics, that’s invariably one of the questions that they ask.  (Other favorites include “What applications does it have to the Real World™?” and “Do you want to become a professor?”  But that’s for another time.)

The type of math that I study (homotopy theory!) is a rather abstract and esoteric subfield of mathematics that is perhaps difficult to motivate or explain without a lot of prerequisite machinery and/or time.

So when trying to explain the math that I study to someone with little or no background in mathematics at all, I often find myself attempting to explain the broader subject area of topology, which encompasses what I do.

The classic explanation that I’ve heard many times is that topology is the study “rubber-sheet” geometry, in the sense that we can stretch and bend and deform the mathematical objects that we study, as this somehow does not affect the intrinsic properties that we consider.  We must be careful, however, not to puncture and tear them, as these actions do change the intrinsic properties of our objects.

The quintessential example that goes along with this explanation is the mumbling that in this setting, coffee mugs are “the same” as donuts, and that both are “different” from spheres.

I hate it.

The first reason is that it’s both hard to explain and hard to understand.  For a seasoned topologist, it’s easy to visualize the necessary contortions to transform a coffee mug to a donut.  For someone who has never thought about it before, it usually doesn’t make much sense.  And while there are some great GIFs on Wikipedia to help illustrate what’s going on, I’d rather not need to pull out my phone every time I need to explain what I do.

The more serious issue I have with this explanation, however, is that while it gives some insight into the way topologists think, it presents topology as an abstract novelty, and it isn’t clear at all why one would care about it. So instead let me present a different explanation of topology which has worked better for me.

One misconception that a lot of non-mathematicians have is that we already know everything to know in mathematics.  They’re used to having equations and formulas that allow you to say “this quantity is that”.  They’re used to dealing with numbers and concrete objects and data.

However, the truth is that we are not nearly so lucky. (If we did, then algebraic topology would not be so interesting!) It is often the case that we must deal with mysterious mathematical objects that are given to us by abstract definitions, existence theorems, universal properties, etc.  Therefore, the questions we deal with tend to be of the form

“Are these two things ‘the same’?”

and/or

“How can we tell them apart?”

It is then our job to find ways to demystify and understand these objects, as we usually cannot access them directly – much like an archaeologist would like to study an ancient civilization, or a paleontologist a species of dinosaur, without being able to travel back in time to see them firsthand.

Nevertheless, both paleontologists and archaeologists have various tools that they can use to indirectly understand the objects of their study, such as fossils, fossil records, trace fossils, or radiocarbon dating.  In other words, both paleontologists and archaeologists study the data that they can access, and can obtain from this data a picture of what is going on.

In the same way, topologists have analogous tools: we calculate what are called (homotopy) invariants such as Euler characteristic, co/homology groups, homotopy groups, etc.  The best of these invariants are both computable (or at least tractable), as well as powerful, in the sense that we can recover much of the information about the object we wish to study just by understanding the invariant.

That, in my eyes, is at the heart of what topologists do.  My particular branch of topology, algebraic topology, brings topological questions (when are two things “the same” or “different”?) into a realm (algebra!) where we can compute things and answer questions more easily.  And the particular subfield of algebraic topology that I study, homotopy theory, determines what it means for two things to be “the same” or “different”.  It turns out that this is both useful in the real world and interesting mathematically, and I hope to tell you about next time.

# My Favorite Number

A few weeks ago, one of my friends asked me the following question:

As a mathematician, what is your favorite number?

My first response was tongue-in-cheek.  I said that I would probably pick 2 or 3.  After all, I rarely work with large numbers in any of my mathematics classes.  Furthermore, as this blog post on Gödel’s Lost Letter shows, there’s a lot of mathematically interesting things about two and three. However, after some more talking and thinking, I remembered a story that I had heard in my number theory class about a particularly special large number,

$M_{67} = 2^{67} - 1$

$M_{67}$ is a special type of number called a Mersenne number, which are the numbers that can be expressed in the form $2^n - 1$ for an integer $n$.  In particular, if a Mersenne number is prime, we say it is a Mersenne prime.  A000668 is the OEIS entry for the sequence of Mersenne primes, and according to the Great Internet Mersenne Primes Search (GIMPS), there are currently 48 known Mersenne primes.  It is conjectured that there are an infinite number of Mersenne primes.

Why are we interested in figuring out when Mersenne numbers are prime?  We will soon see that it is (relatively) easy to generate a list of potential Mersenne primes.  Furthermore, by the Lucas-Lehmer primality test, there is an extremely simple algorithm for determining whether or not a Mersenne number is prime.  These two facts combined means that finding Mersenne primes is the quickest way to finding the largest prime numbers.  Indeed, the current top ten largest known primes are all Mersenne primes.

Theorem.  If $M_n = 2^n - 1$ is prime, then $n$ must be prime.

We prove the contrapositive statement: If $n$ is composite, then $M_n = 2^n - 1$ is composite.  Assuming $n$ is composite, we can write $n = ab$, with $a, b > 1$.  We then claim that $(2^a - 1)$ properly divides $(2^{ab} - 1)$.  This is clear, since $(2^a - 1) > 1$, and

$2^{ab} - 1 = (2^a - 1)(2^{(b-1)a} + 2^{(b-2)a} + \dots + 2^a + 1)$

Therefore, Mersenne primes must be of the form $M_p = 2^p-1$, where $p$ is prime.  Since we can find primes up to 9 digits through other means without too much effort, we now have a list of potential Mersenne primes, with digits numbering in the millions.

However, notice it is not the case that if $p$ is prime, then $M_p = 2^p-1$ is prime.  In fact, for $p = 11$, we have that $2^{11} - 1 = 2047 = 23*89$.  So, back in the 17th century, Marin Mersenne, for whom these numbers are named after, came up with the following list of primes $p$ for which he claimed $M_p$ was prime:

Claim (Mersenne). For $n =2 , 3, 5, 7, 13, 17, 19, 31, 67, 127$, and $257$, $M_n$ is prime.  $M_n$ is composite for all other $n < 257$.

At the time, his claim could not have been verified or proven, and it turns out that his claim was incorrect:  missing from his list were $M_{61}, M_{89},$ and $M_{107}$, which were shown to be prime.  Furthermore, $M_{67}$ and $M_{257}$ were shown to be composite.  This sets the stage for the following anecdote:

In 1903, Frank Cole was to give a talk.  Without a word, he went up to the blackboard, and began to raise 2 to the 67th power, and then subtracted 1.  Then, on the other side of the board, he multiplied 193,707,721 by 761,838,257,287, and got the same number.  He then returned to seat with applause, not having said a single word.

And that’s the story of my favorite number.  There are a bunch of other interesting large numbers in mathematics, as these posts on mathoverflow, math.stackeschange, and reddit indicate.  What’s your favorite number?