Math3mahttps://www.math3ma.comSat, 20 Apr 2019 00:17:46 GMTWebflowcrumbs!https://www.math3ma.com/blog/crumbs-6https://www.math3ma.com/blog/crumbs-6Recently I've been working on a dissertation proposal, which is sort of like a culmination of five years of graduate school (yay). The first draft was rough, but I sent it to my advisor anyway. A few days later I walked into his office, smiled, and said hello. He responded with a look of regret. [Advisor]: I've been... remiss about your proposal. [I think: Remiss? Oh no. I can't remember what the word means, but it sounds really bad. The solemn tone must be a context clue. My heart sinks. I feel so embarrassed, so mortified. He's been remiss at me for days! Probably years! I think back to all the times I should've worked harder, all the exercises I never did. I knew This Day Would Come. I fight back the lump in my throat.] [Me]: Oh no... oh no. I'm sorry. I shouldn't have sent it. It wasn't ready. Oh no.... [Advisor]: What? [Me]: Hold on. What does remiss mean? [Advisor, confused, Googles remiss]: I think I just mean I haven't read your proposal.Fri, 19 Apr 2019 23:38:02 GMTComparing Topologieshttps://www.math3ma.com/blog/comparing-topologieshttps://www.math3ma.com/blog/comparing-topologiesIt'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? Fri, 05 Apr 2019 13:43:40 GMTViewing Matrices & Probability as Graphshttps://www.math3ma.com/blog/matrices-probability-graphshttps://www.math3ma.com/blog/matrices-probability-graphsToday I'd like to share an idea. It's a very simple idea. It's not fancy and it's certainly not new. In fact, I'm sure many of you have thought about it already. But if you haven't—and even if you have!—I hope you'll take a few minutes to enjoy it with me. Here's the idea: Every matrix corresponds to a graph. So simple! But we can get a lot of mileage out of it.To start, I'll be a little more precise: every matrix corresponds to a weighted bipartite graph. By "graph" I mean a collection of vertices (dots) and edges; by "bipartite" I mean that the dots come in two different types/colors; by "weighted" I mean each edge is labeled with a number.Thu, 07 Mar 2019 04:26:50 GMTLimits and Colimits Part 3 (Examples)https://www.math3ma.com/blog/limits-and-colimits-part-3https://www.math3ma.com/blog/limits-and-colimits-part-3Once upon a time, we embarked on a mini-series about limits and colimits in category theory. Part 1 was a non-technical introduction that highlighted two ways mathematicians often make new mathematical objects from existing ones: by taking a subcollection of things, or by gluing things together. The first route leads to a construction called a limit, the second to a construction called a colimit.The formal definitions of limits and colimits were given in Part 2. There we noted that one speaks of "the (co)limit of [something]." As we've seen previously, that "something" is a diagram—a functor from an indexing category to your category of interest. Moreover, the shape of that indexing category determines the name of the (co)limit: product, coproduct, pullback, pushout, etc. In today's post, I'd like to solidify these ideas by sharing some examples of limits. Next time we'll look at examples of colimits. What's nice is that all of these examples are likely familiar to you—you've seen (co)limits many times before, perhaps without knowing it! The newness is in viewing them through a categorical lens. Wed, 06 Mar 2019 14:23:10 GMTLimits and Colimits, Part 2 (Definitions)https://www.math3ma.com/blog/limits-and-colimits-part-2https://www.math3ma.com/blog/limits-and-colimits-part-2Welcome 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.Sun, 03 Feb 2019 13:20:33 GMTA Math Blog? Say What?https://www.math3ma.com/blog/a-math-blog-say-whathttps://www.math3ma.com/blog/a-math-blog-say-whatYes! I'm writing about math. No! Don't close your browser window. Hear me out first... I know very well that math has a bad rap. It's often taught or thought of as a dry, intimidating, unapproachable, completely boring, who-in-their-right-mind-would-want-to-think-about-this-on-purpose kind of subject. I get it. Math was the last thing on earth I thought I'd study. Seriously.Thu, 24 Jan 2019 03:45:25 GMTLimits and Colimits, Part 1 (Introduction)https://www.math3ma.com/blog/limits-and-colimits-part-1https://www.math3ma.com/blog/limits-and-colimits-part-1I'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.Thu, 