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. Suppose that the square behaves like a billiard table, so that any ray (a.k.a "shot") from the assassin will bounce off the sides of the square, with the angle of incidence equaling the angle of reflection. Puzzle: Is it possible to block any possible shot from A to T by placing a finite number of points in the square?

One day while doing a computation on the board in front of my students, I accidentally wrote 1 + 1 = 1. (No idea why.)

Student: Um, don't you mean 1 + 1 = 2?

Me (embarrassed): Oh right, thanks.

[Erases mistake. Pauses.] 

Wait. Is there a universe in which 1 + 1 = 1?

Welcome back to our mini-series on categorical limits and colimits! In Part 1 we gave an intuitive answer to the question, "What are limits and colimits?" As we saw then, there are two main ways that mathematicians construct new objects from a collection of given objects: 1) take a "sub-collection," contingent on some condition or 2) "glue" things together. The first construction is usually a limit, the second is usually a colimit. Of course, this might've left the reader wondering, "Okay... but what are we taking the (co)limit of ?" The answer? A diagram. And as we saw a couple of weeks ago, a diagram is really a functor.

Today I'd like to talk about Brouwer's Fixed Point Theorem. Literally! It's the subject of this week's episode on PBS Infinite Series. Brouwer's Fixed Point Theorem is a result from topology that says no matter how you stretch, twist, morph, or deform a disc (so long as you don't tear it), there's always one point that ends up in its original location.

Last week was the start of a mini-series on limits and colimits in category theory. We began by answering a few basic questions, including, "What ARE (co)limits?" In short, they are a way to construct new mathematical objects from old ones. For more on this non-technical answer, be sure to check out Limits and Colimits, Part 1. Towards the end of that post, I mentioned that (co)limits aren't really related to limits of sequences in topology and analysis (but see here). There is however one similarity. In analysis, we ask for the limit of a sequence. In category theory, we also ask for the (co)limit OF something. But if that "something" is not a sequence, then what is it?

Answer: a diagram.

I'd like to embark on yet another mini-series here on the blog. The topic this time? Limits and colimits in category theory! But even if you're not familiar with category theory, I do hope you'll keep reading. Today's post is just an informal, non-technical introduction. And regardless of your categorical background, you've certainly come across many examples of limits and colimits, perhaps without knowing it! They appear everywhere--in topology, set theory, group theory, ring theory, linear algebra, differential geometry, number theory, algebraic geometry. The list goes on. But before diving in, I'd like to start off by answering a few basic questions.

This blog post is a continuation of today's episode on PBS Infinite Series, "Topology vs. 'a' Topology." My hope is that this episode and post will be helpful to anyone who's heard of topology and thought, "Hey! This sounds cool!" then picked up a book (or asked Google) to learn more, only to find those formidable three axioms of 'a topology' that, admittedly do not sound cool. But it turns out those axioms are what's "under the hood" of the whole topological business! So without further ado, let's pick up where we left off in the video.

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. My examples of multiplying non-numbers were vectors and matrices, which come from the land of algebra. Today I'd like to highlight another example: We can multiply shapes!

Hi everyone! Here's a bit of exciting news: As of today, I'll be extending my mathematical voice from the blogosphere to the videosphere! In addition to Math3ma, you can now find me over at PBS Infinite Series, a YouTube channel dedicated to the wonderful world of mathematics.

Last week we introduced the definition of an operad: it's a sequence of sets or vector spaces or topological spaces or most anything you like (whose elements we think of as abstract operations), together with composition maps and a way to permute the inputs using symmetric groups. We also defined an algebra over an operad, which a way to realize each abstract operation as an actual operation. Now it's time for some examples!

If you browse through the research of your local algebraist, homotopy theorist, algebraic topologist or―well, anyone whose research involves an operation of some type, you might come across the word "operad." But what are operads? And what are they good for? Loosely speaking, operads―which come in a wide variety of types―keep track of various "flavors" of operations. 

☺️ I love math

😀 It's so cool.

Welcome to our third and final installment on the Yoneda lemma! In the past couple of weeks, we've slowly unraveled the mathematics behind the Yoneda perspective, i.e. the categorical maxim that an object  is completely determined by its relationships to other objects. Last week we divided this maxim into two points...

