This blog is good for topology and category theory amongst other mathematical concepts.
# About
My name is Tai-Danae Bradley, and I am a research mathematician at Alphabet, Inc. My research interests, publications, and contact info are listed on my research page. This website was originally created in 2015 as tool to help me transition from undergraduate to graduate level mathematics- math3ma.com 
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Results
Language, Statistics, & Category Theory, Part 3
Wed, Jul 28, 2021
Welcome to the final installment of our mini-series on the new preprint "An Enriched Category Theory of Language," joint work with John Terilla and Yiannis Vlassopoulos (https://arxiv.org/abs/2106.07890). Last time we discussed a way to assign sets to expressions in language ” words like "red" or "blue" “ which served as a first approximation to the meanings of those expressions. Motivated by elementary logic, we then found ways to represent combinations of expressions ” "red or blue" and "red and blue" and "red implies blue" ” using basic constructions from category theory. In today's short post, I'll share the statistical version of these ideas. [Read more on Math3ma!]- math3ma.com
Entropy + Algebra + Topology = ?
Wed, Jul 28, 2021
Today I'd like to share some math connecting ideas from information theory, algebra, and topology. It's all in a new paper I've recently uploaded to the arXiv (https://arxiv.org/abs/2107.09581), which describes a correspondence between Shannon entropy and functions on topological simplices that obey a version of the Leibniz rule from Calculus. The paper is short ” just 11 pages!Even so, I thought it'd be nice to stroll through some of the surrounding ideas here on the blog. [Read more on Math3ma.]- math3ma.com
Language, Statistics, & Category Theory, Part 2
Wed, Jul 21, 2021
Part 1 of this mini-series opened with the observation that language is an algebraic structure. But we also mentioned that thinking merely algebraically doesn't get us very far. The algebraic perspective, for instance, is not sufficient to describe the passage from probability distributions on corpora of text to syntactic and semantic information in language that wee see in today's large language models. This motivated the category theoretical framework presented in a new paper I shared last time But even before we bring statistics into the picture, there are some immediate advantages to using tools from category theory rather than algebra. One example comes from elementary considerations of logic, and that's where we'll pick up today. Let's start with a brief recap. [Read more on Math3ma. - math3ma.com
Language, Statistics, & Category Theory, Part 1
Wed, Jul 7, 2021
In the previous post I mentioned a new preprint that John Terilla, Yiannis Vlassopoulos, and I recently posted on the arXiv. In it, we ask a question motivated by the recent successes of the world's best large language models: "What's a mice mathematical framework in which to explain the passage from probability distributions on text to syntactic and semantic information in language?" To understand the motivation behind this question, and to recall what a "large language model" is, I'll encourage you to read the opening article from last time. In the next few blog posts ” starting today ” I'll give a tour of mathematical ideas presented in the paper towards answering the question above. I like the narrative we give, so I'll follow it closely here on the blog. You might think of the next few posts as an informal tour through the formal ideas found in the paper- math3ma.com
A Nod to Non-Traditional Applied Math
Tue, Jun 29, 2021
What is applied mathematics? The phrase might bring to mind historical applications of analysis to physical problems, or something similar. I think that's often what folks mean when they say "applied mathematics." And yet there's a much broader sense in which mathematics is applied, especially nowadays. I like what mathematician Tom Leinster once had to say about this: "I hope mathematicians and other scientists hurry up and realize that theres a glittering array of applications of mathematics in which non-traditional areas of mathematics are applied to non-traditional problems. It does no one any favours to keep using the term 'applied mathematics' in its current overly narrow sense." I'm all in favor of rebranding the term "applied mathematics" to encompass this wider notion. I certainly enjoy applying non-traditional areas of mathematics to non-traditional problems ” it's such a vibrant place to be! It's especially fun to take ideas that mathematicians already know lots about, then repurpose those ideas for potential applications in other domains. In fact, I plan to spend some time sharing one such example with you here on the blog.But before sharing the math” which I'll do in the next couple of blog posts ” I want to first motivate the story by telling you about an idea from the field of artificial intelligence (AI)- math3ma.com
Linear Algebra for Machine Learning
