Commutative Diagrams Explained

Have you ever come across the words "commutative diagram" before? Perhaps you've read or heard someone utter a sentence that went something like

"For every [bla bla] there exists
the following diagram commutes."

and perhaps it left you wondering what it all meant.

The Most Obvious Secret in Mathematics

Yes, I agree. The title for this post is a little pretentious. It's certainly possible that there are other mathematical secrets that are more obvious than this one, but hey, I got your attention, right? Good. Because I'd like to tell you about an overarching theme in mathematics - a mathematical mantra, if you will. A technique that mathematicians use all the time to, well, do math.

In recent months, several of you have asked me to recommend resources for various subjects in mathematics. Well, folks, here it is! I've finally rounded up a collection of books, PDFs, videos, and websites that I found helpful while studying for my intro-level graduate courses.

Automorphisms of the Riemann Sphere

This is the last in a four-part series in which we prove that the automorphisms of the unit disc, upper half plane, complex plane, and Riemann sphere each take on a different form. Today our focus is on the Riemann sphere.

Automorphisms of the Complex Plane

This is part three of a four-part series in which we prove that the automorphisms of the unit disc, upper half plane, complex plane, and Riemann sphere each take on a different form. Today our focus is on the complex plane.

Automorphisms of the Upper Half Plane

This is part two of a four-part series in which we prove that the automorphisms of the unit disc, upper half plane, complex plane, and Riemann sphere each take on a different form. Today our focus is on the upper half plane.

Automorphisms of the Unit Disc

This is part one of a four-part series in which we prove that the automorphisms of the unit disc, upper half plane, complex plane, and Riemann sphere each take on a different form. Today our focus is on the unit disc.

Three Important Riemann Surfaces

In this post we ramble on about Riemann surfaces, the uniformization theorem, universal covers, and two secret (or not-so-secret!) techniques that mathematicians use to study a given space. Our intent is to provide motivation for an upcoming mini-series on the automorphisms of the unit disc, upper half plane, complex plane, and Riemann sphere.

The Pseudo-Hyperbolic Metric and Lindelöf's Inequality (cont.)

Last time, we proved that the pseudo-hyperbolic metric on the unit disc in is indeed a metric. In today’s post, we use this fact to verify Lindelöf’s inequality which says, "Hey! Want to apply Schwarz's Lemma but don't know if your function fixes the origin? Here's what you do know...."