Last week we began a discussion about the Yoneda lemma. Though rather than stating the lemma (sans motivation), we took a leisurely stroll through an implication of its corollaries - the Yoneda perspective, as we called it:

 An object is completely determined by its relationships to other objects,

i.e.

by what the object "looks like" from the vantage point of each object in the category.

 But this left us wondering, What are the mathematics behind this idea? And what are the actual corollaries? In this post, we'll work to discover the answers.

In the words of Dan Piponi, it "is the hardest trivial thing in mathematics." The nLab catalogues it as "elementary but deep and central," while Emily Riehl nominates it as "arguably the most important result in category theory." Yet as Tom Leinster has pointed out, "many people find it quite bewildering."

And what are they referring to?

The Yoneda lemma.

"But," you ask, "what is the Yoneda lemma? And if it's just a lemma then - my gosh - what's the theorem?"

Dear Autocorrect,

No.

"Topos theory" is not the theory of tops. Or coats or shoes or hats or socks or gloves or slacks or scarves or shorts or skorts or--um, actually, what is topos theory?

“Zorn’s lemma” is not a result attributed to corn. Neither boiled corn, grilled corn, frozen corn, fresh corn, canned corn, popped corn, nor unicorns. Though I'm sure one of these is equivalent to the Axiom of Choice.

Mathematicians are a creative bunch, especially when it comes to naming things. And category theorists are no exception. So here's a little spin on this xkcd comic. It's inspired by a recent conversation I had on Twitter and, well, every category theory book ever.

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 existsa [yadda yadda] such thatthe following diagram commutes." and perhaps it left you wondering what it all meant.

Not too long ago, my college-algebra students and I were chatting about graphing polynomials. At one point during our lesson, I quickly drew a smooth, wavy curve on the board and asked,

"How many roots would a polynomial with this graph have? Five? It crosses the x-axis five times."

A quick browse through my Instagram account and you might guess that I take notes. Lots of notes. And you'd be spot on! For this reason, I suppose, I am often asked the question, "How do you do it?!" Now while I don't think my note-taking strategy is particularly special, I am happy to share! I'll preface the information by stating what you probably already know: I LOVE to write.* I am a very visual learner and often need to go through the physical act of writing things down in order for information to "stick." So while some people think aloud (or quietly), 

I think on paper.

Once upon a time I wrote a six-part blog series on why the fundamental group of the circle is isomorphic to the integers. (You can read it here, though you may want to grab a cup of coffee first.) Last week, I shared a proof* of the same result. In one line. On Twitter. I also included a fewer-than-140-characters explanation. But the ideas are so cool that I'd like to elaborate a little more.

One of my students recently said to me, "I'm not good at math because I'm really slow." Right then and there, she had voiced what is one of many misconceptions that folks have about math.

But friends, speed has nothing to do with one's ability to do mathematics. In particular, being "slow" does not mean you do not have the ability to think about, understand, or enjoy the ideas of math.

Let me tell you....

Physicist Freeman Dyson once observed that there are two types of mathematicians: birds -- those who fly high, enjoy the big picture, and look for unifying concepts -- and frogs -- those who dwell on the ground, find beauty in the scenery close by, and enjoy the details.

Of course, both vantage points are essential to mathematical progress, and I often tend to think of myself as more of a bird.(I'm, uh, bird-brained?)

On Monday we concluded our mini-series on basic category theory with a discussion on natural transformations and functors. This led us to make the simple observation that the elements of any set are really just functions from the single-point set {✳︎} to that set. But what if we replace "set" by "group"? Can we view group elements categorically as well? The answer to that question is the topic for today's post, written by guest-author Arthur Parzygnat. 

Continuing our list of examples of natural transformations, here is Example #2 (double dual space of a vector space) and Example #3 (representability and Yoneda's lemma).

I hope you have enjoyed our little series on basic category theory. (I know I have!) This week we'll close out by chatting about natural transformations which are, in short, a nice way of moving from one functor to another. If you're new to this mini-series, be sure to check out the very first post, What is Category Theory Anyway? as well as What is a Category? and last week's What is a Functor?

