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μάθημα = mathema (n.) that which is learned
μάθημα = mathema (n.)

that which is learned

MATH3MA

a math blog

WHAT IS SUPERPOSITION, REALLY?

March 28, 2023
•
Physics

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 have a technical background and who aren't really sure what those
words mean.

We thought it'd be good to launch the podcast with some basic, fundamental ideas
that can be used as a foundation for discussing real-world application in future
episodes. Last time we introduced the topic of qubits, and today we're focusing
on another basic topic, namely superposition.

So, what is superposition? We spend nearly an hour on this question, so I won't
spoil it all for you! But there are a few remarks I can't resist sharing here.

NEW VIDEO PODCAST: FAQ

March 14, 2023
•
Physics

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 who wants to learn more about these ideas. Our hope is that these
informal chats might help demystify some ideas in math, physics, and their
applications and make the concepts more accessible to wide audiences.

Adam is a biologist by training, an excellent science communicator, and before
joining Sandbox he was the Director of US Academic Content at Khan Academy. He's
now the Head of Science education at Sandbox, and since neither of us are
physicists, we're essentially working together to learn new things and are
inviting anyone to join us!

Our plan is to spend the first few episodes discussing fundamental ideas, just
to lay down some ground work, and then we'll see where things go from there. So,
without any further ado, here's a short seven minute "trailer" video we made to
introduce the podcast.

SYMPOSIUM AT THE MASTER'S UNIVERSITY

February 9, 2023
•
Other

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 (Duke University) for a two-day symposium,
which invites folks with vocations in a wide range of scientific disciplines
from academia, industry, and government for a time of fellowship, encouragement,
and the opportunity for dialogue and discussion. We're also honored to be joined
by theologian Abner Chou, the president of TMU, as well as John MacArthur, the
chancellor of TMU and the pastor of Grace Community Church in Los Angeles.

If you're interested to learn more, details and registration are now available
at: www.masters.edu/math3ma.

WHAT IS QUANTUM TECHNOLOGY?

January 10, 2023
•
Physics

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 about
this, including this one from the Wall Street Journal and this one from Forbes.
More press coverage is listed on the company website here.

But what is SandboxAQ exactly? I recently asked Sandbox founder and CEO Jack
Hidary that question, and you can now check it out on our new YouTube channel.
Take a look!

A NEW PERSPECTIVE OF ENTROPY

February 22, 2022
•
Probability

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 entropy and functions on topological simplices
that obey a version of the Leibniz rule from calculus, which I call "derivations
of the operad of topological simplices," hence the title.

By what do those words mean? And why is such a theorem interesting?

‍To help make the ideas more accessible, I've recently written a new article
aimed at a wide audience to explain it all from the ground up. I'm very excited
to share it with you! It's entitled "A New Perspective of Entropy," and a
trailer video is below:

As mentioned in the video, the reader is not assumed to have prior familiarity
with the words "information theory" or "abstract algebra" or "topology" or even
"Shannon entropy." All these ideas are gently introduced from the ground up.

INTRODUCING THE MATH3MA INSTITUTE

November 1, 2021
•
Other

Today I am excited 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 personal 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 The Master's University (TMU), a small private Christian
university in southern California, as a visiting research professor of
mathematics. I am still a full-time research mathematician in the tech world,
but I've also been collaborating part time with the math, science, and
engineering faculty at TMU to launch a little research hub on the university's
campus and online.

We are calling it The Math3ma Institute, and the website is now live:
www.math3ma.institute.

WHAT IS THE MATH3MA INSTITUTE?

I've branded our little venture an "institute," though its function is notably
different from that of other research institutions. Our vision is that The
Math3ma Institute will grow into a place where TMU faculty and advanced
undergraduates, along with external colleagues, can engage in research
activities in various STEM fields—not just mathematics—and where those results
will then be made accessible to a broad audience in easy-to-understand ways.
This is summarized in the trifecta boldly displayed on our homepage: discover.
share. repeat.

To this end, we are aiming to produce several publicly-available resources,
including the launch of semi-annual journal described below.

MATH3MA: BEHIND THE SCENES (3B1B PODCAST)

September 27, 2021
•
Other

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. Enjoy!

LANGUAGE, STATISTICS, & CATEGORY THEORY, PART 3

July 28, 2021
•
Category Theory

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. In Part 2 of this series, 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.

I like to think of Part 2 as a commercial advertising the benefits of a category
theoretical approach to language, rather than a merely algebraic one. But as we
observed in Part 1, algebraic structure is not all there is to language. There's
also statistics! And far from being an afterthought, those statistics play an
essential role as evidenced by today's large language models discussed in Part
0.

Happily, category theory already has an established set of tools that allow one
to incorporate statistics in a way that's compatible with the considerations of
logic discussed last time. In fact, the entire story outlined in Part 2 has a
statistical analogue that can be repeated almost verbatim. In today's short
post, I'll give lightning-quick summary.

