Published:
The goal of this post is to discuss Rokhlin’s Theorem and also the Rokhlin invariant which is for 3-manifolds. Theorem (Rokhlin): Let $M^4$ be a closed, smooth, spin 4-manifold. Then its signature $\sigma(M)$ is divisible by 16.
Published:
In this post, I’ll like to provide two proofs of the Pythagorean theorem which I find elegant and also, don’t really require anything other than pictures though, of course, I’ll discuss them with words as well. Now, recall that if we have a right triangle in the Euclidean plane with side lengths $a,b,c$ where $c$ is the length of the hypotenuse (the longest side), the Pythagoreans understood that the three numbers are related to each other by the equation $a^2 + b^2 = c^2$.
Published:
The goal of this post is to discuss Rokhlin’s Theorem and also the Rokhlin invariant which is for 3-manifolds. Theorem (Rokhlin): Let $M^4$ be a closed, smooth, spin 4-manifold. Then its signature $\sigma(M)$ is divisible by 16.
Published:
I’m not a homotopy theorist but ever since the paper by Abouzaid-McLean-Smith was released which showed the power of homotopy theory towards answering questions in symplectic geometry, I’ve been interested in stable homotopy theory. In particular, what the role of the Morava K-theories play. Of course, the Abouzaid-Blumberg paper is an earlier application of Morava K-theory to the Arnold conjecture. In that paper, one would like to define Hamiltonian Floer theory over $\mathbb{F}_p$ but because our counts of curves ought to divide by the order of finite stabilizer groups, it is too naive to do mod $p$ counts since $p$ would not be invertible. Instead, one uses the hierarchy of generalized cohomology theories $K_p(n)$ which approximates $\mathbb{F}_p$ as $n$ gets larger. The notation here with the subscript $p$ is unconventional but I wanted to emphasize that we’ve fixed a prime $p$. I’ll continue to do this for a while but will eventually suppress that part of the notation.
Published:
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.
Published:
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.”
Published:
I’m teaching Calculus 1 this semester and one of my former students introduced me to the following problem which needs only Calculus 1 ideas.
Published:
Let $p(n)$ denote the number of unordered partitions of $n$. For example, if $n=5$, we can partition this as $1+1+1+1+1,1+1+1+2,1+1+3,1+4,5,1+2+2,2+3$. So $p(5)=7$. We can write down a generating function $P(x) = \sum^\infty_{n=0} p(n)x^n$. But also, observe that if we wrote out $(1+x+x^2+x^3+…)(1+x^2+x^4+x^6…)…(1+x^k+x^{2k}+x^{3k})…$ and start multiplying things out, it amounts to picking an element from each “bubble.” If we want a finite product, we eventually only choose 1’s. So pick something from the first bubble, say $x^2$, $x^6$ from the second, and $x^6$ from the third, then all 1’s. The product is $x^{14}$ and these choices gave us a partition $14=1+1+2+2+2+3+3$. How did we produce this? Our choice of $x^2$ from the first bubble says we want two contributions of 1. Our choice of $x^6 = (x^2)^3$ from the second bubble says we want three contributions of 2. And our choice of $x^6=(x^3)^2$ from the third bubble says we want two contributions of 3.
Published:
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.
Published:
Consider the following system of nonlinear ODEs. \(\begin{cases} \frac{dx}{dt} = 2-x-y \\ \frac{dy}{dt} = y-x^2 \end{cases}\)
Published:
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: \(\begin{cases} x'(t) = f(x,y) \\ y'(t) = g(x,y) \end{cases}\)
Published:
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.
Published:
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).
Published:
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.
Published:
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.
Published:
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.”
Published:
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.
Published:
In this post, we’ll consider an example related to the Divergence Theorem and gravity. I think it’s a nice example to show students taking multivariable/vector calculus (which I’m teaching this semester). Let’s first recall the statement of the Divergence Theorem. Let $W \subset \mathbb{R}^3$ be a 3-dim region with boundary being a piecewise smooth, closed surface $S$. It’s a somewhat nontrivial fact that all closed hypersurfaces of $\mathbb{R}^n$ are orientable but in this case, one can use the cross product on $\mathbb{R}^3$ to produce an orientation on a closed surface. Note the closedness is essential; the Möbius band is a nonorientable surface in $\mathbb{R}^3$ but it is not closed. Moreover, in $\mathbb{R}^3$, having a unit normal vector field on $S$ induces an orientation on $S$. So let’s suppose that $W$ is oriented and $S$ has the induced orientation from $W$. If $F$ is a vector field defined on a neighborhood of $W$, we may find the divergence of $F$ which is a scalar function that intuitively, measures the net flow of $F$ at a given point.