24 Jan 2019 03:01:00 GMTThe Yoneda Perspectivehttps://www.math3ma.com/blog/the-yoneda-perspectivehttps://www.math3ma.com/blog/the-yoneda-perspectiveIn 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?Mon, 21 Jan 2019 13:03:29 GMTAnnouncing Applied Category Theory 2019https://www.math3ma.com/blog/act-2019https://www.math3ma.com/blog/act-2019Hi everyone. Here's a quick announcement: the Applied Category Theory 2019 school is now accepting applications! As you may know, I participated in ACT2018, had a great time, and later wrote a mini-book based on it. This year, it's happening again with new math and new people! As before, it consists of a five-month long, online school that culminates in a week long conference (July 15-19) and a week long research workshop (July 22-26, described below). Last year we met at the Lorentz Center in the Netherlands; this year it'll be at Oxford. Daniel Cicala and Jules Hedges are organizing the ACT2019 school, and they've spelled out all the details in the official announcement, which I've copied-and-pasted it below. Read on for more! And please feel free to spread the word. Do it quickly, though. The deadline is soon! APPLICATION DEADLINE: JANUARY 30, 2019Wed, 09 Jan 2019 20:39:30 GMTThe Tensor Product, Demystifiedhttps://www.math3ma.com/blog/the-tensor-product-demystifiedhttps://www.math3ma.com/blog/the-tensor-product-demystifiedPreviously on the blog, we've discussed a recurring theme throughout mathematics: making new things from old things. Today, I'd like to focus on a particular way to build a new vector space from old vector spaces: the tensor product. This construction often come across as scary and mysterious, but I hope to shine a little light and dispel a little fear. In particular, we won't talk about axioms, universal properties, or commuting diagrams. Instead, we'll take an elementary, concrete look: Given two vectors $\mathbf{v}$ and $\mathbf{w}$, we can build a new vector, called the tensor product $\mathbf{v}\otimes \mathbf{w}$. But what is that vector, really? Likewise, given two vector spaces $V$ and $W$, we can build a new vector space, also called their tensor product $V\otimes W$. But what is that vector space, really? Sun, 06 Jan 2019 02:15:43 GMTNotes on Applied Category Theoryhttps://www.math3ma.com/blog/notes-on-acthttps://www.math3ma.com/blog/notes-on-actHave you heard the buzz? Applied category theory is gaining ground! But, you ask, what is applied category theory? Upon first seeing those words, I suspect many folks might think either one of two thoughts: 1. Applied category theory? Isn't that an oxymoron? or 2. Applied category theory? What's the hoopla? Hasn't category theory always been applied? (Visit the blog to read more!)Sun, 06 Jan 2019 02:00:46 GMTEnglish is Not Commutativehttps://www.math3ma.com/blog/english-is-not-commutativehttps://www.math3ma.com/blog/english-is-not-commutativeHere's another unspoken rule of mathematics: English doesn't always commute! Word order is important...Fri, 23 Nov 2018 19:15:55 GMT"Up to Isomorphism"?https://www.math3ma.com/blog/up-to-isomorphismhttps://www.math3ma.com/blog/up-to-isomorphismUp 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!Fri, 23 Nov 2018 19:15:55 GMTWhy are Noetherian Rings Special?https://www.math3ma.com/blog/why-are-noetherian-rings-specialhttps://www.math3ma.com/blog/why-are-noetherian-rings-specialIn 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)Fri, 23 Nov 2018 19:15:55 GMTOne Unspoken Rule of Algebrahttps://www.math3ma.com/blog/an-unspoken-rule-of-algebrahttps://www.math3ma.com/blog/an-unspoken-rule-of-algebraHere'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!Fri, 23 Nov 2018 19:15:55 GMTBorel-Cantelli Lemma (Pictorially)https://www.math3ma.com/blog/borel-cantelli-lemma-pictoriallyhttps://www.math3ma.com/blog/borel-cantelli-lemma-pictoriallyThe 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…Fri, 23 Nov 2018 19:15:55 GMTOperator Norm, Intuitivelyhttps://www.math3ma.com/blog/operator-norm-intuitivelyhttps://www.math3ma.com/blog/operator-norm-intuitivelyIf $X$ 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…Fri, 23 Nov 2018 19:15:55 GMTNeed to Prove Your Ring is NOT a UFD?https://www.math3ma.com/blog/need-to-prove-your-ring-is-not-a-ufdhttps://www.math3ma.com/blog/need-to-prove-your-ring-is-not-a-ufdYou'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….Fri, 