Matt Macauley
@VisualAlgebra
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Associate Professor (Clemson) | AIMS Lecturer (South Africa) | Author: "Visual Algebra" (forthcoming) | YouTuber | First Gen | Homesteader | Dad to Ida & Felix
Clemson, SC
Joined July 2009
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Pinned Tweet
#Week1
of
#GraduateVisualAlgebra
in the books! As promised in my ugrad
#VisualAlgebra
meta-thread (), I'll post summaries for each day -- LOTS of never-before-seen materials!
We'll return to Felix & his topological 🍼 later in the class.
1/8 Wed 🧵👇
Now that my
#VisualAlgebra
class is in the books, I want to post a long "meta thread" of all 16 weekly threads, with daily summaries. Here's my entire class, including lectures, HW, & exams, in one convenient place.
And stayed tuned for some surprise announcements below!👇🧵
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The look my wife game me when I immediately shouted “epsilon”, without really thinking it through…
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@CPierre67
Exactly, it’s not a math question at all. It’s actually a disadvantage (misleading) if you know some math.
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@Smackledorf
I actually use it a lot for a maps that are supposed to be injective, especially in a commutative diagram or universal property.
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In today's Abstract Algebra class, I introduced automorphisms and semidirect products, despite us not having defined a homomorphism yet. Students should see the intuition behind the concepts, the motivation for them, and the inherent BEAUTY that is too often missed.
#MathTwitter
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Here is the full-color preprint of my forthcoming paper "Dihedralizing the quaternions", which will appear in the Amer. Math. Monthly in early 2024. A fun read, especially for undergrad algebra students.
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Now that my
#VisualAlgebra
class is in the books, I want to post a long "meta thread" of all 16 weekly threads, with daily summaries. Here's my entire class, including lectures, HW, & exams, in one convenient place.
And stayed tuned for some surprise announcements below!👇🧵
6
76
493
Last week I did the Sylow theorems in class, and I want to share how I do them with my visual approach to groups.
To start, here are the 5 groups of order 12. Note how there are "towers of p-groups", for p=2 and p=3.
This is what the 1st Sylow theorem guarantees.
1/17
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I want to share some fun unique problems from my Graduate
#VisualAlgebra
Final Exam yesterday.
This first one involves quotients of the Gaussian integers. Note how one quotient has order 10, and the other has order 13.
What does this tell us about irreducibility?
1/10
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I woke up a few days ago with the sobering realization: actually, I do NOT really understand groups actions.
Spoiler: I do now, but it took some work. And now I realize how incomplete my understanding was. 😳
Let me explain, I think some of you might enjoy this!
1/12 🧵👇
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Caught a glimpse of my daughter's CS homework tonight and came to the realization that I wasted a LOT of time learning this stuff years ago. I have no idea what I’m looking at here. Really, what is the purpose of this…whatever it is? Anyone use this?
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Today was the 1st day of my Spring 2022 Abstract Algebra class! This semester, I'll be tweeting class summaries & visual highlights, every MWF in weekly threads, and fun creative HW exercises. Feel free to engage! Can anybody think of a good hashtag? (Also need a book title.)
I wish I had taken Abstract Algebra from
@mattmacauley
when I was in college. Here’s how he teaches it: . In contrast, when I asked my teacher what I should think about when I think about groups, he said, “Think about the axioms.”
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What does it really mean for a group to be "nilpotent"?
This year, I've asked many people to describe it in simple, memorable terms, and have yet to get a good answer.
Usually: something something about an ascending series. But what exactly, and WHY? Let's dig in! 🧵👇
1/17
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We began
#Week10
of
#GraduateVisualAlgebra
with functors. Here's a motivating example from topology.
Yes, technically this should be Top_•, but it's not worth nitpicking like that at this point.
1/7 Mon 🧵👇
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Hey
#MathTwitter
, let's talk (group) Isomorphism Theorems, which I did in class yesterday.
I want to CONVINCE you that they're always in the wrong order, and there should be (gasp!) 5 of them.
I'll end this thread with a question for all of you. Let's get started! 🧵👇
1/12
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Teaching abstract algebra w/o pictures feels as wrong as teaching calculus w/o graphs, chemistry w/o diagrams, or crystallography w/o drawing shapes. Yet it's almost universally done. In 10 years, I want that approach to be obsolete from undergraduate classrooms.