It all begins with a small, yet crucial, twist.

ENTROPY + ALGEBRA + TOPOLOGY = ?

July 21, 2021
•
Topology

Today I'd like to share a bit of math involving ideas from information theory,
algebra, and topology. It's all in a new paper I've recently uploaded to the
arXiv, whose abstract you can see on the right. The paper is short — just 11
pages! Even so, I thought it'd be nice to stroll through some of the surrounding
mathematics here.

To introduce those ideas, let's start by thinking about the function
d:[0,1]→Rd:[0,1]→R defined by d(x)=−xlogxd(x)=−xlog⁡x when x>0x>0 and
d(x)=0d(x)=0 when x=0x=0. Perhaps after getting out pencil and paper, it's easy
to check that this function satisfies an equation that looks a lot like the
product rule from Calculus:

Functions that satisfy an equation reminiscent of the "Leibniz rule," like this
one, are called derivations, which invokes the familiar idea of a derivative.
The nonzero term −xlogx−xlog⁡x above may also look familiar to some of you. It's
an expression that appears in the Shannon entropy of a probability distribution.
A probability distribution on a finite set {1,…,n}{1,…,n} for n≥1n≥1 is a
sequence p=(p1,…,pn)p=(p1,…,pn) of nonnegative real numbers satisfying
∑ni=1pi=1∑i=1npi=1, and the Shannon entropy of pp is defined to be

Now it turns out that the function dd is nonlinear, which means we can't pull it
out in front of the summation. In other words, H(p)≠d(∑ipi).H(p)≠d(∑ipi). Even
so, curiosity might cause us to wonder about settings in which Shannon entropy
is itself a derivation. One such setting is described in the paper above, which
shows a correspondence between Shannon entropy and derivations of (wait for
it...) topological simplices!

LANGUAGE, STATISTICS, & CATEGORY THEORY, PART 2

July 7, 2021
•
Category Theory

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.

LANGUAGE, STATISTICS, & CATEGORY THEORY, PART 1

June 29, 2021
•
Category Theory

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 nice 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, 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.

Now, where shall we begin? What math are we talking about?

Let's start with a simple fact about language.

LANGUAGE IS ALGEBRAIC.

By "algebraic," I mean the basic sense in which things combine to form a new
thing. We learn about algebra at a young age: given two numbers xx and yy we can
multiply them to get a new number xyxy. We can do something similar in language.
Numbers combine to give new numbers, and words and phrases in a language combine
to give new expressions. Take the words red and firetruck, for example. They can
be "multiplied" together to get a new phrase: red firetruck.

Here, the "multiplication" is just concatenation — sticking things side by side.
This is a simple algebraic structure, and it's inherent to language. I'm
concatenating words together as I type this sentence. That's algebra! Another
word for this kind of structure is compositionality, where things compose
together to form something larger.

So language is algebraic or compositional.

A NOD TO NON-TRADITIONAL APPLIED MATH

June 24, 2021
•
Other

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
(emphasis mine):

> "I hope mathematicians and other scientists hurry up and realize that there’s
> 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).

‍

LINEAR ALGEBRA FOR MACHINE LEARNING

June 17, 2021
•
Algebra

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, too, in case it may be of interest to anyone!

I imagine the content here might be helpful for undergraduate students who are
in their first exposure to linear algebra and/or to ML, or for anyone else who's
new to the topic and wants to get an idea for what it is and some ways it's
used.

The video covers three basic concepts — vectors and matrix factorizations and
eigenvectors/eigenvalues — and explains a few ways these concepts arise in ML —
namely, as data representations, to find vector embeddings, and for
dimensionality reduction techniques, respectively.

Enjoy!

WARMING UP TO ENRICHED CATEGORY THEORY, PART 2

June 10, 2021
•
Category Theory

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}{0,1} and the unit interval [0,1][0,1] and the nonnegative extended
reals [0,∞][0,∞] 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,…X,Y,… where there is at most one arrow between any
pair XX and YY, 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. The formal definition — and the
intuition behind it — will then pave the way for the notion of a category
enriched over an arbitrary (and sufficiently nice) category, not just a
preorder.

En route to this goal, it will help to make a couple of opening remarks.

TWO THINGS TO THINK ABOUT.

First, take a closer look at the picture on the right. I've written
"hom(X,Y)hom(X,Y)" in quotation marks because the notation hom(−,−)hom(−,−) is
often used for a set of morphisms in ordinary category theory. But the point of
this discussion is that we're not just interested in sets! So we should use
better notation: let's refer to the number associated to a pair of objects
XYXY and YY as C(X,Y)C(X,Y), where the letter "CC" reminds us there's an
(enriched) CCategory being investigated.

Second, for the theory to work out nicely, it turns out that preorders need a
little more added to them.