23 Nov 2018 19:15:55 GMTOne Unspoken Rule of Measure Theoryhttps://www.math3ma.com/blog/one-unspoken-rule-of-measure-theoryhttps://www.math3ma.com/blog/one-unspoken-rule-of-measure-theoryHere'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.Fri, 23 Nov 2018 19:15:55 GMTWhat's a Transitive Group Action?https://www.math3ma.com/blog/whats-a-transitive-group-actionhttps://www.math3ma.com/blog/whats-a-transitive-group-actionLet 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…Fri, 23 Nov 2018 19:15:55 GMTCompleting a Metric Space, Intuitivelyhttps://www.math3ma.com/blog/completing-a-metric-space-intuitivelyhttps://www.math3ma.com/blog/completing-a-metric-space-intuitivelyAn incomplete metric space is very much like a golf course: it has a lot of missing points!Fri, 23 Nov 2018 19:15:55 GMTLearning How to Learn Mathhttps://www.math3ma.com/blog/learning-mathhttps://www.math3ma.com/blog/learning-mathOnce upon a time, while in college, I decided to take my first intro-to-proofs class. I was so excited. "This is it!" I thought, "now I get to learn how to think like a mathematician." You see, for the longest time, my mathematical upbringing was very... not mathematical. As a student in high school and well into college, I was very good at being a robot. Memorize this formula? No problem. Plug in these numbers? You got it. Think critically and deeply about the ideas being conveyed by the mathematics? Nope. It wasn't because I didn't want to think deeply. I just wasn't aware there was anything to think about. I thought math was the art of symbol-manipulation and speedy arithmetic computations. I'm not good at either of those things, and I never understood why people did them anyway. But I was excellent at following directions. So when teachers would say "Do this computation," I would do it, and I would do it well. I just didn't know what I was doing. By the time I signed up for that intro-to-proofs class, though, I was fully aware of the robot-symptoms and their harmful side effects. By then, I knew that math not just fancy hieroglyphics and that even people who aren't super-computers can still be mathematicians because—would you believe it?—"mathematician" is not synonymous with "human calculator." There are even—get this—ideas in mathematics, which is something I could relate to. ("I know how to have ideas," I surmised one day, "so maybe I can do math, too!") One of my instructors in college was instrumental in helping to rid me of robot-syndrome. One day he told me, "To fully understand a piece of mathematics, you have to grapple with it. You have to work hard to fully understand every aspect of it." Then he pulled out his cell phone, started rotating it about, and said, "It's like this phone. If you want to understand everything about it, you have to analyze it from all angles. You have to know where each button is, where each ridge is, where each port is. You have to open it up and see how it the circuitry works. You have to study it—really study it—to develop a deep understanding." "And that" he went on to say, "is what studying math is like."Sun, 18 Nov 2018 18:16:55 GMTIs the Square a Secure Polygon?https://www.math3ma.com/blog/is-the-square-a-secure-polygonhttps://www.math3ma.com/blog/is-the-square-a-secure-polygonIn 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?Sat, 13 Oct 2018 16:17:48 GMTThe Yoneda Lemmahttps://www.math3ma.com/blog/the-yoneda-lemmahttps://www.math3ma.com/blog/the-yoneda-lemmaWelcome 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...Sat, 13 Oct 2018 16:15:19 GMTThe Yoneda Embeddinghttps://www.math3ma.com/blog/the-yoneda-embeddinghttps://www.math3ma.com/blog/the-yoneda-embeddingLast 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,Sat, 13 Oct 2018 16:14:28 GMTTopological Magic: Infinitely Many Primeshttps://www.math3ma.com/blog/topological-magic-infinitely-many-primeshttps://www.math3ma.com/blog/topological-magic-infinitely-many-primesA 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!Sun, 07 Oct 2018 19:33:16 GMTWhat is a Category? Definition and Exampleshttps://www.math3ma.com/blog/what-is-a-categoryhttps://www.math3ma.com/blog/what-is-a-categoryAs 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...Thu, 20 Sep 2018 13:01:14 GMTOn Constructing Functions, Part 