Thu, Jun 24, 2021
The TensorFlow channel on YouTube recently uploaded a video I made on some elementary ideas from linear algebra and how they're used in machine learning (ML).It's a very nontechnical introduction ” more of a bird's-eye view of some basic concepts and standard applications ” with the simple goal of whetting the viewer's appetite to learn more. I've decided to share it here on the blog, too, in case it may be of interest to anyone!- math3ma.com
Warming Up to Enriched Category Theory, Part 2
Thu, Jun 17, 2021
Let's jump right in to where we left off in part 1 of our warm-up to enriched category theory. If you'll recall from last time, we saw that the set of truth values $\{0, 1\}$ and the unit interval $[0,1]$ and the nonnegative extended reals $[0,\infty]$ were not just sets but actually preorders and hence categories. We also hinted at the idea that a "category enriched over" one of these preorders (whatever that means ” we hadn't defined it yet!) looks something like a collection of objects $X,Y,\ldots$ where there is at most one arrow between any pair $X$ and $Y$, and where that arrow can further be "decorated with" ”or simply replaced by ” a number from one of those three exemplary preorders.With that background in mind, my goal in today's article is to say exactly what a category enriched over a preorder is- math3ma.com
Warming Up to Enriched Category Theory, Part 1
Thu, Jun 10, 2021
It's no secret that Ilike category theory. It's a common theme on this blog, and it provides a nice lens through which to view old ideas in new ways ” and to view new ideas in new ways!Speaking of new ideas, my coauthors and I are planning to upload a new paper on the arXiv soon. I've really enjoyed the work and can't wait to share it with you. But first, you'll have to know a little something about enriched category theory. (And before that, you'll have to know something about ordinary category theory... you'll find an intro elsewhere on the blog!) So that's what I'd like to introduce today. A warm up, if you will- math3ma.com
The Fibonacci Sequence as a Functor
Tue, Jun 8, 2021
Over the years, the articles on this blog have spanned a wide range of audiences, from fun facts (Multiplying Non-Numbers), to undergraduate level (The First Isomorphism Theorem, Intuitively), to graduate level (What is an Operad?), to research level. This article is more on the fun-fact side of things, along with”like most articles here”an eye towards category theory. So here's a fun fact about greatest common divisors and the Fibonacci sequence $F_1,F_2,F_3,\ldots$, where $F_1=F_2=1$and $F_n:=F_{n-1} +F_{n-2}$ for $n>1$. Namely, for all $n,m\geq 1$, the greatest common divisor of the $n$th and $m$th Fibonacci numbers is the Fibonacci number whose index is the greatest common divisor of $n$ and $m$.... [Continue reading on Math3ma!]- math3ma.com
Limits and Colimits Part 3 (Examples)
Tue, Dec 29, 2020
Once upon a time, we embarked on a mini-series about limits and colimits in category theory.Part 1was a non-technical introduction that highlighted two ways mathematicians often make new mathematical objects from existing ones: bytaking 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 weregiven inPart 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 categoryto 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 newnessis in viewing them through a categorical lens.- math3ma.com
Language Modeling with Reduced Densities
Tue, Dec 29, 2020
Today I'd like to share with you a new paper on the arXiv [2007.03834]”my latest project in collaboration with mathematician Yiannis Vlassopoulos (Tunnel, IHES). In it, We present a framework for modeling words, phrases, and longer expressions in a natural language with linear operators. We show these operators capture something of the meaning of these expressions and also preserve both a simple form of entailment and the relevant statistics therein. Pulling back the curtain, the assignment is shown to be a functor between categories enriched over probabilities. [Read more on Math3ma!]- math3ma.com
Topology Book Launch
Tue, Dec 29, 2020
This is the official launch week of our new book, "Topology:ACategorical Approach," which is now available for purchase! We are also happy to offer a free open access version through MITPress at topology.mitpress.mit.edu. [Read more on Math3ma!]- math3ma.com
What's Next? (An Update)
Tue, Dec 29, 2020
Before introducing today's post, I'd like to first thank everyone who's reached out to me about my thesis and video posted last week. Thanks! Iappreciate all the generous feedback. Now onto the topic of the day: I'd like to share an update about what's coming next, both for me and for the blog. First, a word on the blog. [Read more on Math3ma.]- math3ma.com
At the Interface of Algebra and Statistics
Tue, Dec 29, 2020