Published:
In a class on vector calculus, the students are presented with a few theorems such as the Fundamental Theorem of Line Integrals, Green’s Theorem, Stokes’ Theorem, and the Divergence Theorem. The first one relates differentiation and integration in the following way. Take the gradient of a multivariable function $f:\mathbb{R}^n \to \mathbb{R}$ and consider a smooth path parametrized by $\vec{r}(t)$ with $t\in [a,b]$. Then the definite integral of the dot product $\nabla f \cdot d\vec{r} \, dt$, a quantity arising from a 1-dimensional object has a 0-dim interpretation: $\int^a_b \nabla f \cdot d \vec{r}\,dt = f(\vec{r}(b)) - f(\vec{r}(a))$. The proof of this really is just to use the usual Fundamental Theorem of Calculus.
Published:
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.
Published:
I’m not a homotopy theorist but ever since the paper by Abouzaid-McLean-Smith was released which showed the power of homotopy theory towards answering questions in symplectic geometry, I’ve been interested in stable homotopy theory. In particular, what the role of the Morava K-theories play. Of course, the Abouzaid-Blumberg paper is an earlier application of Morava K-theory to the Arnold conjecture. In that paper, one would like to define Hamiltonian Floer theory over $\mathbb{F}_p$ but because our counts of curves ought to divide by the order of finite stabilizer groups, it is too naive to do mod $p$ counts since $p$ would not be invertible. Instead, one uses the hierarchy of generalized cohomology theories $K_p(n)$ which approximates $\mathbb{F}_p$ as $n$ gets larger. The notation here with the subscript $p$ is unconventional but I wanted to emphasize that we’ve fixed a prime $p$. I’ll continue to do this for a while but will eventually suppress that part of the notation.
Published:
I am not a number theorist but this topic has fascinated me and I’ve decided to write about it as one of the first blog posts. The classification of definite forms is difficult but also appears in the study of 4-manifolds since Freedman proved that closed, oriented, simply-connected, topological 4-manifolds are classified by their intersection form. This post isn’t about that so much. Much of what I write here can be found in A First Course in Modular Forms by Fred Diamond and Jerry Shurman.
Published:
Recently, a friend introduced me to a problem posed by Presh Talwalkar in a video here. The problem is as follows. Take an equilateral triangle and inscribe a circle into it. Pick any point $P$ on the circle and connect line segments from $P$ to each of the vertices. The challenge is to show that the sum of the squares of the lengths is actually a constant, independent of the placement of $P$. The video gives a nice proof of the fact but when one studies it more closely, we see that we can actually generalize the statement quite a bit.
Published:
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.
Published:
I am not a number theorist but this topic has fascinated me and I’ve decided to write about it as one of the first blog posts. The classification of definite forms is difficult but also appears in the study of 4-manifolds since Freedman proved that closed, oriented, simply-connected, topological 4-manifolds are classified by their intersection form. This post isn’t about that so much. Much of what I write here can be found in A First Course in Modular Forms by Fred Diamond and Jerry Shurman.
Published:
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.
Published:
In this post, we’ll consider an example related to the Divergence Theorem and gravity. I think it’s a nice example to show students taking multivariable/vector calculus (which I’m teaching this semester). Let’s first recall the statement of the Divergence Theorem. Let $W \subset \mathbb{R}^3$ be a 3-dim region with boundary being a piecewise smooth, closed surface $S$. It’s a somewhat nontrivial fact that all closed hypersurfaces of $\mathbb{R}^n$ are orientable but in this case, one can use the cross product on $\mathbb{R}^3$ to produce an orientation on a closed surface. Note the closedness is essential; the Möbius band is a nonorientable surface in $\mathbb{R}^3$ but it is not closed. Moreover, in $\mathbb{R}^3$, having a unit normal vector field on $S$ induces an orientation on $S$. So let’s suppose that $W$ is oriented and $S$ has the induced orientation from $W$. If $F$ is a vector field defined on a neighborhood of $W$, we may find the divergence of $F$ which is a scalar function that intuitively, measures the net flow of $F$ at a given point.