4https://www.math3ma.com/blog/on-constructing-functions-part-4https://www.math3ma.com/blog/on-constructing-functions-part-4This post is the fourth example in an ongoing list of various sequences of functions which converge to different things in different ways. Here's a sequence of Lebesgue integrable functions which converges uniformly to a function which is not Lebesgue integrable.Tue, 11 Sep 2018 00:56:51 GMTMaximal ≠ Maximum!https://www.math3ma.com/blog/maximal-not-maximumhttps://www.math3ma.com/blog/maximal-not-maximumSuffixes are important! Did you know that the words "maximal" and "maximum" generally do NOT mean the same thing in mathematics?Tue, 11 Sep 2018 00:56:51 GMTOn Constructing Functions, Part 5https://www.math3ma.com/blog/on-constructing-functions-part-5https://www.math3ma.com/blog/on-constructing-functions-part-5This post is the fifth example in an ongoing list of various sequences of functions which converge to different things in different ways. Today we have a sequence of functions which converges to 0 pointwise but does not converge to 0 in $L^1$.Tue, 11 Sep 2018 00:56:51 GMT4 Ways to Show a Group is Not Simplehttps://www.math3ma.com/blog/4-ways-to-show-a-group-is-not-simplehttps://www.math3ma.com/blog/4-ways-to-show-a-group-is-not-simpleYou know the Sylow game. You're given a group of a certain order and are asked to show it's not simple. But where do you start? Here are four options that may be helpful when trying to produce a nontrivial normal subgroup.Tue, 11 Sep 2018 00:56:51 GMTNoetherian Rings = Generalization of PIDshttps://www.math3ma.com/blog/noetherian-rings-generalization-of-pidshttps://www.math3ma.com/blog/noetherian-rings-generalization-of-pidsWhen I was first introduced to Noetherian rings, I didn't understand why my professor made such a big hoopla over these things. What makes Noetherian rings so special? Today's post is just a little intuition to stash in The Back Pocket, for anyone hearing the word "Noetherian" for the first time. Tue, 11 Sep 2018 00:56:51 GMTTwo Tricks Using Eisenstein's Criterionhttps://www.math3ma.com/blog/two-tricks-using-eisensteins-criterionhttps://www.math3ma.com/blog/two-tricks-using-eisensteins-criterionToday we're talking about Eisenstein's (not Einstein's!) Criterion - a well-known test to determine whether or not a polynomial is irreducible. In particular, we'll consider two examples where a not-so-obvious trick is needed in order to apply the criterion.Tue, 11 Sep 2018 00:56:51 GMTA Little Fact From Group Actionshttps://www.math3ma.com/blog/a-little-fact-from-group-actionshttps://www.math3ma.com/blog/a-little-fact-from-group-actionsToday we've got a little post on a little fact relating to group actions. I wanted to write about this not so much to emphasize its importance (it's certainly not a major result) but simply to uncover the intuition behind it.Tue, 11 Sep 2018 00:56:51 GMTFinitely Generated Modules Over a PIDhttps://www.math3ma.com/blog/finitely-generated-modules-over-a-pidhttps://www.math3ma.com/blog/finitely-generated-modules-over-a-pidWe know what it means to have a module $M$ over a (commutative, say) ring $R$. We also know that if our ring $R$ is actually a field, our module becomes a vector space. But what happens if $R$ is "merely" a PID? Answer: A lot. Today we'll look at a proposition, which, thanks to the language of exact sequences, is quite simple and from which the Fundamental Theorem of Finitely Generated Modules over a PID follows almost immediately. The information below is loosely based on section 12.1 of Dummit and Foote' Abstract Algebra.Tue, 11 Sep 2018 00:56:51 GMTRational Canonical Form: A Summaryhttps://www.math3ma.com/blog/rational-canonical-form-a-summaryhttps://www.math3ma.com/blog/rational-canonical-form-a-summaryThis post is intended to be a hopefully-not-too-intimidating summary of the rational canonical form (RCF) of a linear transformation. Of course, anything which involves the word "canonical" is probably intimidating no matter what. But even so, I've attempted to write a distilled version of the material found in (the first half of) section 12.2 from Dummit and Foote's Abstract Algebra.Tue, 11 Sep 2018 00:56:51 GMTRational Canonical Form: Example #1https://www.math3ma.com/blog/rational-canonical-form-example-1https://www.math3ma.com/blog/rational-canonical-form-example-1Last 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. It's