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A list of all the posts and pages found on the site. For you robots out there is an XML version available for digesting as well.
Pages
About
About me
Posts
Optimizing your Wordle Strategy
Published:
Wordle seems to have quickly become quite popular. As word games go, it’s a good option if you want a quick dose. And as I’m beginning to find, it can be rather addictive.
Catalan Numbers in Perturbation Theory
Published:
Catalan numbers are everywhere. Quite famously, in volume 2 of his Enumerative Combinatorics, Richard P. Stanley gives 66 different interpretations of these numbers. They are given by
Jacobi’s Triple Product
Published:
There are some surprising connections between bosonization in quantum field theory, classical theorems in number theory, and characters of the affine algebra $A_1^{(1)} = \widehat{\mathfrak{su}}(2)_1$. All of them connect to the Jacobi triple product identity:
Continued Fractions 2: The Final $F$-coalgebra
Published:
In a previous post, we addressed some questions whether continued fractions can be considered a natural and practical way to represent real numbers. From a decimal standpoint, continued fractions can at first seem awkward, but soon some nice properties start to emerge. For instance, a number is rational if and only if its continued fraction expansion terminates. This is much simpler than in decimal expansions, when rational numbers can either have finite expansions or infinite but eventually periodic ones. Better yet, infinite but eventually periodic expansions also have their place in the land of continued fractions: they correspond to quadratic irrationals, a beautiful fact that is somewhat harder to prove.
Tropical Geometry of ReLU Neural Nets
Published:
A rectified linear unit (ReLU) is a neuron with activation function
Continued Fractions: Introduction and Gosper Arithmetic
Published:
Continued fractions are expressions of the form \[\lbrack a_0;a_1,a_2,\ldots\rbrack = a_0 + \frac{1}{a_1 + \frac{1}{a_2 + \frac{1}{\cdots}}}.\] Every real number has a unique expression as a continued fraction, so long as we exclude representations which end in 1, since \[a_0;a_1,\ldots,a_n,1] = [a_0;a_1,\ldots,a_n+1].\] Note that this is no worse (and probably quite a bit better) than the ambiguity in decimal expansions, for which \[1 = 0.999\cdots.\]
notes
Splash 2017
Published:
Splash is a program in which undergraduate students develop and teach short courses to local high school students. I taught a seven-part seminar on various topics in physics. Here are brief descriptions and the slides.
Relativity, Electromagnetism, and Least Action
Published:
These are some brief notes reviewing some of the most important topics in classical mechanics, and applying them to electromagnetism. The four-potential and the field strength tensor are motivated physically, and the Lagrangian for the electrodynamic field is introduced.
Splash 2018
Published:
Splash is a program in which undergraduate students develop and teach short courses to local high school students. I gave seven lectures on various topics in mathematics and physics. Here are brief descriptions and the slides.
Quantum Mechanics
Published:
The basic principles of quantum mechanics are developed for a first-time student, from the perspective of canonical quantization.
Gauge Theory
Published:
Some unfinished notes on classical gauge theory. There is particular emphasis on the mathematics of connections on principal bundles.
Undergraduate Physics in a Hurry
Published:
A preparation guide for the Princeton University graduate preliminary exams in physics. It is one-third exposition, one-third worked prelim problems, and one-third bad jokes.
publications
Mechanics: An Extended Introduction
Published:
The size–luminosity relationship of quasar narrow-line regions
Published in MNRAS, 2018
In some distant galaxies – many of them so distant that the light we see from them was emitted some 10 billion years ago – a supermassive black hole, containing the mass of millions or billions of Suns, is surrounded by a disk of gas. As this gas falls into the black hole, its energy is converted into radiation across the electromagnetic spectrum, forming an extraordinarily luminous object known as a quasar. Quasars are so luminous that they can outshine entire galaxies, and thus they are a force to be reckoned with in their own host galaxies. Astronomers studying galaxy formation are particularly interested in how they formation of a quasar can lead to feedback on the host galaxy, whereby the quasar is so powerful that it blows away gas in the galaxy and diminishes the formation of more stars.
The spatial extension of extended narrow line regions in MaNGA AGN
Published in MNRAS, 2019
As part of the Sloan Digital Sky Survey, the MaNGA (Mapping Nearby Galaxies at APO) project is producing a catalog of integral field unit (IFU) spectra of nearby galaxies. IFU spectra provide detailed information about the light coming from different points in a galaxy. This is an invaluable tool for astronomy, and in particular, when these galaxies are hosts of Type II quasars, the IFU spectra can be used to calculate the [O III] size and luminosity, the quantities we modeled here. This paper uses MaNGA data to add many more points to the observed size/luminosity relationship, and among many other things, addresses whether that model holds water.
Kerr-Schild Double Copy and Complex Worldlines
Published in JHEP, 2020
The Schwarzschild black hole is “like” a point charge. It’s a mass localized at the origin which sources a field ($h_{\mu\nu} \equiv g_{\mu\nu}-\eta_{\mu\nu}$) which decays as $\frac{1}{r}$, just like a point charge is a charge localized at the origin which sources a field ($A_\mu$) which decays as $\frac{1}{r}$. Is there a precise way in which these two objects are related? In fact, yes: in 2014, Monteiro et al. presented the Kerr-Schild double copy, which provides a prescription for taking certain stationary solutions to the Einstein equations and writing them as “squares” of solutions to the Maxwell equations.
Formation of Orion Fingers
Published in MNRAS, 2020
A short time ago (about 500 – 1000 years), in a galaxy very nearby (our own Milky Way, in fact), something fascinating happened in the Orion Nebula. If only humans had evolved and developed astronomy on a slightly faster schedule, we might have witnessed this event. Regrettably, we procrastinated for several hundred years in the Middle Ages, and missed the show. All we can see is the aftermath, known as the Orion fingers.