@mathillustrated
@stevenstrogatz
@mattmacauley
Wait...you can use PICTURES with abstract algebra??? I still have "Topics in Algebra" by Herstein from my college Abstract Algebra course. (This was the course that made me decide to major in history instead of math.) Not a single picture (or number) in it!
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Over the years, I've heard many students and faculty (!) say things like "I never really fully understood group actions". Given how they're treated in nearly all books, this isn't surprising.
I just started this topic in class. Here's how to get EVERYONE on board. 🧵👇
1/14
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@reedgalen
@ProjectLincoln
@realDonaldTrump
They’re in the 1st inning,
@aoc
. It’s a movement and they’re worth every penny.
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During a recent deep house clean, I found a folder, containing a forgotten 20-year-old (well-deserved) rejection letter from an Ivy League PhD program, signed by their grad-director, a name I didn’t know at the time.
I just finished refereeing one of his papers. 😂
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@EmiDoesMath
I usually Google a bunch of papers, open them up in about 20 tabs, leave them there for at least 4 months, certain that I’ll eventually read them, all the while terrified that if I close one, I’ll never find it again.
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Happy Monday! It's Week 3 of
#VisualAlgebra
, and only today did we see the official definition of a group.
Last week, we classified all 7 frieze groups. We began today with all 17 wallpaper groups (w/o proof).
1/11 Mon
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A beautiful theme across mathematics is "conjugation preserves structure".
How do we impart this knowledge in a memorable way that students will truly understand?
My attempt, after introducing conjugacy classes yesterday. Feel free to use/adapt in your own class!
#MathTwitter
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@UsmanBaghi
The ship stuck in the Suez Canal is blocking so much Important Stuff, that they‘ve decided to Rename it
‘THE McCONNELL’
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Raise your hand if you struggled with group actions as a student. 🙋♂️<-- that's my hand!
I genuinely wonder how differently things would have gone, for so many, if the following simple picture had been given right after the def'n. Just finished making this for tomorrow's class.
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This is SO COOL! 57 pages of
#VisualAlgebra
written by a 4-time Putnam Fellow and Stanford Professor. I’m telling you, visual learning is the way of the future.
I can’t wait to dive into this in more detail later.
This, by Ravi Vakil, on depicting homological algebra with puzzle pieces, is insanely cool:
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We're 4 weeks into my Abstract Algebra class, but have only done 3 proofs in lecture. However, I suspect that my students have a holistic understanding of groups, in ways that most don't.
#MathTwitter
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We started
#Week13
of
#VisualAlgebra
with the 3rd Sylow theorem, and then finished Chapter 5 (group actions) with a classic musical number of
@matthematician
.
Here's a nice example illustrating the "towers of p-subgroups" guaranteed by the Sylow theorems.
1/9 Mon 🧵👇
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Question for
#MathTwitter
: What are your favorite examples of infinite groups for an undergraduate abstract algebra class? Specifically looking for things that illustrate key concepts well.
I have a few favorites, but feel like I'm missing some really good ones. Thank you!
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We started
#Week6
of
#GraduateVisualAlgebra
with the three Sylow theorems, which tell us:
1. p-subgroups come in "towers"
2. there's a unique tower for each p, and the "tops" are conjugate.
3. the size of the top of the tower is 1 mod p.
Picture is 1000 words.
1/11 Mon 🧵👇
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@3blue1brown
Yes. Almost all finite groups are 2-groups. In fact, 99.2% of all groups of order < 2000 have order exactly 1024.
I'm writing a book related to this that has hundreds of visuals. I think you'd really like it,
@3blue1brown
, I'd be happy to email you a current draft.
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I really like how this
#VisualAlgebra
exam turned out. Here's my favorite problem, which shows the subgroup lattice of a finite simple group of order 2,588,772, and its 1,058,074 subgroups.
The full exam is here:
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How do you motivate the isomorphism theorem that says (G/N)/(H/N)≅H/N?
This weekend, I came up with a new (to me) way to visualize it that I really like. Stayed up way too late last night making this 1st image, that I'm gonna test-drive in
#VisualAlgebra
class on Wednesday.
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@MitchellLerner
This is incorrect, and it’s even worse — the $3.4 billion endowment is from June 2019.