this fact which allows us to classify distinct linear transformations on a given $F$-vector space $V$ for some field $F$. Today, to illustrate this, we'll work through a concrete example: Find representatives for the distinct conjugacy classes of matrices of finite order in the multiplicative group of 2x2 matrices with rational entries.Tue, 11 Sep 2018 00:56:51 GMTRational Canonical Form: Example #2 (with Galois Theory)https://www.math3ma.com/blog/rational-canonical-form-example-2-with-galois-theoryhttps://www.math3ma.com/blog/rational-canonical-form-example-2-with-galois-theoryLast week we saw an example of how to use the rational canonical form (RCF) to classify matrices of a given order in $GL_2(\mathbb{Q})$. Today we have a similar example (taken from CUNY's spring 2015 qualifying exam) where now our matrices have entires in the finite field $F_13$. The fact that our field is $F_13$ instead of $\mathbb{Q}$ actually makes little difference in how to approach the solution, but I think this problem is particularly nice because part of it calls on some Galois Theory.Tue, 11 Sep 2018 00:56:51 GMTStone Weierstrass Theoremhttps://www.math3ma.com/blog/stone-weierstrass-theoremhttps://www.math3ma.com/blog/stone-weierstrass-theoremThe Stone Weierstrass Theorem is a generalization of the familiar Weierstrass Approximation Theorem. In this post, we introduce the Stone Weierstrass Theorem and, by looking at counterexamples, discover why each of the hypotheses of the theorem are necessary.Tue, 11 Sep 2018 00:56:51 GMTStone Weierstrass Theorem (Example)https://www.math3ma.com/blog/stone-weierstrass-theorem-examplehttps://www.math3ma.com/blog/stone-weierstrass-theorem-exampleThis week we continue our discussion on the Stone Weierstrass Theorem with an example. This exercise is taken from Rudin's Principles of Mathematical Analysis (affectionately known as "Baby Rudin"), chapter 7 #20.Tue, 11 Sep 2018 00:56:51 GMTOn Constructing Functions, Part 6https://www.math3ma.com/blog/on-constructing-functions-part-6https://www.math3ma.com/blog/on-constructing-functions-part-6This post is the sixth example in an ongoing list of various sequences of functions which converge to different things in different ways. Today we have a sequence of functions on $[0,1]$ which converges to 0 in $L^1$, but does not converge anywhere on $[0,1]$.Tue, 11 Sep 2018 00:56:51 GMTNeed Some Disjoint Sets? (A Measure Theory Trick)https://www.math3ma.com/blog/need-some-disjoint-sets-a-measure-theory-trickhttps://www.math3ma.com/blog/need-some-disjoint-sets-a-measure-theory-trickGiven a countable collection of measurable sets, is it possible to construct a new collection of sets which are pairwise disjoint and have the same union as the original? Yes! Here's the trick....Tue, 11 Sep 2018 00:56:51 GMTContinuous Functions, Discontinuous Supremumhttps://www.math3ma.com/blog/continuous-functions-discontinuous-supremumhttps://www.math3ma.com/blog/continuous-functions-discontinuous-supremumA 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. Tue, 11 Sep 2018 00:56:51 GMTBaire Category & Nowhere Differentiable Functions (Part One)https://www.math3ma.com/blog/baire-category-nowhere-differentiable-functions-part-onehttps://www.math3ma.com/blog/baire-category-nowhere-differentiable-functions-part-oneThe 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.Tue, 11 Sep 2018 00:56:51 GMTLebesgue Measurable But Not Borelhttps://www.math3ma.com/blog/lebesgue-but-not-borelhttps://www.math3ma.com/blog/lebesgue-but-not-borelOur 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.Tue, 11 Sep 2018 00:56:51 GMTA Non-Measurable Sethttps://www.math3ma.com/blog/a-non-measurable-sethttps://www.math3ma.com/blog/a-non-measurable-setToday 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.)Tue, 11 Sep 2018 00:56:51 GMTBaire Category & Nowhere Differentiable Functions (Part Two)https://www.math3ma.com/blog/baire-category-nowhere-differentiable-functions-part-twohttps://www.math3ma.com/blog/baire-category-nowhere-differentiable-functions-part-twoWelcome 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.Tue, 11 Sep 2018 00:56:51 GMTWhat is Galois Theory Anyway?https://www.math3ma.com/blog/what-is-galois-theory-anywayhttps://www.math3ma.com/blog/what-is-galois-theory-anywayPerhaps 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...Tue, 11 