Math3ma

Last time we discussed the rational canonical form (RCF) of a linear transformation, and we mentioned that any two similar linear transformations have the same RCF.

The notion of a topological space being Hausdorff or normal identifies the degree to which points or sets can be "separated."

One of my students recently said to me, "I'm not good at math because I'm really slow."

Welcome back to our mini-series on categorical limits and colimits!

A while ago I wrote a post in response to a great Slate article reminding us that math - like writing - isn't something that anyone is good at without (at least a little!)

Today we talk tensor products.

In any area of math, it's always good idea to keep a few counterexamples in your back pocket.

Have you ever come across the words "commutative diagram" before?

It's possible that a set $X$ can be endowed with two or more topologies that are comparable.

Yes!

I'd like to embark on yet another mini-series here on the blog.

Hi everyone.

Have you heard the buzz?

If $X$ and $Y$ are normed vector spaces, a linear map $T:X\to Y$ is said to be bounded if $\|T\|< \infty$ where $$\|T\|=\sup_{\underset{x\neq 0}{x\in X}}\left\{\frac{|T(x)|}{|x|}\right\}.$$ (Note that $|T(x)|$ is the norm in $Y$ whereas $|x|$ is the norm…

Up to isomorphism” is a phrase that seems to get thrown around a lot without ever being explained.

The Borel-Cantelli Lemma says that if $(X,\Sigma,\mu)$ is a measure space with $\mu(X)<\infty$ and if $\{E_n\}_{n=1}^\infty$ is a sequence of measurable sets such that $\sum_n\mu(E_n)<\infty$, then $$\mu\left(\bigcap_{n=1}^\infty \bigcup_{k=n}^\infty E_k\…

Here's another unspoken rule of mathematics: English doesn't always commute!

Let a group $G$ act on a set $X$.

You're given a ring $R$ and are asked to show it's not a UFD.

In short, "Noetherian-ness" is a property which generalizes "PID-ness."

Here's an algebra tip!

An incomplete metric space is very much like a golf course: it has a lot of missing points!

Here's a measure theory trick: when asked to prove that a set of points in $\mathbb{R}$ (or some measure space $X$) has a certain property, try to show that the set of points which does NOT have that property has measure 0!

Once upon a time, while in college, I decided to take my first intro-to-proofs class.

In this week's episode of PBS Infinite Series, I shared the following puzzle: Consider a square in the xy-plane, and let A (an "assassin") and T (a "target") be two arbitrary-but-fixed points within the square.

A while ago, I wrote about the importance of open sets in topology and how the properties of a topological space $X$ are highly dependent on these special sets.

As promised, here is the first in our triad of posts on basic category theory definitions: categories, functors, and natural transformations.

Our goal for today is to construct a Lebesgue measurable set which is not a Borel set.

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.

You know the routine.

Mathematicians are a creative bunch, especially when it comes to naming things.

Welcome to part three of a six-part series where we prove that the fundamental group of the circle $\pi_1(S^1)$ is isomorphic to $\mathbb{Z}$.

Welcome to part five of a six-part series where we prove that the fundamental group of the circle $\pi_1(S^1)$ is isomorphic to $\mathbb{Z}$.

Yesterday I received a disheartening 44/50 on a homework assignment.

Dear Autocorrect, No.

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 we're resuming our informal chat on quotient groups.

Have you ever read Tristan Needham’s Visual Complex Analysis?

The Baire Category Theorem is a powerful result that relates a metric space to its underlying topology.

This post is the fifth example in an ongoing list of various sequences of functions which converge to different things in different ways.

Today we're looking at a fairly simple proof of a standard result in measure theory: Theorem: Any measurable subset $A$ of the real line with positive measure contains a non-measurable subset.

This post is the fourth example in an ongoing list of various sequences of functions which converge to different things in different ways.

A function $f$ is said to be continuous if the preimage of any open set is open.

Welcome back to our little discussion on quotient groups!

My goal for today is to provide a step-by-step guideline for classifying closed surfaces.

Hello friends!

Welcome to part two of a six-part series where we prove that the fundamental group of the circle $\pi_1(S^1)$ is isomorphic to $\mathbb{Z}$.

In last last week's episode of PBS Infinite Series, we talked about different flavors of multiplication (like associativity and commutativity) to think about when multiplying things that aren't numbers.

In this post, we visually explore the definition of a transitive group action and see how it relates to the phrase, "Where there's a will, there's a way!"

In this post we ramble on about Riemann surfaces, the uniformization theorem, universal covers, and two secret (or not-so-secret!)