Last week was the start of a mini-series on limits and colimits in category theory. We began by answering a few basic questions, including, "What ARE (co)limits?" In short, they are a way to construct new mathematical objects from old ones. For more on this non-technical answer, be sure to check out Limits and Colimits, Part 1. Towards the end of that post, I mentioned that (co)limits aren't…
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.
Over the past couple of years, I've been learning a little about the world of quantum machine learning (QML) and the sorts of things people are thinking about there. I recently gave an high-level talk on some of these ideas in connection to a December 2024 preprint called "Towards Structure-Preserving Quantum Encodings", coauthored with collaborators at Deloitte (Andrew Vlasic and Anh Pham) and…
Next up on Good Reads: The Princeton Companion to Mathematics, edited by Fields medalist Timothy Gowers. This book is an exceptional resource! With over 1,000 pages of mathematics explained by the experts for the layperson, it's like an encyclopedia for math, but so much more. Have you heard about category theory but aren't sure what it is? There's a chapter for that! Seen the recent headlines…
It's hard for me to believe, but Math3ma is TEN YEARS old today. My first entry was published on February 1, 2015 and is entitled "A Math Blog? Say What?" As evident from that post, I was very unsure about creating this website. At the time, writing about mathematics seemed very niche — though, I guess it still is — and the thought of sharing my musings publicly was quite intimidating. But as I…
The next episode in the fAQ video podcast is now up! As mentioned last time, this is a new project I've embarked on with Adam Green where we chat about different ideas in quantum physics and (at some point) AI. Our primary goal is simply to help make these ideas more accessible to wide audiences — especially to folks who may've heard about certain words in, say, the popular media, but who may not…
In a bit of fun news, I've just launched a new video podcast with my coworker Adam Green. This new video series, which we're calling fAQ, consists of casual conversations between me and Adam on basic ideas in quantum physics and eventually some topics in AI. (Hence the "A" and "Q," which is also a hat tip to our employer, SandboxAQ.) The target audience is very broad and includes any curious human…
Today I'm excited to share a few new videos with you. But first, a little background. As you may know, I started working at Alphabet, Inc. just after finishing graduate school in 2020. I was on a team of amazing people that formed the core of what is now SandboxAQ, a new company focusing on AI and quantum technologies, which spun out of Alphabet in March 2022. There were several news articles…
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…
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…
Recently on The Math3ma Institute's blog, I announced an upcoming event that will be hosted at The Master's University (TMU), which is a small private university in Santa Clarita, California. I wanted to briefly mention it here, too, in case it might be of interest to any readers. This summer on June 9–10, I'll be joined by NASA astronaut Jeffrey Williams and molecular geneticist Beth Sullivan…
Hello world! Last summer I wrote a short paper entitled "Entropy as a Topological Operad Derivation," which describes a small but interesting connection between information theory, abstract algebra, and topology. I blogged about it here in June 2021, and the paper was later published in an open-access journal called Entropy in September 2021. In short, it describes a correspondence between Shannon…
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...
Today I'm happy to share that the Math3ma platform has recently grown in a small yet personal way. This new endeavor is in its early stages, but it is one that is close to my heart and gives life to the reasons I started this blog six years ago. A more formal announcement can be found in a new article I wrote for the university, but I'd like to give an update here as well. This semester I joined…
I recently had the pleasure of chatting with Grant Sanderson on the 3Blue1Brown podcast about a variety of topics, including what first drew me to math and physics, my time in graduate school, thoughts on category theory, basketball, and lots more. We also chatted a bit about Math3ma and its origins, so I thought it'd be fun to share this "behind the scenes" peek with you all here on the blog.…
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…
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…
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.
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?
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…
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…
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…
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…
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…
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…
It's no secret that I like 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…
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…
This is the official launch week of our new book, "Topology: A Categorical Approach," which is now available for purchase! We are also happy to offer a free open access version through MIT Press at topology.mitpress.mit.edu. [Read more on Math3ma!]
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…
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! I appreciate 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.]
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 I realize is…
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: A Categorical 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 MIT Press and will be…
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…
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 I thought it'd be appropriate to share the answers here, just as a quick update! First, I'm graduating this semester!
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…
Last time I shared 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…
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…
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…
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 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…
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…
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...
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,…
Once 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…
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…
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…
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!
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?
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:
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?
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,