Sep 2018 00:56:51 GMTOn Connectedness, Intuitivelyhttps://www.math3ma.com/blog/on-connectedness-intuitivelyhttps://www.math3ma.com/blog/on-connectedness-intuitivelyToday'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!Tue, 11 Sep 2018 00:56:51 GMTOpen Sets Are Everythinghttps://www.math3ma.com/blog/open-sets-are-everythinghttps://www.math3ma.com/blog/open-sets-are-everythingIn 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." Tue, 11 Sep 2018 00:56:51 GMTThe Borel-Cantelli Lemmahttps://www.math3ma.com/blog/the-borel-cantelli-lemmahttps://www.math3ma.com/blog/the-borel-cantelli-lemmaToday 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?Tue, 11 Sep 2018 00:56:51 GMTFatou's Lemmahttps://www.math3ma.com/blog/fatous-lemmahttps://www.math3ma.com/blog/fatous-lemmaFatou'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").Tue, 11 Sep 2018 00:56:51 GMTMonotone Convergence Theoremhttps://www.math3ma.com/blog/monotone-convergence-theoremhttps://www.math3ma.com/blog/monotone-convergence-theoremFatou'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!Tue, 11 Sep 2018 00:56:51 GMTGood Reads: The Shape of Spacehttps://www.math3ma.com/blog/good-reads-the-shape-of-spacehttps://www.math3ma.com/blog/good-reads-the-shape-of-spaceHave 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?!)Tue, 11 Sep 2018 00:56:51 GMTDominated Convergence Theoremhttps://www.math3ma.com/blog/dominated-convergence-theoremhttps://www.math3ma.com/blog/dominated-convergence-theoremFatou'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.Tue, 11 Sep 2018 00:56:51 GMTTransitive Group Actions: "Where There's a Will, There's a Way!"https://www.math3ma.com/blog/transitive-group-actions-where-theres-a-will-theres-a-wayhttps://www.math3ma.com/blog/transitive-group-actions-where-theres-a-will-theres-a-wayIn 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!"Tue, 11 Sep 2018 00:56:51 GMTReal Talk: Math is Hard, Not Impossiblehttps://www.math3ma.com/blog/real-talk-math-is-hard-not-impossiblehttps://www.math3ma.com/blog/real-talk-math-is-hard-not-impossibleThe 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."Tue, 11 Sep 2018 00:56:51 GMTThe Fundamental Group of the Circle, Part 1https://www.math3ma.com/blog/the-fundamental-group-of-the-circle-part-1https://www.math3ma.com/blog/the-fundamental-group-of-the-circle-part-1Welcome 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.Tue, 11 Sep 2018 00:56:51 GMTThe Fundamental Group of the Circle, Part 2https://www.math3ma.com/blog/the-fundamental-group-of-the-circle-part-2https://www.math3ma.com/blog/the-fundamental-group-of-the-circle-part-2Welcome 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.Tue, 11 Sep 2018 00:56:51 GMTThe Fundamental Group of the Circle, Part 3https://www.math3ma.com/blog/the-fundamental-group-of-the-circle-part-3https://www.math3ma.com/blog/the-fundamental-group-of-the-circle-part-3Welcome 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.Tue, 11 Sep 2018 00:56:51 GMTThe Fundamental Group of the Circle, Part 4https://www.math3ma.com/blog/the-fundamental-group-of-the-circle-part-4https://www.math3ma.com/blog/the-fundamental-group-of-the-circle-part-4Welcome 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.Tue, 11 Sep 2018 00:56:51 GMTGood Reads: Visual Complex Analysishttps://www.math3ma.com/blog/good-reads-visual-complex-analysishttps://www.math3ma.com/blog/good-reads-visual-complex-analysisHave 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 knowTue, 11 Sep 2018 00:56:51 GMTThe Fundamental Group of the Circle, Part 5https://www.math3ma.com/blog/the-fundamental-group-of-the-circle-part-5https://www.math3ma.com/blog/the-fundamental-group-of-the-circle-part-5Welcome 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.Tue, 11 Sep 2018 00:56:51 GMTA Recipe for the Universal Cover of X⋁Yhttps://www.math3ma.com/blog/a-recipe-for-the-universal-cover-of-x-yhttps://www.math3ma.com/blog/a-recipe-for-the-universal-cover-of-x-yBelow 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. Tue, 11 Sep 2018 00:56:51 GMTThe Fundamental Group of the Circle, Part 6https://www.math3ma.com/blog/the-fundamental-group-of-the-circle-part-6https://www.math3ma.com/blog/the-fundamental-group-of-the-circle-part-6Welcome 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.Tue, 11 Sep 2018 00:56:51 GMTAbsolute Continuity (Part