According to Bloomberg, their endowment gained 41% last year.
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Just finished Chapter 7 of
#VisualAlgebra
this evening (or a very rough draft of it). Imported it back into the main doc, and hit compile...553 pages!
I debating stopping here, but I think I'll write one more chapter, on Galois Theory. Too much rich material to pass up!
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@ZXORA2012
@amandacarpenter
He might have just done that...
NEW: Testimonial ad from Trump's Former DHS Chief of Staff
@MilesTaylorUSA
, declaring his support for Joe Biden and describing Trump's presidency as "terrifying" and "actively doing damage to our security."
WATCH & go to for more.
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@DynamicsSIAM
@stevenstrogatz
There is none. I'm writing my own. About 450 pages through, should be done by Summer 2022. My goal if for this to change how algebra is taught -- every future book will have some visual component. Students in 20 yrs will be amazing that it used be taught non-visually.
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It is a travesty that one of the most beautiful subjects in all of undergrad mathematics is routinely taught in such a dry and unmotivated way that so many students find it incredibly boring (for good reason!) This is way too common. We need to do better as educators. No excuses.
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One week ago today, this little guy arrived, weighing 6 lbs 5 oz, and making our family a group of order 4.
Welcome to the world, Felix Coen Macauley. We’re so excited you’re here.
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@ProjectLincoln
@DonaldJTrumpJr
Wait, his gripe is that DEMOCRATS are trying to sneak in tax breaks for millionaires?!?!? 🤣😂🤣😂
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On Monday, I introduced automorphisms in class (see ). Yesterday, we did inner and outer automorphisms. These visuals really capture the essence of these concepts.
So, what is HW like in an algebra class taught this way?
I'm glad you asked! 😀 1/4
I introduced automorphisms today. Starting with cyclic groups:
σ_2 = "doubling map" (r-arrow goes to 2 r-arrows)
σ_3 = "tripling map"
σ_2 * σ_3 = "the sextupling map"
On Wednesday, it REALLY gets fun -- inner vs. outer. Stay tuned!
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@drinkderiver
Is this a “wrong answers only” thread? If so, can I plug my own paper?
This article by Matthew Macauley and Nora Youngs highlights many interesting applications of applied algebraic geometry to biology (genomics, neuroscience, systems biology, ...)
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Today was
#Week16
-- the
#VisualAlgebra
#FinalExam
! As promised, I'll post the problems here, with commentary.
I spent a whole day writing it, and REALLY like how it turned out!
Here's my favorite problem. What sort of things do YOU think I asked about A_6?
1/10 Final 🧵👇
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We finally finished up "Chapter 2: Examples of Groups" today, in
#Week6
of
#VisualAlgebra
. We also held a fun CONTEST, which I'll describe below!
Last week, I showed the following 3 Cayley diagrams, and asked: which one is *not* a (nontrivial) direct product?
1/14 Mon 🧵👇
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@BillKristol
LOL. Looks like you didn’t read the whole article, Bill. It ends with author saying the following about you: “there is nothing Republican or conservative, or frankly, decent, about you.”
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@BillKristol
Stopped reading when I got to: "Wood voted for Trump in 2020..."
Sit down! Next?
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Feel-good anecdote: many of you remember the story of Christopher Jackson, who reached out to
@mathyawp
from about learning math from federal prison, and helped inspire
#MathForHumanFlourishing
.
Francis sent me an email, saying that Christopher...
1/2
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...is learning the Sylow theorems (!) and recently said:
"I wish there were a way to visualize what was going on..."
So, en route to Florida, is an (almost) camera-ready copy of the first 5 chapters of
#VisualAlgebra
-- 353 pages with 356 figures.
He's going to LOVE this.
2/2
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Maybe this will get me to finally learn spectral sequences once and for all.
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@cunat_n
@madrid_mike
I can't like this tweet enough, Marc. I would say the exact same thing about my father. Also a WWII vet (1st Marine Division, Guadalcanal), and who died a few weeks before yours. Thank you for sharing.
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@matthematician
Let X & Y be finite dim'l vector spaces.