Published:
The standard definition for integrable system, which, for example, can be found on the Wikipedia article, is due to Liouville. Given a Poisson manifold $(P,\{ \})$ parametrising the states of a mechanical system, a Hamiltonian function $H$ defines a vector field $X_H := \{H,-\}$, whose flows are the classical trajectories of the system. A function $f$ which Poisson-commutes with $H$ is constant along the classical trajectories and hence is called a conserved quantity. For example, if $(P,\omega)$ is a symplectic manifold, then there is a Poisson bracket given by ${f,g} = \omega(X_f,X_g)$ where $X_f,X_g$ are Hamiltonian vector fields; i.e. $\iota_{X_f} \omega = df$ and similarly for $g$. Note that $\{f,H\} = \omega(X_f,X_H) = df(X_H)$ and so if this is zero, then $f$ is constant in the direction of $X_H$ which are the classical trajectories.
Published:
Imagine two parallel straight copper rods set a centimeter apart that extend infinitely out in both directions and suppose there are positive unit charges spaced out evenly on the two rods. We’ll also suppose that the charges on one rod are lined up with charges on the other. You’re an observer some distance away and you take note that the rods want to repel each other because everything is positively charged; call this amount $\alpha$. We’ll suppose the rods are magical and can never be more than one centimeter apart. So they aren’t actually repelled.
Published:
Yesterday, I attended the last summer Sunday at Brookhaven National Labs and in particular, did a brief tour of the engineering department that worked on building the RHIC (relativistic hadron ion collider) as well as hear a lecture by Gene van Buren on a solenoidal tracker attached to RHIC called the STAR detector. The RHIC is able to accelerate gold nuclei to 99.995% the speed of light; the track has about a 2.5 mile circumference and the particles travel around it about 80,000 per second. At speeds like this, the particles are very massive and experience very little time. This statement is not something we usually say in our everyday experiences but if you have some experience with special relativity, you know that in order to accelerate something, you need to apply force. At speeds near the speed of light, there is a diminishing return. In order to gain a fraction of speed, one needs to apply more and more force and asymptotically, one needs an infinite amount of force to get to the speed of light. If we view mass as a property which resists being moved (faster), then one way to view relativistic particles is that they’re very massive; cf. moment of inertia.
Published:
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.
Published:
This is yet another post in the series of “I am not at all an expert in this but enjoyed thinking about it.” Also, I wanted to write something shorter, that doesn’t require a big time commitment to read. This seems to fit the bill. The problem that will be posed is basic in the sense that it is easy to understand the question but I did not solve it right away.
Published:
This post is about a short story by Ted Chiang entitled Anxiety is the Dizziness of Freedom. It is found in his collection Exhalation which has several other short stories, all worth reading. In case you don’t know, Ted Chiang is a Chinese-American writer and his short story Story of Your Life is possibly his most well-known work since it was adapted into the movie Arrival by the visionary Denis Villeneuve. Though Chiang doesn’t write science fiction exclusively, many of his stories can be considered science fiction though I’d argue that his focus is often not so much on the science side of things; I’ll try to convey what I mean in this post. A fun fact (for me anyways) is that he was born in Port Jefferson, NY where I lived for a year. His father is a professor at Stony Brook University in mechanical engineering though we never crossed paths while I was at SBU.
Published:
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).
Published:
Consider the following system of nonlinear ODEs. \(\begin{cases} \frac{dx}{dt} = 2-x-y \\ \frac{dy}{dt} = y-x^2 \end{cases}\)
Published:
I’m teaching Multivariable and Vector Calculus at Trinity College this fall semester and have students with all kinds of backgrounds. For some of them, this is their first semester of college and are used to year-long math classes rather than a whole course in one semester. The calculus they had was high-school AP Calculus which is at a different level than college-level calculus and can vary a lot depending on, say, whether it’s a public high school or a private science and technology magnet school. Some of them finished high school on Zoom because of COVID and for many, distance learning is not as effective as in-person learning. And then they took a break from math for a full year. Some of them don’t speak English as their first language. For others, they’ve taken the first two college-level calculus courses and have also taken Physics: Mechanics and Physics: Electromagnetism where vector calculus already appears. So this class is a breeze for them. One of my students went the first three months of his first year before losing a single point in any of his classes. English is not his first language and so he seems to appreciate that in a math course, the most important language to understand is mathematics (though of course that happens by means of English in my classroom).
Published:
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).