I was at the grocery store earlier today, minding my own business, and while I was intently studying the lentil beans (Why are there so many options?) a man came down the aisle, pushing a cart with him. He then stopped in front of me, turned, looked me directly in the eyes and said,

Continuing yesterday's list of examples of functors, here is Example #3 (the chain rule from multivariable calculus), Example #4 (contravariant functors), and Example #5 (representable functors).

Next up in our mini series on basic category theory: functors! We began this series by asking What is category theory, anyway? and last week walked through the precise definition of a category along with some examples. As we saw in example #3 in that post, a functor can be viewed an arrow/morphism between two categories.

Hello friends! I've decided to launch a new series on the blog called crumbs! Every now and then, I'd like to share little stories -- crumbs, if you will -- from behind the scenes of Math3ma. To start us off, I posted (a slightly modified version of) the story below on January 23 on Facebook/Twitter/Instagram, so you may have seen this one already. Even so, I thought it'd be a good fit for the blog as well. I have a few more of these quick, soft-topic blurbs that I plan to share throughout the year. So stay tuned! I do hope you'll enjoy this newest addition to Math3ma.

As promised, here is the first in our triad of posts on basic category theory definitions: categories, functors, and natural transformations. If you're just now tuning in and are wondering what is category theory, anyway? be sure to follow the link to find out!

A category $\mathsf{C}$ consists of some data that satisfy certain properties...

A quick browse through my Twitter or Instagram accounts, and you might guess that I've had category theory on my mind. You'd be right, too! So I have a few category-theory themed posts lined up for this semester, and to start off, I'd like to (attempt to) answer the question, What is category theory, anyway? for anyone who may not be familiar with the subject.

Now rather than give you a list of definitions--which are easy enough to find and may feel a bit unmotivated at first--I thought it would be nice to tell you what category theory is in the grand scheme of (mathematical) things. You see, it's very different than other branches of math....

If you've been following this blog for a while, you'll know that I have strong opinions about the misconception that "math is only for the gifted." I've written about the importance of endurance and hard work several times. Naturally, these convictions carried over into my own classroom this past semester as I taught a group of college algebra students.

Whether they raised their hand during a lecture and gave a "wrong" answer, received a less-than-perfect score on an exam or quiz, or felt completely confused during a lesson, I tried to emphasize that things aren't always as bad as they seem...

Over the past few weeks, we've been chatting about quotient groups  in hopes of answering the question, "What's a quotient group, really?" In short, we noted that the quotient of a group $G$ by a normal subgroup $N$ is a means of organizing the group elements according to how they fail---or don't fail---to satisfy the property required to belong to $N$. The key point was that there's only one way to belong to $N$, but generally there may be several ways to fail to belong. 

Last week we took an intuitive peek into the First Isomorphism Theorem as one example in our ongoing discussion on quotient groups. Today we'll explore another quotient that you've likely come across, namely that of a group by its center.

Welcome back to our little discussion on quotient groups! (If you're just now tuning in, be sure to check out "What's a Quotient Group, Really?" Part 1 and Part 2!) We're wrapping up this mini series by looking at a few examples. I'd like to take my time emphasizing intuition, so I've decided to give each example its own post. Today we'll take an intuitive look at the quotient given in the First Isomorphism Theorem.

Today we're resuming our informal chat on quotient groups.  Previously we said that belonging to a (normal, say) subgroup $N$ of a group $G$ just means you satisfy some property. For example, $5\mathbb{Z}\subset\mathbb{Z}$ means "You belong to $5\mathbb{Z}$ if and only if you're divisible by 5". And the process of "taking the quotient" is the simple observation that every element in $G$ either

#1) belongs to N            or            #2) doesn't belong to N

and noting that...

I realize that most of my posts for the past, er, few months have been about some pretty hefty duty topics. Today, I'd like to dial it back a bit and chat about some basic group theory! So let me ask you a question: When you hear the words "quotient group," what do you think of? In case you'd like a little refresher, here's the definition...