X ≅ Span(1, x, x^2, ... , x^{n-1})
Y ≅ Span(1, y, y^2, ... , y^{m-1})
Then X⊗Y ≅ Span(x^i*y^j : i<n, j<m}
The "bilinear" part here just means
(3x^i) * y^j = x^i * (3y^j) = 3(x^i * y^j).
I usually think of tensors as polynomials.
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Last week in
#VisualAlgebra
we analyzed conjugacy classes of subgroups. We'll spend much of
#Week8
looking at conjugacy classes of elements.
Big idea throughout mathematics "conjugacy preserves structure".
One of my favorite examples of that is this frieze group.
1/3 Mon 🧵👇
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@amazingmap
Michigan is farther East than most people recognize. My favorite variant of this is:
"The entirety of continental South America lies east of Disney World."
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We started
#Week9
of
#GraduateVisualAlgebra
with what it means for a map to "factor through" another, and when existence implies uniqueness. This topic is often glossed over.
We'll also get a taste of categories, and even see a surprise ε>0 this week (gasp!).
1/10 Mon 🧵👇
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@MeghanMcCain
Don't overthink this, there are PLENTY of other things to grill your husband for. 😀
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We started
#Week4
of
#GraduateVisualAlgebra
with the (last 3) isomorphism theorems.
Normally, these are ordered as arbitrarily as gifts from last days of the "12 Days of Christmas."
And my ordering is different, but for good reason! Here's what should come next.
1/13 Mon 🧵👇
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This is Google’s coolest Easter egg ever! By the Brouwer fixed point theorem, there’s a unique point on this map directly above where it corresponds to in MN.
Whoever put this entry into Google Maps cleverly placed the pin in that exact point! 😂
1/2
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We began
#Week2
of
#GraduateVisualAlgebra
with direct products, and the first proof of the class: Z_n x Z_m is cyclic iff gcd(n,m)=1. Am I supposed to wait until Day 3 of a graduate algebra class to do a proof?? 🤔
1/9 Mon 🧵👇
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We began
#Week11
of
#GraudateVisualAlgebra
by formalizing group presentations.
Big idea: adding relations corresponds to taking quotients, i.e., collapsing by cosets.
Here's a picture of this for D_4. Each node (left) represents a conjugacy class of subgroups.
1/10 Mon 🧵👇
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Dropping on the arXiv tomorrow! Come read about snakes on a plane, an ouroboros on a torus, scales, slithers & co-slithers, and much more! Can you find other puns and hidden surprises?
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We started
#Week11
of
#VisualAlgebra
with a new diagram of one of the isomorphism theorems. I made this over spring break. The concept is due to Douglas Hofstadter (author of "Gödel, Escher, Bach"), who calls this a "pizza diagram".
1/14 Mon 🧵👇
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Buried in any algebra book is a tidbit: "the number of conjugate subgroups is the index of the normalizer", or worse, as "|xHx^{-1}|=[G:N_G(H)]"...but what does that really mean? My favorite example beautifully illustrates it. From class today, enjoy!
#MathTwitter
1/3
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@ass_deans
So he decided not to return to the faculty, teach 1 class a semester, and keep his $400k/year salary?
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From class today: my favorite example illustrating the 1st isomorphism theorem.
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@Lgirl47F
@WalshFreedom
@glangendorf01
OMG, the headlines at the bottom are hilarious, almost missed em!
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So why is this? People teach subjects the way they learned them. There's a huge void in the literature that approaches materials in such a memorable way.
Let's change things, and make math more inclusive and accessible! For more materials, see:
17/17
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Posting my weekly
#VisualGroupTheory
puzzle for
#MathTwitter
one day late this week. If you construct the left Cayley diagram of a group, and collapse the left cosets of a non-normal subgroup, then the result is not a group. This is shown on the left, for the subgroup...
1/3
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We started
#Week9
of
#VisualAlgebra
with conjugacy classes in symmetric groups.
Two permutations are conjugate in S_n iff they have the same cycle type. In this example, people are allowed to rotate seats, or those in chairs 2 & 3 can swap seats.
1/6 Mon
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Last week we introduced central ascents and descents. We can visualize all of them conveniently with what I like to call the "chutes and ladders diagram", which is how we started
#Week8
of
#GraduateVisualAlgebra
.
Stick around for an important 🇬🇧 poll below!