One)https://www.math3ma.com/blog/absolute-continuity-part-onehttps://www.math3ma.com/blog/absolute-continuity-part-oneThere 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.Tue, 11 Sep 2018 00:56:51 GMTAbsolute Continuity (Part Two)https://www.math3ma.com/blog/absolute-continuity-part-twohttps://www.math3ma.com/blog/absolute-continuity-part-twoThere 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.Tue, 11 Sep 2018 00:56:51 GMTThe Fundamental Group of the Real Projective Planehttps://www.math3ma.com/blog/the-fundamental-group-of-the-real-projective-planehttps://www.math3ma.com/blog/the-fundamental-group-of-the-real-projective-planeThe 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....Tue, 11 Sep 2018 00:56:51 GMTGood Reads: Love and Mathhttps://www.math3ma.com/blog/good-reads-love-and-mathhttps://www.math3ma.com/blog/good-reads-love-and-mathLove 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...Tue, 11 Sep 2018 00:56:51 GMTThe Pseudo-Hyperbolic Metric and Lindelöf's Inequalityhttps://www.math3ma.com/blog/the-pseudo-hyperbolic-metric-and-lindelofs-inequalityhttps://www.math3ma.com/blog/the-pseudo-hyperbolic-metric-and-lindelofs-inequalityIn 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.Tue, 11 Sep 2018 00:56:51 GMTThe Pseudo-Hyperbolic Metric and Lindelöf's Inequality (cont.)https://www.math3ma.com/blog/the-pseudo-hyperbolic-metric-and-lindelofs-inequality-conthttps://www.math3ma.com/blog/the-pseudo-hyperbolic-metric-and-lindelofs-inequality-contLast 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...."Tue, 11 Sep 2018 00:56:51 GMTGood Reads: Real Analysis by N. L. Carothershttps://www.math3ma.com/blog/good-reads-real-analysis-by-n-l-carothershttps://www.math3ma.com/blog/good-reads-real-analysis-by-n-l-carothersHave 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.Tue, 11 Sep 2018 00:56:51 GMTGraduate School: Where Grades Don't Matterhttps://www.math3ma.com/blog/graduate-school-where-grades-dont-matterhttps://www.math3ma.com/blog/graduate-school-where-grades-dont-matterYesterday 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.Tue, 11 Sep 2018 00:56:51 GMTClassifying Surfaces (CliffsNotes Version)https://www.math3ma.com/blog/classifying-surfaceshttps://www.math3ma.com/blog/classifying-surfacesMy 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:Tue, 11 Sep 2018 00:56:51 GMT(Co)homology: A Poemhttps://www.math3ma.com/blog/co-homology-a-poemhttps://www.math3ma.com/blog/co-homology-a-poemI 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!Tue, 11 Sep 2018 00:56:51 GMTSnippets of Mathematical Candorhttps://www.math3ma.com/blog/snippets-of-mathematical-candorhttps://www.math3ma.com/blog/snippets-of-mathematical-candorA 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.Tue, 11 Sep 2018 00:56:51 GMTClever Homotopy Equivalenceshttps://www.math3ma.com/blog/clever-homotopy-equivalenceshttps://www.math3ma.com/blog/clever-homotopy-equivalencesYou 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? Tue, 11 Sep 2018 00:56:51 GMTGood Reads: The Princeton Companion to Mathematicshttps://www.math3ma.com/blog/good-reads-the-princeton-companion-to-mathematicshttps://www.math3ma.com/blog/good-reads-the-princeton-companion-to-mathematicsNext 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.Tue, 11 Sep 2018 00:56:51 GMTThree Important Riemann Surfaceshttps://www.math3ma.com/blog/three-important-riemann-surfaceshttps://www.math3ma.com/blog/three-important-riemann-surfacesIn 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.Tue, 11 Sep 2018 00:56:51 GMTAutomorphisms of the Unit Dischttps://www.math3ma.com/blog/automorphisms-of-the-unit-dischttps://www.math3ma.com/blog/automorphisms-of-the-unit-discThis 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.Tue, 11 Sep 2018 00:56:51 GMTAutomorphisms of the Upper Half Planehttps://www.math3ma.com/blog/automorphisms-of-the-upper-half-planehttps://www.math3ma.com/blog/automorphisms-of-the-upper-half-planeThis 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.Tue, 11 Sep 2018 00:56:51 GMTAutomorphisms of the Complex Planehttps://www.math3ma.com/blog/automorphisms-of-the-complex-planehttps://www.math3ma.com/blog/automorphisms-of-the-complex-planeThis 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.Tue, 11 Sep 2018 00:56:51 GMTAutomorphisms of the Riemann Spherehttps://www.math3ma.com/blog/automorphisms-of-the-riemann-spherehttps://www.math3ma.com/blog/automorphisms-of-the-riemann-sphereThis 