I'm happy to share that I've successfully defended my PhD thesis, and my dissertation”"At the Interface of Algebra and Statistics"”is now available online at arXiv:2004.05631. In a few words, my thesis uses basic tools from quantum physics to investigate mathematical structure that is both algebraic and statistical. What do I mean?Well, the dissertation is about 130 pages long, which Irealize is a lot to chew. So Imade a 10-minute introductory video!It gives a brief tour of the paper and describes what I think is the quickest way to get a feel for what's inside. [Read more on Math3ma.]- math3ma.com
Topology: A Categorical Approach
Tue, Dec 29, 2020
I've been collaborating on an exciting project for quite some time now, and today I'm happy to share it with you. There is a new topology book on the market! Topology:ACategorical Approach is a graduate-level textbook that presents basic topology from the modern perspective of category theory. Coauthored with Tyler Bryson and John Terilla, Topology is published through MITPress and will be released on August 18, 2020. But you can pre-order on Amazon now!- math3ma.com
crumbs!
Tue, Dec 29, 2020
There are a couple of questions that I'm asked quite frequently these days: "How far along are you in graduate school?" "What's your research about anyways?" I created Math3ma precisely for my time in graduate school, so Ithought it'd be appropriate to share the answers here, just as a quick update!First, I'm graduating this semester!- math3ma.com
Modeling Sequences with Quantum States
Tue, Dec 29, 2020
In the past few months, I've shared a few mathematical ideas that I think are pretty neat: drawing matrices as bipartite graphs, picturing linear maps as tensor network diagrams, and understanding the linear algebraic (or "quantum") versions of probabilities.These ideas are all related by a project I've been working on with Miles Stoudenmire”a research scientist at the Flatiron Institute”and John Terilla”a mathematician at CUNY and Tunnel. We recently posted a paper on the arXiv: "Modeling sequences with quantum states: a look under the hood" (arXiv: 1910.07425) and today I'd like to tell you a little about it. [Continue reading on Math3ma!]- math3ma.com
What is an Adjunction? Part 1 (Motivation)
Tue, Dec 29, 2020
Some time ago, I started a blog series introducing the basics of category theory (categories, functors, natural transformations). Today, adjunctions are now on the list! So, what *is* an adjunction? Here's the start to a leisurely stroll through the ideas...- math3ma.com
What is an Adjunction? Part 2 (Definition)
Tue, Dec 29, 2020
Last time Ishared a light introduction to adjunctions in category theory. As we saw then, an adjunction consists of a pair of opposing functors $F$ and $G$ together with natural transformations $\text{id}\to\ GF$ and $FG\to\text{id}$ that interact nicely. Behind "interact nicely" is an idea that can be made precise. Unwinding this idea, and the formal definition of an adjunction, is what we'll do in today's post. [Continue reading on Math3ma!]- math3ma.com
What is an Adjunction? Part 3 (Examples)
Tue, Dec 29, 2020
Welcome to the last installment in our mini-series on adjunctions in category theory. We motivated the discussion in Part 1 and walked through formal definitions in Part 2. Today I'll share some examples. In Mac Lane's well-known words, "adjoint functors arise everywhere," so this post contains only a tiny subset of examples. Even so, I hope they'll help give you an eye for adjunctions and enhance your vision to spot them elsewhere.... [Continue reading on Math3ma!]- math3ma.com
A First Look at Quantum Probability, Part 1
Tue, Dec 29, 2020
In this article and the next, I'd like to share some ideas from the world of quantum probability. The word "quantum" is pretty loaded, but don't let that scare you. We'll take a first”not second or third”look at the subject, and the only prerequisites will be linear algebra and basic probability. In fact, I like to think of quantum probability as another name for "linear algebra + probability," so this mini-series will explore the mathematics, rather than the physics, of the subject- math3ma.com
A First Look at Quantum Probability, Part 2
Tue, Dec 29, 2020
Welcome back to our mini-series on quantum probability! Last time, we motivated the series by pondering over a thought from classical probability theory, namely that marginal probability doesn't have memory. That is, the process of summing over of a variable in a joint probability distribution causes information about that variable to be lost. But as we saw then, there is a quantum version of marginal probability that behaves much like "marginal probability with memory." It remembers what's destroyed when computing marginals in the usual way. In today's post, I'll unveil the details. Along the way, we'll take an introductory look at the mathematics of quantum probability theory.... [Read more on Math3ma!]- math3ma.com
Viewing Matrices & Probability as Graphs
Tue, Dec 29, 2020
Today 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 youhave!”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 aweighted 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- math3ma.com
crumbs!