WARMING UP TO ENRICHED CATEGORY THEORY, PART 1

June 8, 2021
•
Category Theory

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 about enriched category theory. (And before that, you'll have to know
something about ordinary category theory... here's an intro!) So that's what I'd
like to introduce today.

A warm up, if you will.

WHAT IS ENRICHED CATEGORY THEORY?

As the name suggests, it's like a "richer" version of category theory, and it
all starts with a simple observation. (Get your category theory hats on, people.
We're jumping right in!)

In a category, you have some objects and some arrows between them, thought of as
relationships between those objects. Now in the formal definition of a category,
we usually ask for a set's worth of morphisms between any two objects, say XX
and YY. You'll typically hear something like, "The hom set hom(X,Y)hom(X,Y) bla
bla...."
‍

Now here's the thing. Quite often in mathematics, the set hom(X,Y)hom(X,Y) may
not just be a set. It could, for instance, be a set equipped with extra
structure. You already know lots of examples. Let's think about about linear
algebra, for a moment.

THE FIBONACCI SEQUENCE AS A FUNCTOR

December 14, 2020
•
Category Theory

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. Today's 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 (GCDs) and the Fibonacci
sequence F1,F2,F3,…F1,F2,F3,…, where F1=F2=1F1=F2=1 and
Fn:=Fn−1+Fn−2Fn:=Fn−1+Fn−2 for n>1n>1. For all n,m≥1n,m≥1,

In words, the greatest common divisor of the nnth and mmth Fibonacci numbers is
the Fibonacci number whose index is the greatest common divisor of nn and mm.
(Here's a proof.) Upon seeing this, your "spidey senses" might be
tingling. Surely there's some structure-preserving map FF lurking in the
background, and this identity means it has a certain nice property. But what is
that map? And what structure does it preserve? And what's the formal way to
describe the nice property it has?

The short answer is that the natural numbers N={1,2,3,…}N={1,2,3,…} form a
partially ordered set (poset) under division, and the function F:N→NF:N→N
defined by n↦Fn:=F(n)n↦Fn:=F(n) preserves meets: Fn∧Fm=F(n∧m)Fn∧Fm=F(n∧m).

UNDERSTANDING ENTANGLEMENT WITH SVD

September 7, 2020
•
Algebra

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.

SINGULAR VALUE DECOMPOSITION

SVD is arguably one of the most important, well-known tools in linear algebra.
You are likely already very familiar with it, but here's a lightening-fast
recap. Every matrix MM can be factored as M=UDV†M=UDV† as shown below, called
the singular value decomposition of MM. The entries of the diagonal matrix DD
are nonnegative numbers called singular values, and the number of them is equal
to the rank of MM, say kk. What's more, UU and VV have exactly kk columns,
called the left and right singular vectors, respectively.

There are different ways to think about this, depending on which applications
you have in mind. I like to think of singular vectors as encoding meaningful
"concepts" inherent to MM, and of singular values as indicating how important
those concepts are.

TOPOLOGY BOOK LAUNCH

August 20, 2020
•
Topology

This is the official launch week of our new book, Topology: A Categorical
Approach, which is now available for purchase! We're also happy to offer a free
open access version through MIT Press at topology.mitpress.mit.edu.

Inside, you'll find a presentation of basic, point-set topology from the
perspective of category theory, targeted at graduate students in a
first-semester course on topology. The idea is that most of these students are
already somewhat familiar with the point-set ideas through a course on analysis
or undergraduate topology. For this reason, many graduate-level instructors are
tempted to rush through point-set topology or to skip it altogether to reach
algebraic topology, which can be more fun to learn and to teach.

Our book presents an alternative to this approach. Rather than skipping over the
basic ideas, we view this as an excellent opportunity to introduce students to
the modern viewpoint of category theory.

LANGUAGE MODELING WITH REDUCED DENSITIES

July 9, 2020
•
Category Theory

Today I'd like to share with you a new paper on the arXiv—my latest project in
collaboration with mathematician Yiannis Vlassopoulos (Tunnel, IHES). To whet
your appetite, let me first set the stage. A few months ago I made a 10-minute
introductory video to my PhD thesis, which was an investigation into
mathematical structure that is both algebraic and statistical. In the video,
I noted that natural language is an example of where such mathematical structure
can be found.

> Language is algebraic, since words can be concatenated to form longer
> expressions. Language is also statistical, since some expressions occur more
> frequently than others.

As a simple example, take the words "orange" and "fruit." We can stick them
together to get a new phrase, "orange fruit." Or we could put "orange" together
with "idea" to get "orange idea." That might sound silly to us, since the phrase
"orange idea" occurs less frequently in English than "orange fruit." But that's
the point. These frequencies contribute something to the meanings of these
expressions. So what is this kind of mathematical structure? As I mention in the
video, it's helpful to have a set of tools to start exploring it, and basic
ideas from quantum physics are one source of inspiration. I won't get into this
now—you can watch the video or read the thesis! But I do want to emphasize the
following: In certain contexts, these tools provide a way to see that statistics
can serve as a proxy for meaning. I didn't explain how in the video. I left it
as a cliffhanger.