Last time we chatted about a pervasive theme in mathematics, namely that objects are determined by their relationships with other objects, or more informally, you can learn a lot about an object by studying its interactions with other things. Today I'd to give an explicit illustration of this theme in the case when "objects" = topological spaces and "relationships with other objects" = continuous functions. The goal of this post, then, is to convince you that the topology on a space  is completely determined by the set of all continuous functions to it.

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. 

It's possible that a set $X$ can be endowed with two or more topologies that are comparable. Over the years, mathematicians have used various  words to describe the comparison: a topology $\tau_1$ is said to be coarser than another topology $\tau_2$, and we write $\tau_1\subseteq\tau_2$, if every open set in $\tau_1$ is also an open set in $\tau_2$. In this scenario, we also say $\tau_2$ is finer than $\tau_1$. But other folks like to replace "coarser" by "smaller" and "finer" by "larger." Still others prefer to use "weaker" and "stronger." But how can we keep track of all of this? 

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. 

Today I'm talking about about qualifying exams! But no, I won't be dishing out advice on preparing for these exams. Tons of excellent advice is readily available online, so I'm not sure I can contribute much that isn't already out there. However, it's that very web-search that has prompted me to write this post.

You see, before I started graduate school I had heard of these rites-of-passage called the qualifying exams.* And to be frank, I thought they sounded terrifying.

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.

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.

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.

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.

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.

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

Word order is important...

Next up on Good Reads: The Princeton Companion to Mathematics, edited by Fields medalist Timothy Gowers. This book is an exceptional resource! With over 1,000 pages of mathematics explained by the experts for the layperson, it's like an encyclopedia for math, but so much more. Have you heard about category theory but aren't sure what it is? There's a chapter for that! Seen the recent headlines about the abc conjecture but don't know what it's about? There's a chapter for that! Need a crash course in general relativity and Einstein's equations, or the P vs. NP conjecture, or C*-algebras, or the Riemann zeta function, or Calabi-Yau manifolds? There are chapters for all of those and more.

You know the routine. You come across a topological space $X$ and you need to find its fundamental group. Unfortunately, $X$ is an unfamiliar space and it's too difficult to look at explicit loops and relations. So what do you do? You look for another space $Y$ that is homotopy equivalent to $X$ and whose fundamental group $\pi_1(Y)$ is much easier to compute. And voila! Since $X$ and $Y$ are homotopy equivalent, you know $\pi_1(X)$ is isomorphic to $\pi_1(Y)$. Mission accomplished.

 Below is a list of some homotopy equivalences which I think are pretty clever and useful to keep in your back pocket for, say, a qualifying exam or some other pressing topological occasion.

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!) effort. As the article's author put it, "no one is born knowing the axiom of completeness." Since then, I've come across a few other snippets of mathematical candor that I found both helpful and encouraging. And since final/qualifying exam season is right around the corner, I've decided to share them here on the blog for a little morale-boosting.

I was recently (avoiding doing my homology homework by) reading through some old poems by Shel Silverstein, author of The Giving Tree, A Light in the Attic, and Falling Up to name a few. Feeling inspired, I continued to procrastinate by writing a little poem of my own - about homology, naturally!

My goal for today is to provide a step-by-step guideline for classifying closed surfaces. (By 'closed,' I mean a surface that is compact and has no boundary.) The information below may come in handy for any topology student who needs to know just the basics (for an exam, say, or even for other less practical (but still mathematically elegant) endeavors) so there won't be any proofs today. Given a polygon with certain edges identified, we can determine the surface that it represents in just three easy steps:

Yesterday I received a disheartening 44/50 on a homework assignment. Okay okay, I know. 88% isn't bad, but I had turned in my solutions with so much confidence that admittedly, my heart dropped a little (okay, a lot!) when I received the grade. But I quickly had to remind myself, Hey! Grades don't matter.

Have you been on the hunt for a good introductory-level real analysis book? Look no further! The underrated Real Analysis by N. L. Carothers is, in my opinion, one of the best out there. Real analysis has a reputation for being a fearful subject for many students, but this text by Carothers does a great job of mitigating those fears. Aimed towards advanced undergraduate and early graduate students, it is written in a fantastically warm and approachable manner without sacrificing too much rigor. The text is intentionally conversational (which I love!) and includes plenty of exercises and illustrations, all the while informing the reader of context and historical background along the way. 