Tue, Dec 29, 2020
Recently 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- math3ma.com
Applied Category Theory 2020
Tue, Dec 29, 2020
Hi all, just ducking in to help spread the word:the annual applied category theory conference (ACT2020) is taking place remotely this summer!Be sure to check out the conference website for the latest updates. As you might know, I was around for ACT2018, which inspired my 'What is Applied Category Theory?' booklet. This year I'm on the program committee and plan to be around for the main conference in July. Speaking of, here are the dates to know: Adjoint School:June 29 -- July 3; Tutorial Day: July 5; Main Conference: July 6 -- 10. Check out the conference website for the latest updates! http://act2020.mit.edu/- math3ma.com
The Tensor Product, Demystified
Tue, Dec 29, 2020
Previously 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?- math3ma.com
Matrices as Tensor Network Diagrams
Tue, Dec 29, 2020
In the previous post, I described a simple way to think about matrices, namely as bipartite graphs. Today I'd like to share a different way to picture matrices”one which is used not only in mathematics, but also in physics and machine learning. Here's the basic idea. An $m\times n$ matrix $M$ with real entries represents a linear map from $\mathbb{R}^n\to\mathbb{R}^m$. Such a mapping can be pictured as a node with two edges. One edge represents the input space, the other edge represents the output space.That's it! We can accomplish much with this simple idea. But first, a few words about the picture... (Continue reading on Math3ma!)- math3ma.com
Understanding Entanglement With SVD
Tue, Dec 29, 2020
Quantum entanglement is, as you know, a phrase that's jam-packed with meaning in physics. But what you might not know is that the linear algebra behind it is quite simple.If you're familiar with singular value decomposition (SVD), then you're 99% there. My goal for this post is to close that 1% gap. In particular, I'd like to explain something called the Schmidt rank in the hopes of helping the math of entanglement feel a little less... tangly. And to do so, I'll ask that you momentarily forget about the previous sentences. Temporarily ignore the title of this article. Forget we're having a discussion about entanglement. Forget I mentioned that word. And let's start over. Let's just chat math. Let's talk about SVD. [Read more on Math3ma!]- math3ma.com
Finitely Generated Modules Over a PID
Sat, Dec 12, 2020
We 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- math3ma.com
What do Polygons and Galois Theory Have in Common?
Thu, Oct 15, 2020
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!- math3ma.com
The Borel-Cantelli Lemma
Fri, Aug 14, 2020
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?- math3ma.com
The Yoneda Perspective
Tue, Apr 21, 2020
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?- math3ma.com
Monotone Convergence Theorem
Tue, Apr 21, 2020
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!- math3ma.com
Dominated Convergence Theorem
Tue, Apr 21, 2020
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- math3ma.com
Ways to Show a Group is Abelian
Fri, Feb 7, 2020
After some exposure to group theory, you quickly learn that when trying to prove a group $G$ is abelian, checking if $xy=yx$ for arbitrary $x,y$ in $G$ is not always the most efficient - or helpful! - tactic. Here is a (not comprehensive) running tab of other ways you may be able to prove your group is abelian:- math3ma.com
What is a Natural Transformation? Definition and Examples
Thu, Jan 30, 2020
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?- math3ma.com
The Yoneda Embedding
Wed, Jan 29, 2020
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,- math3ma.com
The Yoneda Lemma
Wed, Jan 29, 2020
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...- math3ma.com
Rational Canonical Form: Example #1
Fri, Dec 27, 2019
Last time we discussed the rational canonical form (RCF) of a linear transformation, and we mentioned that any two similar linear transformations have the same RCF. 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- math3ma.com
Compact + Hausdorff = Normal
Tue, Dec 3, 2019
The notion of a topological space being Hausdorff or normal identifies the degree to which points or sets can be "separated." In a Hausdorff space, it's guaranteed that if you pick any two distinct points in the space -- say $x$ and $y$ -- then you can always find an open set containing $x$ and an open set containing $y$ such that those two sets don't overlap- math3ma.com
crumbs!