But I'll tell you the rest of the story now.

WHAT'S NEXT? (AN UPDATE)

April 18, 2020
•
Other

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.

WHAT'S NEXT FOR MATH3MA?

I created Math3ma in early 2015 as an aid in transitioning from
undergraduate-level to graduate-level mathematics. And it worked! The blog has
been a sort of public math journal for more than five years, and I'm so glad I
started it. My appreciation extends to all readers and to everyone who has
contacted me through the site over the years. I'm always delighted to hear that
the blog has been helpful for you, as well.

So since Math3ma has accomplished its purpose, my use for it will change. As you
may have noticed already, I blog less frequently than I used to, and the content
of my more recent articles is slightly different than the content in 2015--2017.

So what will happen to Math3ma going forward? I don't know, and I plan to be
flexible about this. Of course I'll leave the website up, but going forward I
plan to be flexible with how often I'll blog (not often, probably) and with what
I'll blog about (my research, probably). I don't have any big plans for upcoming
blog posts, so I may just take a break for a while. On the other hand, I do plan
to post a paper or two on the arXiv in the near future, and I might decide to
blog about it since I find sharing math irresistible.

See? Flexible.

AT THE INTERFACE OF ALGEBRA AND STATISTICS

April 13, 2020
•
Probability

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 a lot to chew. So I made 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.

Now, let me highlight an important point that I make in the video:

> I wrote my dissertation with a wide audience in mind.

In particular, there is a great deal of exposition woven into the mathematics
that provides intuition and motivation for the ideas. I’ve also sprinkled
several “behind the scenes” snippets throughout, and alongside the propositions,
lemmas, and corollaries there are Takeaways that summarize key ideas. Several of
these key ideas are introduced through simple examples that are placed
before—not after—the theory they're meant to illustrate. And in a happy turn of
events, there is a low entrance fee for following the mathematics. The main
tools are linear algebra and basic probability theory. And yes, there is some
category theory, too!

TOPOLOGY: A CATEGORICAL APPROACH

April 3, 2020
•
Topology

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 released on August 18, 2020. But you can pre-order on
Amazon now!

Here is the book's official description:

This graduate-level textbook on topology takes a unique approach: it
reintroduces basic, point-set topology from a more modern, categorical
perspective. Many graduate students are familiar with the ideas of point-set
topology and they are ready to learn something new about them. Teaching the
subject using category theory—a contemporary branch of mathematics that provides
a way to represent abstract concepts—both deepens students' understanding of
elementary topology and lays a solid foundation for future work in advanced
topics.

After presenting the basics of both category theory and topology, the book
covers the universal properties of familiar constructions and three main
topological properties—connectedness, Hausdorff, and compactness. It presents a
fine-grained approach to convergence of sequences and filters; explores
categorical limits and colimits, with examples; looks in detail at adjunctions
in topology, particularly in mapping spaces; and examines additional
adjunctions, presenting ideas from homotopy theory, the fundamental groupoid,
and the Seifert van Kampen theorem. End-of-chapter exercises allow students to
apply what they have learned. The book expertly guides students of topology
through the important transition from undergraduate student with a solid
background in analysis or point-set topology to graduate student preparing to
work on contemporary problems in mathematics.

APPLIED CATEGORY THEORY 2020

March 2, 2020
•
Other

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

The Adjoint School is a months-long online reading group, where participants
paired with researchers work through some of the major papers in the field. It
culminates in an in-person meet-up a week before the main conference.
Unfortunately, the deadline apply to the school is already closed. But this
year, the conference has a Tutorial Day, too! I love this idea. It's open to
anyone (first-come first serve) who's new to applied category theory and wants a
little more background to get the most out of the talks during the main
conference. You'll get to meet with Paolo Perrone, David Spivak, and Emily Riehl
and learn math! And here's a quick blurb about the main conference, taken from
this year's website.

CRUMBS!

February 6, 2020
•
crumbs!

How far along are you in graduate school? What exactly is it that you do? These
are two questions I'm asked frequently these days and am happy to answer. I
created Math3ma precisely for my time in graduate school, so I thought it'd be
appropriate to share the answers here, too, as a quick update! First,

I'm graduating this semester!

It's exciting. I've really enjoyed my time as a graduate student and have been
looking forward to the future for a while now. (I'll share more on my
post-graduate plans in another post.) In the mean time, I'm in the midst of
writing up my dissertation. The title is pending. Stay tuned!

Now to the second question: "What's your research about anyways?"

My work doesn't fit into any one mathematical label (geometry, topology,
algebra, category theory, etc.), so it's hard to answer this question with just
one or two words. I'm using different ideas to make connections across different
things! And I love it for that reason. I'll elaborate.