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. In that post, we discovered that the real line $\mathbb{R}$ can either be compact or non-compact, depending on which topological glasses we choose to view $\mathbb{R}$ with. Today, I’d like to show you another such example - one which has a surprising consequence!

Last time we proved that the pseudo-hyperbolic metric on the unit disc in ℂ 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...."

In this post, we define the pseudo-hyperbolic metric on the unit disc in ℂ and prove it does indeed satisfy the conditions of a metric.

Love and Math by Edward Frenkel is an excellent book about the hidden beauty and elegance of mathematics. It is primarily about Frenkel’s work on the Langlands Program (a sort of grand unified theory of mathematics) and its recent connections to quantum physics. Yet the author's goal is not merely to inform but rather to convert the reader into a lover of math. While Frenkel acknowledges that many view mathematics as an “insufferable torment… pure torture, or a nightmare that turns them off,” he also feels that math is “too precious to be given away to the ‘initiated few.’” In the preface he writes...

The goal of today's post is to prove that the fundamental group of the real projective plane, is isomorphic to $\mathbb{Z}/2\mathbb{Z}$ And unlike our proof for the fundamental group of the circle, today's proof is fairly short, thanks to the van Kampen theorem! To make our application of the theorem a little easier, we start with a simple observation: projective plane - disk = Möbius strip. Below is an excellent animation which captures this quite clearly....

There are two definitions of absolute continuity out there. One refers to an absolutely continuous function and the other to an absolutely continuous measure. And although the definitions appear unrelated, they are in fact very much related, linked together by Lebesgue's Fundamental Theorem of Calculus. This is the second of a two-part series where we explore that relationship.

There are two definitions of absolute continuity out there. One refers to an absolutely continuous function and the other to an absolutely continuous measure. And although the definitions appear unrelated, they are in fact very much related, linked together by Lebesgue's Fundamental Theorem of Calculus. This is part one of a two-part series where we explore that relationship.

In sum, the sufficient condition (a.k.a. the "if" direction) allows you to get what you want. That is, if you assume the sufficient condition, you'll obtain your desired conclusion. It's enough. It's sufficient. 

On the other hand, the necessary condition (a.k.a. the "only if" direction) is the one you must assume in order to get what you want. In other words, if you don't have the necessary condition then you can't reach your desired conclusion. It is necessary. 

Welcome to the final post in a six-part series where we prove that the fundamental group of the circle $\pi_1(S^1)$ is isomorphic to $\mathbb{Z}$. Today we prove two lemmas (the path- and homotopy-lifting properties) that were used in parts four and five. The proof follows that found in Hatcher's Algebraic Topology section 1.1.

Below is a general method —a recipe, if you will —for computing the universal cover of the wedge sum $X\vee Y$ of arbitrary topological spaces $X$ and $Y$. This is simply a short-and-quick guideline that my prof mentioned in class, and I thought it'd be helpful to share on the blog. To help illustrate each step, we'll consider the case when $X=T^2$ is the torus and $Y=S^1$ is the circle.       

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 $\mathbb{Z}$. In this post we prove that our homomorphism from $\mathbb{Z}$ to $\pi_1(S^1)$ is injective. The proof follows that found in Hatcher's Algebraic Topology section 1.1.

Have you ever read Tristan Needham’s Visual Complex Analysis? I highly recommend this book as a supplement to a standard undergrad/grad course in complex analysis. It's nothing (nothing!) like your usual textbook. The author writes to build your intuition and insight, so it's warm like a conversation and not cold like some math texts. It’s also loaded with illustrations (hence the title), historical background, and context. For example, did you know

Welcome to part four 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 this post we prove that our homomorphism from $\mathbb{Z}$ to $\pi_1(S^1)$ is surjective. The proof follows that found in Hatcher's Algebraic Topology section 1.1.

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 $\mathbb{Z}$. In this post we prove that our map from $\mathbb{Z}$ to $\pi_1(S^1)$ is a group homomorphism. The proof follows that found in Hatcher's Algebraic Topology section 1.1.