Mon, Aug 19, 2019
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....- math3ma.com
Limits and Colimits, Part 2 (Definitions)
Sat, Jun 29, 2019
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- math3ma.com
Snippets of Mathematical Candor
Fri, Jun 7, 2019
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- math3ma.com
Constructing the Tensor Product of Modules
Wed, May 15, 2019
Today we talk tensor products. Specifically this post covers the construction of the tensor product between two modules over a ring. But before jumping in, I think now's a good time to ask, "What are tensor products good for?" Here's a simple example where such a question might arise...- math3ma.com
The Integral Domain Hierarchy, Part 2
Wed, May 15, 2019
In any area of math, it's always good idea to keep a few counterexamples in your back pocket. This post continues part 1 with examples/non-examples from some of the different subsets of integral domains- math3ma.com
Commutative Diagrams Explained
Mon, May 6, 2019
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- math3ma.com
Comparing Topologies
Fri, Apr 5, 2019
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?- math3ma.com
A Math Blog? Say What?
Thu, Jan 24, 2019
Yes! 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- math3ma.com
Limits and Colimits, Part 1 (Introduction)
Thu, Jan 24, 2019
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- math3ma.com
Announcing Applied Category Theory 2019
Wed, Jan 9, 2019
Hi 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, 2019- math3ma.com
Notes on Applied Category Theory
Sun, Jan 6, 2019
Have 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!)- math3ma.com
Why are Noetherian Rings Special?
Fri, Nov 23, 2018
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)- math3ma.com
What's a Transitive Group Action?
Fri, Nov 23, 2018
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¦- math3ma.com
English is Not Commutative
Fri, Nov 23, 2018
Here's another unspoken rule of mathematics: English doesn't always commute! Word order is important...- math3ma.com
One Unspoken Rule of Algebra
Fri, Nov 23, 2018
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!- math3ma.com
One Unspoken Rule of Measure Theory
Fri, Nov 23, 2018
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- math3ma.com
Completing a Metric Space, Intuitively
Fri, Nov 23, 2018
An incomplete metric space is very much like a golf course: it has a lot of missing points!- math3ma.com
"Up to Isomorphism"?
Fri, Nov 23, 2018
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 œup to isomorphism if theyre 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 œsame." Instead we prefer to say œsame up to isomorphism. Voila!- math3ma.com
Operator Norm, Intuitively
Fri, Nov 23, 2018
If $X$ and $Y$ are normed vector spaces, a linear map $T:X\to Y$ is said to be bounded if $\|T\|< \infty$ where $$\|T\|=\sup_{\underset{x\neq 0}{x\in X}}\left\{\frac{|T(x)|}{|x|}\right\}.$$ (Note that $|T(x)|$ is the norm in $Y$ whereas $|x|$ is the norm 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¦- math3ma.com
Borel-Cantelli Lemma (Pictorially)
Fri, Nov 23, 2018
The Borel-Cantelli Lemma says that if $(X,\Sigma,\mu)$ is a measure space with $\mu(X)- math3ma.com
Need to Prove Your Ring is NOT a UFD?
Fri, Nov 23, 2018
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¦- math3ma.com
Learning How to Learn Math
Sun, Nov 18, 2018
Once 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."- math3ma.com
Is the Square a Secure Polygon?