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.Tue, 11 Sep 2018 00:56:51 GMTA Ramble About Qualifying Examshttps://www.math3ma.com/blog/a-ramble-about-qualifying-examshttps://www.math3ma.com/blog/a-ramble-about-qualifying-examsToday 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.Tue, 11 Sep 2018 00:56:51 GMTResources for Intro-Level Graduate Courseshttps://www.math3ma.com/blog/resources-for-intro-level-graduate-courseshttps://www.math3ma.com/blog/resources-for-intro-level-graduate-coursesIn 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. Tue, 11 Sep 2018 00:56:51 GMTThe Most Obvious Secret in Mathematicshttps://www.math3ma.com/blog/the-most-obvious-secret-in-mathematicshttps://www.math3ma.com/blog/the-most-obvious-secret-in-mathematicsYes, 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. Tue, 11 Sep 2018 00:56:51 GMTThe Sierpinski Space and Its Special Propertyhttps://www.math3ma.com/blog/the-sierpinski-space-and-its-special-propertyhttps://www.math3ma.com/blog/the-sierpinski-space-and-its-special-propertyLast 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.Tue, 11 Sep 2018 00:56:51 GMTWhat's a Quotient Group, Really? Part 1https://www.math3ma.com/blog/whats-a-quotient-group-really-part-1https://www.math3ma.com/blog/whats-a-quotient-group-really-part-1I 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...Tue, 11 Sep 2018 00:56:51 GMTWhat's a Quotient Group, Really? Part 2https://www.math3ma.com/blog/whats-a-quotient-group-really-part-2https://www.math3ma.com/blog/whats-a-quotient-group-really-part-2Today 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 $G$ eitherTue, 11 Sep 2018 00:56:51 GMTThe First Isomorphism Theorem, Intuitivelyhttps://www.math3ma.com/blog/the-first-isomorphism-theorem-intuitivelyhttps://www.math3ma.com/blog/the-first-isomorphism-theorem-intuitivelyWelcome 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.Tue, 11 Sep 2018 00:56:51 GMTA Group and Its Center, Intuitivelyhttps://www.math3ma.com/blog/a-group-and-its-center-intuitivelyhttps://www.math3ma.com/blog/a-group-and-its-center-intuitivelyLast 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.Tue, 11 Sep 2018 00:56:51 GMTA Quotient of the General Linear Group, Intuitivelyhttps://www.math3ma.com/blog/a-quotient-of-the-general-linear-group-intuitivelyhttps://www.math3ma.com/blog/a-quotient-of-the-general-linear-group-intuitivelyOver 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. Tue, 11 Sep 2018 00:56:51 GMT#TrustYourStrugglehttps://www.math3ma.com/blog/trustyourstrugglehttps://www.math3ma.com/blog/trustyourstruggleIf 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.Tue, 11 Sep 2018 00:56:51 GMTWhat is Category Theory Anyway?https://www.math3ma.com/blog/what-is-category-theory-anywayhttps://www.math3ma.com/blog/what-is-category-theory-anywayA 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.Tue, 11 Sep 2018 00:56:51 GMTIntroducing... crumbs!https://www.math3ma.com/blog/introducing-crumbshttps://www.math3ma.com/blog/introducing-crumbsHello 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.Tue, 11 Sep 2018 00:56:51 GMTWhat is a Functor? Definition and Examples, Part 1https://www.math3ma.com/blog/what-is-a-functor-part-1https://www.math3ma.com/blog/what-is-a-functor-part-1Next 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.Tue, 11 Sep 2018 00:56:51 GMTWhat is a Functor? Definitions and Examples, Part 2https://www.math3ma.com/blog/what-is-a-functor-part-2https://www.math3ma.com/blog/what-is-a-functor-part-2Continuing 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).Tue, 11 Sep 2018 00:56:51 GMTcrumbs!https://www.math3ma.com/blog/crumbshttps://www.math3ma.com/blog/crumbsI 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,Tue, 11 Sep 2018 00:56:51 GMTWhat is a Natural Transformation? Definition and Exampleshttps://www.math3ma.com/blog/what-is-a-natural-transformationhttps://www.math3ma.com/blog/what-is-a-natural-transformationI 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?Tue, 11 Sep 2018 00:56:51 GMTWhat is a Natural Transformation? Definition and Examples, Part 2https://www.math3ma.com/blog/what-is-a-natural-transformation-2https://www.math3ma.com/blog/what-is-a-natural-transformation-2Continuing 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).Tue, 11 Sep 2018 00:56:51 GMT