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 this post we justify a shortcut that we never actually use in the remainder of this series, so the reader is welcome to skip this post. But I've included it since, in this series, we're closely following section 1.1 of Hatcher's Algebraic Topology.

Welcome to part one 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 this post we define a map from $\mathbb{Z}$ to $\pi_1(S^1)$ and make some simple observations via pictures and an animation! The proof follows that found in Hatcher's Algebraic Topology</a>, section 1.1.

The quote above comes from an excellent Slate article by Chase Felker on why students shouldn't be afraid of or intimidated by mathematics. I posted the quote on Instagram not too long ago, and since it addresses a topic that is near-and-dear to my own heart, I decided to include it on the blog as well. Felker prefaces the quote by saying, "Giving up on math means you don't believe that careful study can change the way you think."

Up to isomorphism” is a phrase that seems to get thrown around a lot without ever being explained. Simply put, we say two groups (or any other algebraic structures) are the same “up to isomorphism” if they’re isomorphic! In other words, they share the exact same structure and therefore they are essentially indistinguishable. Hence we consider them to be one and the same! But, you see, we mathematicians are very precise, and so we really don't like to use the word “same." Instead we prefer to say “same up to isomorphism.” Voila!

Here's a chart to help keep track of some of the different "flavors" of continuity in real analysis.

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!"

Fatou's Lemma, the Monotone Convergence Theorem, and the Dominated Convergence Theorem are three major results in the theory of Lebesgue integration which, when given a sequence of functions $\{f_n\}$, answer the question, "When can I switch the limit symbol and the integral symbol?" In this post, we discuss the Dominated Convergence Theorem and see why "domination" is necessary.

Fatou's Lemma, the Monotone Convergence Theorem, and the Dominated Convergence Theorem are three major results in the theory of Lebesgue integration which, when given a sequence of functions $\{f_n\}$ answer the question, "When can I switch the limit symbol and the integral symbol?" In this post, we discuss the Monotone Convergence Theorem and solve a nasty-looking problem which, thanks to the theorem, is actually quite trivial!

Have you read Jeffrey Weeks' The Shape of Space before? What a great book! It explores the geometry of spheres, tori, Möbius strips, Klein bottles, projective planes and other spaces in an engaging, this-is-definitely-not-a-textbook kind of way. Other topics include: gluing, orientability, connected sums, Euler number, hyperspace, bundles, and more! (Have I whet your appetite yet?!)

Fatou's Lemma, the Monotone Convergence Theorem, and the Dominated Convergence Theorem are three major results in the theory of Lebesgue integration which, when given a sequence of functions $\{f_n\}$ answer the question, "When can I switch the limit symbol and the integral symbol?" In this post, we discuss Fatou's Lemma and solve a problem from Rudin's Real and Complex Analysis (a.k.a. "Big Rudin").

Today we're chatting about the Borel-Cantelli Lemma. When I first came across this lemma, I struggled to understand what it meant "in English." What does $\mu(\cup\cap E_k)=0$ really signify?? There's a pretty simple explanation if $(X,\Sigma,\mu)$ is a probability space, but how are we to understand the result in the context of general measure spaces?

In today's post I want to emphasize a simple - but important - idea in topology which I think is helpful for anyone new to the subject, and that is: Open sets are everything! What do I mean by that? Well, for a given set $X$, all the properties of $X$ are HIGHLY dependent on how you define an "open set."

Today's post is a bit of a ramble, but my goal is to uncover the intuition behind one of the definitions of a connected topological space. Ideally, this is just a little tidbit I'd like to stash in The Back Pocket. But as you can tell already, the length of this post isn't so "little"! Oh well, here we go!

Perhaps you've heard of Évariste Galois? (Pronounced "GAL-wah.") You know, the French mathematician who died tragically in 1832 in a duel at the tender age of 20? (Supposedly over a girl! C'est romantique, n'est-ce pas?)  Well, today we're taking a bird's-eye view of his most well-known contribution to mathematics: the appropriately named Galois theory. The goal of this post is twofold...

Welcome to part two of our discussion on Baire's Category Theorem. Today we'll sketch the proof that we can find a continuous function on $[0,1]$ which is nowhere differentiable.