Sat, Oct 13, 2018
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?- math3ma.com
Topological Magic: Infinitely Many Primes
Sun, Oct 7, 2018
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, Id like to show you another such example - one which has a surprising consequence!- math3ma.com
What is a Category? Definition and Examples
Thu, Sep 20, 2018
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...- math3ma.com
Motivation for the Tensor Product
Tue, Sep 11, 2018
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.)- math3ma.com
On Connectedness, Intuitively
Tue, Sep 11, 2018
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!- math3ma.com
Some Notes on Taking Notes
Tue, Sep 11, 2018
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),- math3ma.com
crumbs!
Tue, Sep 11, 2018
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?- math3ma.com
Transitive Group Actions: "Where There's a Will, There's a Way!"
Tue, Sep 11, 2018
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!"- math3ma.com
Continuous Functions, Discontinuous Supremum
Tue, Sep 11, 2018
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- math3ma.com
A Little Fact From Group Actions
Tue, Sep 11, 2018
Today 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- math3ma.com
Baire Category & Nowhere Differentiable Functions (Part One)
Tue, Sep 11, 2018
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- math3ma.com
The Pseudo-Hyperbolic Metric and Lindelöf's Inequality (cont.)
Tue, Sep 11, 2018
Last time we proved that the pseudo-hyperbolic metric on the unit disc in „‚ is indeed a metric. In todays post, we use this fact to verify Lindelöfs 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...."- math3ma.com
The Pseudo-Hyperbolic Metric and Lindelöf's Inequality
Tue, Sep 11, 2018
In this post, we define the pseudo-hyperbolic metric on the unit disc in „‚ and prove it does indeed satisfy the conditions of a metric- math3ma.com
Group Elements, Categorically
Tue, Sep 11, 2018
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 {œ³ï¸Ž} 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- math3ma.com
What is an Operad? Part 2
Tue, Sep 11, 2018
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!- math3ma.com
What is an Operad? Part 1
Tue, Sep 11, 2018
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- math3ma.com
Noetherian Rings = Generalization of PIDs
Tue, Sep 11, 2018
When 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- math3ma.com
Open Sets Are Everything
Tue, Sep 11, 2018
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."- math3ma.com
Good Reads: Real Analysis by N. L. Carothers
Tue, Sep 11, 2018
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- math3ma.com
Four Flavors of Continuity
Tue, Sep 11, 2018
Here's a chart to help keep track of some of the different "flavors" of continuity in real analysis- math3ma.com
A Non-Measurable Set
Tue, Sep 11, 2018
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.)- math3ma.com
A Group and Its Center, Intuitively
Tue, Sep 11, 2018
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- math3ma.com
What is a Natural Transformation? Definition and Examples, Part 2
Tue, Sep 11, 2018
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)- math3ma.com
#TrustYourStruggle
Tue, Sep 11, 2018
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- math3ma.com
A Ramble About Qualifying Exams
Tue, Sep 11, 2018
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- math3ma.com
A Recipe for the Universal Cover of X⋁Y
Tue, Sep 11, 2018
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- math3ma.com
Stone Weierstrass Theorem
Tue, Sep 11, 2018
The 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- math3ma.com
Clever Homotopy Equivalences
Tue, Sep 11, 2018
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?- math3ma.com
Baire Category & Nowhere Differentiable Functions (Part Two)
Tue, Sep 11, 2018
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- math3ma.com
Introducing... crumbs!
Tue, Sep 11, 2018
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- math3ma.com
Resources for Intro-Level Graduate Courses
Tue, Sep 11, 2018
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- math3ma.com
Classifying Surfaces (CliffsNotes Version)
Tue, Sep 11, 2018
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:- math3ma.com
Topology vs. "A Topology" (cont.)
Tue, Sep 11, 2018
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- math3ma.com
The Fundamental Group of the Real Projective Plane
Tue, Sep 11, 2018
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....- math3ma.com
crumbs!
Tue, Sep 11, 2018
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?)- math3ma.com
Good Reads: The Shape of Space
Tue, Sep 11, 2018
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?!)- math3ma.com
Lebesgue Measurable But Not Borel
Tue, Sep 11, 2018
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- math3ma.com
Absolute Continuity (Part Two)
Tue, Sep 11, 2018
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- math3ma.com