Recently, my friends and I were talking about the following problem. Let $n$ be a positive integer and consider sequences of $+$ and $-$ signs of length $n$. To each sequence, we can assign the $i$ th term to $i$ and hence, get a series which sums to some integer. For example, with $n=4$, the sequence $+,-,-,+$ gives $1-2-3+4 =0$. In fact, for any consecutive 4 integers, this sequence gives 0: $k-(k+1)-(k+2)+(k+3)=0$.
I was talking with a friend recently about a logic puzzle. Back in college, when first learning about the Principle of Mathematical Induction, we were told the following puzzle concerning green-eyed dragons.
In October 2023 ago, I wrote a post about the Three Body Problem and how it’s an example of a chaotic system and moreover, is not completely integrable. Recently, I’ve been reading Agustin Moreno’s wonderful lecture notes on the subject, particularly because his aim is to show how symplectic and contact topology apply to the study of the Three Body Problem. I wanted to write down some of what I learned; nothing here is original to me. In particular, I want to write about the Circular Restricted Three Body Problem (CR3BP). This is a version of the traditional problem where three point masses live in a system where only Newtonian gravity is at play. One of the masses is assumed to be much smaller than the others so that it is essentially negligible; i.e. it’s movement is influenced by the two primary masses but it does not affect the movement of the primaries. Moreover, it is assumed that the other two masses move in circles around their common center of mass.
This is a somewhat different post from my previous ones about applying topology. However, the spirit of those posts was to illustrate unexpected applications and this post will remain in this spirit, if only because we’ll be doing things in a backwards and needlessly complicated way which perhaps breaks expectations. Here is a question: given the ellipse defined by $(ax+by)^2+(cd+dy)^2 = r^2$ in $\mathbb{R}^2$, what are the lines which contain its major and minor axes? I actually never took a formal geometry class in high school nor college so I don’t know what the usual pedagogical approach is to learning this.
Here is an interesting application of topology that I learned from Cumran Vafa. In astronomy, there is the phenenomenon of gravitational lensing which is when light shining from some star or galaxy gets bent as it travels towards us on earth. The bending is due to a strongly gravitating object such as a black hole (the light doesn’t “fall in” if it doesn’t cross the event horizon). Thus, we sometimes see multiple images of the same object because of this effect. We may even see the images inverted, similar to how when you look at your own reflection in a spoon, depending on the distance you hold it from yourself, you might see your image flipped. Here is an image, courtesy of NASA. Note that around the center, we have four bright white spots which look similar. I believe these should actually all be the same object, just seen more than once due to this effect.
This is another post where I give an application of topology to something outside of math. In my freshmen year of college, some friends and I were too tired to do homework after dinner but too awake to go to bed. So we began cutting up strips of paper and then taping them to gether with various twists and seeing what would happen when we cut these up again. The most well-known of these creations is a cylinder where you simply take a long rectangular strip of paper and tape the two ends together without any flourish. But the second-most well known is the Möbius band which is made from taking a long rectangular strip of paper, making half a twist in the strip, and then taping the two ends together. If you then cut the cylinder by following the midline of the strip, you’ll get an unsurprising result: you cut it into two cylinders. However, what happens if you cut the Möbius band in a similar way down the midline? If you’ve never done this before and want to give it a try, pause here and don’t read on. I’ll put a picture of the Lofoten Islands in Norway here for you to enjoy while you do some “experimental mathematics.”
I recently learned of a fun “application” of topology though this has been widely known for decades, probably. I’ll present it from my own perspective. A gimbal is a tool built from three nested rings, meant to model rotations in $\mathbb{R}^3$ and also used by older systems for navigation and orientation. Let $SO(3)$ denote the Lie group of $3\times 3$ real matrices with $\det = +1$; these give rotations on $\mathbb{R}^3$ and together, they form a space which one can identify as being diffeomorphic to $\mathbb{RP}^3$, real projective space; I’ll describe why later.
Another topic I’ve introduced in my differential equations class regards nonlinear systems. Let’s consider just systems of two equations of the following form: \(\begin{cases} x'(t) = f(x,y) \\ y'(t) = g(x,y) \end{cases}\)
I’m teaching an introductory differential equations class this semester and one of the topics is about studying harmonic oscillators in various settings. They’re a great source of 2nd order ODEs and the general form they take is $x’’ + p x’ + qx = f(t)$ where the LHS of the equation comes from Hooke’s Law. For a moment, suppose $f(t)=0$ and $p,q > 0$. Then we have $x’’ = -px’ -qx$; if we think of $x(t)$ as describing the position of some object along a line on a spring with mass normalized to 1, then the $-qx$ term tells us that the force of the spring is always towards the equilibrium position. If $x(t)>0$, this means the spring is stretched and the force wants to pull it back. If $x(t) < 0$, then the spring is compressed and wants to push. Thus, $q$ is called the spring constant. The other constant is related to factors that depend on the velocity. For example, if the object moves quickly, then it may experience greater forces related to friction. Thus, the constant $p$ is called the damping coefficient. When $f(t) \neq 0$, we view it as an extra term telling us how the forces are affected. So we’ll call it a forcing term; note that we’re only asking it to depend on time so it’s an external force that doesn’t care about the position of the object.
I’ll be teaching a class on differential equations and another on statistics this coming Spring 2024. I’ve been wondering what similarities or relations there are between the two classes. One of the important theorems of differential equations is that given a 1st order differential equation $\frac{dy}{dt} = f(t,y)$ where $f$ and $\partial_y f$ are continuous and has some initial conditions, there exists a unique solution on a small neighborhood of the initial condition. One of the important theorems of statistics is the Central Limit Theorem which, in a very weak form, says: Suppose that $X_i$ are independent, identically distributed random variables with zero mean and variance $\sigma^2$. Then $\frac{1}{\sqrt{N}}\sum^N_{i=1}X_i \to \mathcal{N}(0,\sigma^2)$ as $N \to \infty$. Here, $\mathcal{N}(0,\sigma^2)$ means a normal distribution with mean $\mu = 0$ and variance $\sigma^2$. It’s not so obvious but there is some similarities between these two theorems in that there are proofs for them which involve the notion of a fix point. The goal of this post is to spell this out a bit more (though not fully rigorously).