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.

(Remark: we used this theorem last week to prove the existence of a Lebesgue measurable set which is not a Borel set.)

In short, "Noetherian-ness" is a property which generalizes "PID-ness." As Keith Conrad so nicely puts it, "The property of all ideals being singly generated is often not preserved under common ring-theoretic constructions (e.g. $\mathbb{Z}$ is a PID but $\mathbb{Z}[x]$ is not), but the property of all ideals being finitely generated does remain valid under many constructions of new rings from old rings. For example... every quadratic ring $\mathbb{Z}[\sqrt{d}]$ is Noetherian, even though many of these rings are not PIDs." (italics added)

In general, if $F$ is a field and $V$ is a vector space over $F$, the tensor product answers the question "How can I define scalar multiplication on $V$ by some larger field which contains $F$?" (Of course this holds if we replace the word "field" by "ring" and consider the same scenario with modules.)

Here's an algebra tip! Whenever you're asked to prove $$A/B\cong C$$ where $A,B,C$ are groups, rings, fields, modules, etc., mostly likely the The First Isomorphism Theorem involved!

Our goal for today is to construct a Lebesgue measurable set which is not a Borel set. In summary, we will define a homeomorphism from $[0,1]$ to $[0,2]$  which will map a (sub)set (of the Cantor set) of measure 0 to a set of measure 1. This set of measure 1 contains a non-measurable subset, say $N$. And the preimage of $N$ will be Lebesgue measurable but will not be a Borel set.

Galois Theory is all about symmetry. So, perhaps not surprisingly, symmetries found among the roots of polynomials (via Galois theory) are closely related to symmetries of polygons in the plane (via geometry). In fact, the two are highly analogous! 

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 $$\mu\left(\bigcap_{n=1}^\infty \bigcup_{k=n}^\infty E_k\right)=\mu\left(\limsup_{n\to\infty} En \right)=0.$$

(For the record, I didn't understand this when I first saw it (or for a long time afterwards). My only thought was, "But what does that mean? In English??") To help our intuition…

If $X$ and $Y$ are normed vector spaces, a linear map $T:X\to Y$ is said to be <b>bounded</b> 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 in $X$.) One can show that this is equivalent to

$$\|T\|=\sup_{x\in X}\{|T(x)|:|x|=1\}.$$ So intuitively (at least in two dimensions), we can think of $\|T\|$ this way…

You're given a ring $R$ and are asked to show it's not a UFD. Where do you begin? One standard trick is to apply the Rational Roots Theorem….

  In real analysis, there are two ways a measurable set $E$ can be small. Either

• the measure of $E$ is 0, OR
• $E$ is nowhere dense.

Intuitively, to say the measure of $E$ is $0$ means that...

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! This technique is used quite often.

Let a group $G$ act on a set $X$. The action is said to be transitive if for any two $x,y\in X$ there is a $g\in G$ such that $g\cdot x=y$. This is equivalent to saying there is an $x\in X$ such that $\text{orb}(x)=X$, i.e. there is exactly one orbit. And all this is just the fancy way of saying that $G$ shuffles all the elements of $X$ among themselves. In other words…

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

The Baire Category Theorem is a powerful result that relates a metric space to its underlying topology. (And sadly no, nothing to do with category theory!) Informally, the theorem says that if you can find a metric with respect to which your topological space is complete, then that space cannot be written as a countable union of nowhere dense sets. In other words, a metric can put a restriction on the topology.

A function $f$ is said to be continuous if the preimage of any open set is open. Analogously, we might say that a function is measurable if the preimage of a measurable set is measurable. It's not hard to show that if $\{f_n\}$ is a sequence of measurable functions, then sup$\{f_n\}$, inf$\{f_n\}$, limsup $\{f_n\}$ and liminf $\{f_n\}$ are also measurable functions. But here the analogy between continuity and measurability breaks down. It is not true that if each $f_n$ is a continuous function, then sup$\{f_n\}$, inf$\{f_n\}$, limsup $\{f_n\}$ and liminf $\{f_n\}$ are continuous as well. Below is a counterexample - a sequence of continuous functions with a discontinuous supremum!