I'm in pure mathematics program, but some of the most exciting mathematics is,
to me personally, that which is inspired by cross-discipline communication. I am
most deeply moved by mathematics that is motivated by some phenomena in nature,
or in physics, or in an applied setting.

Amazingly, my thesis is in this very space! (Just three years ago, I didn't know
I'd be doing what I'm doing now, hence "amazingly." It's a very cool story, but
I'll save it for another day.)

MODELING SEQUENCES WITH QUANTUM STATES

October 20, 2019
•
Probability

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," and today I'd
like to tell you a little about it.

‍

WHAT IS AN ADJUNCTION? PART 3 (EXAMPLES)

September 26, 2019
•
Category Theory

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.

An adjunction, you'll recall, consists of a pair of functors F⊣GF⊣G between
categories CC and DD together with a bijection of sets, as below, for all
objects XX in CC and YY in DD.

In Part 2, we illustrated this bijection using a free-forgetful adjunction in
linear algebra as our guide. So let's put "free-forgetful adjuctions" first on
today's list of examples.

WHAT IS AN ADJUNCTION? PART 2 (DEFINITION)

September 24, 2019
•
Category Theory

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 FF and GG
together with natural transformations id→ GFid→ GF and FG→idFG→id. We compared
this to two stricter scenarios: one where the composite functors equal the
identities, and one where they are naturally isomorphic to the identities. The
first scenario defines an isomorphism of categories. The second defines an
equivalence of categories. An adjunction is third on the list.

In the case of an adjunction, we also ask that the natural
transformations—called the unit and counit—somewhat behave as inverses of each
other. This explains why the arrowsarrows point in opposite directions. (It also
explains the "co.") Except, they can't literally be inverses since they're not
composable: one involves morphisms in CC and the other involves morphisms in DD.
That is, their (co)domains don't match. But we can fix this by applying FF and
GG so that (a modified version of) the unit and counit can indeed be composed.
This brings us to the formal definition of an adjunction.

WHAT IS AN ADJUNCTION? PART 1 (MOTIVATION)

September 19, 2019
•
Category Theory

Some time ago, I started a "What is...?" series introducing the basics of
category theory:

* "What is a category?"
* "What is a functor?" Part 1 and Part 2
* "What is a natural transformation?" Part 1 and Part 2

Today, we'll add adjunctions to the list. An adjunction is a pair of functors
that interact in a particularly nice way. There's more to it, of course, so I'd
like to share some motivation first. And rather than squeezing the motivation,
the formal definition, and some examples into a single post, it will be good to
take our time: Today, the motivation. Next time, the formal definition.
Afterwards, I'll share examples.

> Indeed, I will make the admittedly provocative claim that adjointness is a
> concept of fundamental logical and mathematical importance that is not
> captured elsewhere in mathematics.

> - Steve Awodey (in Category Theory, Oxford Logic Guides)

A FIRST LOOK AT QUANTUM PROBABILITY, PART 2

July 23, 2019
•
Probability

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.

Let's begin with a brief recap of the ideas covered in Part 1: We began with a
joint probability distribution on a product of finite sets
p:X×Y→[0,1]p:X×Y→[0,1] and realized it as a matrix MM by setting
Mij=√p(xi),p(yj)Mij=p(xi),p(yj). We called elements of our set X={0,1}X={0,1}
prefixes and the elements of our set Y={00,11,01,10}Y={00,11,01,10} suffixes so
that X×YX×Y is the set of all bitstrings of length 3.

We then observed that the matrix M⊤MM⊤M contains the marginal probability
distribution of YY along its diagonal. Moreover its eigenvectors define
conditional probability distributions on YY. Likewise, MM⊤MM⊤ contains marginals
on XX along its diagonal, and its eigenvectors define conditional probability
distributions on XX.

The information in the eigenvectors of M⊤MM⊤M and MM⊤MM⊤ is precisely the
information that's destroyed when computing marginal probability in the usual
way. The big reveal last time was that the matrices M⊤MM⊤M and MM⊤MM⊤ are the
quantum versions of marginal probability distributions.

As we'll see today, the quantum version of a probability distribution is
something called a density operator. The quantum version of marginalizing
corresponds to "reducing" that operator to a subsystem. This reduction is a
construction in linear algebra called the partial trace. I'll start off by
explaining the partial trace. Then I'll introduce the basics of quantum
probability theory. At the end, we'll tie it all back to our bitstring example.

A FIRST LOOK AT QUANTUM PROBABILITY, PART 1

July 18, 2019
•
Probability

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.**

In today's post, we'll motivate the discussion by saying a few words about
(classical) probability. In particular, let's spend a few moments thinking about
the following: ‍

What do I mean? We'll start with some basic definitions. Then I'll share an
example that illustrates this idea.