1/6 Mon 🧵👇
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@TaylerSeries
Linear Algebra is that strange topic where most schools don’t offer a 2nd semester course on, most grad programs don’t offer an introductory course on, yet it’s expected that everyone know it well.
Gil Strang’s MIT video lectures are phenomenal, imho.
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We started
#Week3
of
#GraduateVisualAlgebra
with a proof that subgroups of cyclic groups are cyclic, and an example of the subgroup lattice of Z_{24} w/ the divisor lattice of 24.
But that's not very pretty, so here's the next thing we did: cosets!
1/12 Mon 🧵☝️👇
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Excited to announce the arrival of our daughter, Ida Maeve, born Tues Nov 20, at 12:03pm. Her and
@CatherineGurri
(Mom) are doing well. Future member of the
@ClemsonUniv
Class of (20)39?
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I'm trying to finish up a paper on a short proof of the Riemann Hypothesis, but I'm hitting a wall. Can you help? How much would you charge for this? Thank you! 🙏
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Just submitted an arXiv paper to get the valuable 2/22/22 timestamp!
My growing collection also includes 7/7/07, 8/8/08, 9/9/09, 10/10/10, 11/11/11, the last 4 leap days, my anniversary, daughter's birthday, and Easter Eggs (hidden in the pdf abstract!) on 10/21/15 & 11/8/16.
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WEEK 1: first lecture of
#VisualAlgebra
.
"The mathematician's patterns, like the painter's or the poet's must be beautiful; the ideas like the colours or the words, must fit together in a harmonious way."
― G.H. Hardy, A Mathematician's Apology
1/4
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Just spent almost 2.5 hrs with ~8 students at Fri night Zoom “Happy Hour” (=office hrs). HW isn’t due until Monday night, but we had such rich discussions about symmetries, polytopes, Cayley diagrams, and more Beautiful Mathematics. I love seeing their genuine interest and joy!
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This book has 2 of my favorite things: visual mathematics, and novel applications of "traditional pure math topics."
1/ in 2009, i began writing a book on topology, meant to be a short introduction to the core concepts, in the context of lots of interesting applications. every idea would be paired with one or more uses, as much outside of mathematics as possible: “applied topology”.
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Happy New Year
#MathTwitter
! Let's start 2022 w/ Part 1 of a fun series: "Groups you Never Knew Existed...and others you can't POSSIBLY live without!"
Today we'll see the "diquaternions", a term you've never heard of b/c I made it up last month. Let's dig in! 🧵👇
1/17
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The opportunity doesn't arise all that often, but it's fun to spend most of class proving a big theorem, and then constructing an explicit counterexample to said theorem in the last 2 minutes, and calling it a day.
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I took this photo out my dining room window at 12:30am. Full moon directly overhead.
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@azurelunatic
@kristysblake
@AshToTheFuture
I broke the news to my family on April 1st that my wife and I were expecting our first child, and then let them eventually figure out on their own that it wasn't a joke. My daughter is 14 months old now.
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For anyone interested in learning more about this, I'm giving a virtual talk at
#WhatIsASeminar
, Thurs Dec 2 at 10am EST.
This will be very different from the
#TMWYF
talk I gave in Oct. Much more Cayley diagram focused, w/ new visuals! Hope to see you!
I can’t wait to see the Abstract Algebra textbook that
@mattmacauley
eventually publishes, based on this remarkably well-motivated course he is teaching. Have a look at the course page below.
#wow
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Shoutout to
@MotivicKyle
for highlighting the following under-emphasized fact of group actions.
Thm: The equivariant automorphisms of a transitive G-set w/ stabilizers cl(H) forms a group, isomorphic to N_G(H) / H.
Huh? Let's go explore this!!
1/9 🧵👇
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Evening
#VisualAlgebra
office hours on Monday lasted over 2 hours, and at one point, 15 students were on simultaneously. Tuesday evening lasted over 3 hours, and with up to 14 at a time. Almost half the class! The level of engagement is UNREAL — I love it! Here’s what we work on.
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We started
#Week10
of
#VisualAlgebra
with the correspondence theorem, using the familiar dicyclic group G=Dic_6 as an example.
We've seen that G/<r^3>≅D_3. And we're very familiar with the subgroups of D_3. Thus, we know all subgroups of the quotient.
Mon 1/7 🧵👇
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