A probability distribution (or simply, distribution) on a finite set XX is a
function p:X→[0,1]p:X→[0,1] satisfying ∑xp(x)=1∑xp(x)=1. I'll use the term joint
probability distribution to refer to a distribution on a Cartesian product of
finite sets, i.e. a function p:X×Y→[0,1]p:X×Y→[0,1] satisfying
∑(x,y)p(x,y)=1∑(x,y)p(x,y)=1. Every joint distribution defines a marginal
probability distribution on one of the sets by summing probabilities over the
other set. For instance, the marginal distribution pX:X→[0,1]pX:X→[0,1] on XX is
defined by pX(x)=∑yp(x,y)pX(x)=∑yp(x,y), in which the variable yy is summed, or
"integrated," out. It's this very process of summing or integrating out that
causes information to be lost. In other words, marginalizing loses information.
It doesn't remember what was summed away!

I'll illustrate this with a simple example. To do so, I need to give you some
finite sets XX and YY and a probability distribution on them.

MATRICES AS TENSOR NETWORK DIAGRAMS

May 15, 2019
•
Algebra

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×nm×n matrix MM with real entries
represents a linear map from Rn→RmRn→Rm. 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: To specify an m×nm×n matrix MM, one must specify all mnmn entries
MijMij. The index ii ranges from 1 to mm—the dimension of the output space—and
the index jj ranges from 1 to nn—the dimension of the input space. Said
differently, ii indexes the number of rows of MM and jj indexes the number of
its columns. These indices can be included in the picture, if we like:

This idea generalizes very easily. A matrix is a two-dimensional array of
numbers, while an nn-dimensional array of numbers is called a tensor of order nn
or an nn-tensor. Like a matrix, an nn-tensor can be represented by a node with
one edge for each dimension.

A number, for example, can be thought of as a zero-dimensional array, i.e. a
point. It is thus a 0-tensor, which can be drawn as a node with zero edges.
Likewise, a vector can be thought of as a one-dimensional array of numbers and
hence a 1-tensor. It's represented by a node with one edge. A matrix is a
two-dimensional array and hence 2-tensor. It's represented by a node with two
edges. A 3-tensor is a three-dimensional array and hence a node with three
edges, and so on.

VIEWING MATRICES & PROBABILITY AS GRAPHS

March 6, 2019
•
Probability

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:

‍

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.

LIMITS AND COLIMITS PART 3 (EXAMPLES)

January 21, 2019
•
Category Theory

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 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.

‍

CRUMBS!

January 9, 2019
•
crumbs!

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.

[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.

ANNOUNCING APPLIED CATEGORY THEORY 2019

January 5, 2019
•
Category Theory

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

‍

THE TENSOR PRODUCT, DEMYSTIFIED

November 18, 2018
•
Algebra

Previously on the blog, we've discussed a recurring theme throughout
mathematics: making new things from old things. Mathematicians do this all the
time:

* When you have two integers, you can find their greatest common divisor or
least common multiple.
* When you have some sets, you can form their Cartesian product or their union.
* When you have two groups, you can construct their direct sum or their free
product.
* When you have a topological space, you can look for a subspace or a quotient
space.
* When you have some vector spaces, you can ask for their direct sum or their
intersection.
* The list goes on!

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 vv and ww, we can build a new vector, called the tensor
product v⊗wv⊗w. But what is that vector, really? Likewise, given two vector
spaces VV and WW, we can build a new vector space, also called their tensor
product V⊗WV⊗W. But what is that vector space, really?

LEARNING HOW TO LEARN MATH

October 14, 2018
•
Other

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.

NOTES ON APPLIED CATEGORY THEORY

September 18, 2018
•
Category Theory

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?
2. Applied category theory? What's the hoopla? Hasn't category theory always
been applied?

For those thinking thought #1, I'd like to convince you the answer is No way!
It's true that category theory sometimes goes by the name of general abstract
nonsense, which might incline you to think that category theory is too
pie-in-the-sky to have any impact on the "real world." My hope is to convince
you that that's far from the truth.

For those thinking thought #2, yes, it's true that ideas and results from
category theory have found applications in computer science and quantum physics
(not to mention pure mathematics itself), but these are not the only
applications to which the word applied in applied category theory is being
applied.

So what is applied category theory?

IS THE SQUARE A SECURE POLYGON?

May 17, 2018
•
Topology

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?

CRUMBS!

February 7, 2018
•
crumbs!

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?

LIMITS AND COLIMITS, PART 2 (DEFINITIONS)

January 30, 2018
•
Category Theory

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.

BROUWER'S FIXED POINT THEOREM (PROOF)

January 18, 2018
•
Topology

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

A DIAGRAM IS A FUNCTOR

January 10, 2018
•
Category Theory

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

Answer: a diagram.

LIMITS AND COLIMITS, PART 1 (INTRODUCTION)

January 2, 2018
•
Category Theory

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.

TOPOLOGY VS. "A TOPOLOGY" (CONT.)

December 21, 2017
•
Topology

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.

MULTIPLYING NON-NUMBERS

November 27, 2017
•
Geometry

In last last week's episode of PBS Infinite Series, we talked about different
flavors of multiplication (like associativity and commutativity) to think about
when multiplying things that aren't numbers. My examples of multiplying
non-numbers were vectors and matrices, which come from the land of
algebra. Today I'd like to highlight another example: We can multiply shapes!

MATH3MA + PBS INFINITE SERIES!

November 23, 2017
•
Other

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

WHAT IS AN OPERAD? PART 2

October 30, 2017
•
Algebra

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!

WHAT IS AN OPERAD? PART 1

October 23, 2017
•
Algebra

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.

MATH EMOJIS

October 2, 2017
•
Other

☺️ I love math

😀 It's so cool.

THE YONEDA LEMMA

September 14, 2017
•
Category Theory

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...

THE YONEDA EMBEDDING

September 6, 2017
•
Category Theory

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

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

i.e.

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

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

THE YONEDA PERSPECTIVE

August 30, 2017
•
Category Theory

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

And what are they referring to?

The Yoneda lemma.

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

DEAR AUTOCORRECT... (SINCERELY, MATHEMATICIAN)

August 21, 2017
•
Other

Dear Autocorrect,

No.

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

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

NAMING FUNCTORS

July 31, 2017
•
Category Theory

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

COMMUTATIVE DIAGRAMS EXPLAINED

July 5, 2017
•
Other

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.

CRUMBS!

June 12, 2017
•
crumbs!

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

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

SOME NOTES ON TAKING NOTES

May 16, 2017
•
Other

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

I think on paper.

"ONE-LINE" PROOF: FUNDAMENTAL GROUP OF THE CIRCLE

April 24, 2017
•
Topology

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

CRUMBS!

March 13, 2017
•
crumbs!

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....

CRUMBS!

February 22, 2017
•
crumbs!

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?)

GROUP ELEMENTS, CATEGORICALLY

February 16, 2017
•
Category Theory

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

WHAT IS A NATURAL TRANSFORMATION? DEFINITION AND EXAMPLES, PART 2

February 13, 2017
•
Category Theory

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).

WHAT IS A NATURAL TRANSFORMATION? DEFINITION AND EXAMPLES

February 7, 2017
•
Category Theory

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?

CRUMBS!

February 4, 2017
•
crumbs!

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

WHAT IS A FUNCTOR? DEFINITIONS AND EXAMPLES, PART 2

February 2, 2017
•
Category Theory

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

WHAT IS A FUNCTOR? DEFINITION AND EXAMPLES, PART 1

January 31, 2017
•
Category Theory

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

INTRODUCING... CRUMBS!

January 26, 2017
•
crumbs!

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.

WHAT IS A CATEGORY? DEFINITION AND EXAMPLES

January 23, 2017
•
Category Theory

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 CC consists of some data that satisfy certain properties...

WHAT IS CATEGORY THEORY ANYWAY?

January 17, 2017
•
Category Theory

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

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

#TRUSTYOURSTRUGGLE

January 3, 2017
•
Other

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

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

A QUOTIENT OF THE GENERAL LINEAR GROUP, INTUITIVELY

December 15, 2016
•
Algebra

Over the past few weeks, we've been chatting about quotient groups in hopes of
answering the question, "What's a quotient group, really?" In short, we noted
that the quotient of a group GG by a normal subgroup NN 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 NN. The key point was that there's only one way
to belong to NN, but generally there may be several ways to fail to belong.

A GROUP AND ITS CENTER, INTUITIVELY

December 6, 2016
•
Algebra

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.

THE FIRST ISOMORPHISM THEOREM, INTUITIVELY

November 28, 2016
•
Algebra

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

WHAT'S A QUOTIENT GROUP, REALLY? PART 2

November 22, 2016
•
Algebra

Today we're resuming our informal chat on quotient groups. Previously we said
that belonging to a (normal, say) subgroup NN of a group GG just means you
satisfy some property. For example, 5Z⊂Z5Z⊂Z means "You belong to 5Z5Z if and
only if you're divisible by 5". And the process of "taking the quotient" is the
simple observation that every element in GG either

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

and noting that...

WHAT'S A QUOTIENT GROUP, REALLY? PART 1

October 17, 2016
•
Algebra

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

THE SIERPINSKI SPACE AND ITS SPECIAL PROPERTY

October 6, 2016
•
Topology

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

THE MOST OBVIOUS SECRET IN MATHEMATICS

September 12, 2016
•
Category Theory

Yes, I agree. The title for this post is a little pretentious. It's certainly
possible that there are other mathematical secrets that are more obvious than
this one, but hey, I got your attention, right? Good. Because I'd like to tell
you about an overarching theme in mathematics - a mathematical mantra, if you
will. A technique that mathematicians use all the time to, well, do math.

COMPARING TOPOLOGIES

August 26, 2016
•
The Back Pocket

It's possible that a set XX 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 τ1τ1 is said to be coarser than another topology
τ2τ2, and we write τ1⊆τ2τ1⊆τ2, if every open set in τ1τ1 is also an open set in
τ2τ2. In this scenario, we also say τ2τ2 is finer than τ1τ1. 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?

RESOURCES FOR INTRO-LEVEL GRADUATE COURSES

August 22, 2016
•
Other

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.

A RAMBLE ABOUT QUALIFYING EXAMS

August 1, 2016
•
Other

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

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

AUTOMORPHISMS OF THE RIEMANN SPHERE

July 11, 2016
•
Analysis

This is the last in a four-part series in which we prove that the automorphisms
of the unit disc, upper half plane, complex plane, and Riemann sphere each take
on a different form. Today our focus is on the Riemann sphere.

AUTOMORPHISMS OF THE COMPLEX PLANE

June 27, 2016
•
Analysis

This is part three of a four-part series in which we prove that the
automorphisms of the unit disc, upper half plane, complex plane, and Riemann
sphere each take on a different form. Today our focus is on the complex plane.

AUTOMORPHISMS OF THE UPPER HALF PLANE

June 15, 2016
•
Analysis

This is part two of a four-part series in which we prove that the automorphisms
of the unit disc, upper half plane, complex plane, and Riemann sphere each take
on a different form. Today our focus is on the upper half plane.

AUTOMORPHISMS OF THE UNIT DISC

June 6, 2016
•
Analysis

This is part one of a four-part series in which we prove that the automorphisms
of the unit disc, upper half plane, complex plane, and Riemann sphere each take
on a different form. Today our focus is on the unit disc.

THREE IMPORTANT RIEMANN SURFACES

May 30, 2016
•
Analysis

In this post we ramble on about Riemann surfaces, the uniformization theorem,
universal covers, and two secret (or not-so-secret!) techniques that
mathematicians use to study a given space. Our intent is to provide motivation
for an upcoming mini-series on the automorphisms of the unit disc, upper half
plane, complex plane, and Riemann sphere.

ENGLISH IS NOT COMMUTATIVE

May 17, 2016
•
The Back Pocket

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

Word order is important...

GOOD READS: THE PRINCETON COMPANION TO MATHEMATICS

May 9, 2016
•
Good Reads

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

CLEVER HOMOTOPY EQUIVALENCES

April 18, 2016
•
Topology

You know the routine. You come across a topological space XX and you need to
find its fundamental group. Unfortunately, XX is an unfamiliar space and it's
too difficult to look at explicit loops and relations. So what do you do? You
look for another space YY that is homotopy equivalent to XX and whose
fundamental group π1(Y)π1(Y) is much easier to compute. And voila! Since XX and
YY are homotopy equivalent, you know π1(X)π1(X) is isomorphic to π1(Y)π1(Y).
Mission accomplished.

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

SNIPPETS OF MATHEMATICAL CANDOR

April 4, 2016
•
Other

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.

(CO)HOMOLOGY: A POEM

March 28, 2016
•
Topology

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

CLASSIFYING SURFACES (CLIFFSNOTES VERSION)

March 16, 2016
•
Topology

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:

GRADUATE SCHOOL: WHERE GRADES DON'T MATTER

March 9, 2016
•
Other

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

GOOD READS: REAL ANALYSIS BY N. L. CAROTHERS

March 2, 2016
•
Good Reads

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

TOPOLOGICAL MAGIC: INFINITELY MANY PRIMES

February 22, 2016
•
Topology

A while ago, I wrote about the importance of open sets in topology and how the
properties of a topological space XX are highly dependent on these special sets.
In that post, we discovered that the real line RR can either be compact or
non-compact, depending on which topological glasses we choose to view RR with.
Today, I’d like to show you another such example - one which has a surprising
consequence!

THE PSEUDO-HYPERBOLIC METRIC AND LINDELÖF'S INEQUALITY (CONT.)

February 18, 2016
•
Analysis

Last time we proved that the pseudo-hyperbolic metric on the unit disc in ℂ is
indeed a metric. In today’s post, we use this fact to verify Lindelöf’s
inequality which says, "Hey! Want to apply Schwarz's Lemma but don't know if
your function fixes the origin? Here's what you do know...."

THE PSEUDO-HYPERBOLIC METRIC AND LINDELÖF'S INEQUALITY

February 17, 2016
•
Analysis

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

GOOD READS: LOVE AND MATH

February 11, 2016
•
Good Reads

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

THE FUNDAMENTAL GROUP OF THE REAL PROJECTIVE PLANE

February 1, 2016
•
Topology

The goal of today's post is to prove that the fundamental group of the real
projective plane, is isomorphic to Z/2ZZ/2Z 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....

ABSOLUTE CONTINUITY (PART TWO)

January 21, 2016
•
Analysis

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.

ABSOLUTE CONTINUITY (PART ONE)

January 11, 2016
•
Analysis

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

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