Top Tweets for #GraduateVisualAlgebra
I will also record a #GraduateVisualAlgebra playlist, for people who have already taken abstract algebra. In that, I will zip through the earlier content.
I have a makeshift webpage where I'll be posting all of the slides & videos.
https://t.co/aKqsWgkeWP
Please share!
3/3
Fall break meant that #Week12 of #GraduateVisualAlgebra was cut a little short (no class Monday). Just a reminder about how we color-code the subring lattices that you'll see throughout this thread.
Also: <r> is the *subgroup* it generated by r; (r) is the ideal.
1/10 Wed π§΅π
@GaryPython Yes! I post all slides, HW, and exams.
#GraduateVisualAlgebra:
https://t.co/f0ifP8hC3o
#GraduateVisualAlgebra:
https://t.co/4ID3MMAShf
We began #Week10 of #GraduateVisualAlgebra with functors. Here's a motivating example from topology.
Yes, technically this should be Top_β’, but it's not worth nitpicking like that at this point.
1/7 Mon π§΅π
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 Ξ΅>0 this week (gasp!).
1/10 Mon π§΅π
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 π¬π§ poll below!
1/6 Mon π§΅π
We began #Week7 of #GraduateVisualAlgebra with Chapter 6: "extensions and universal constructions." I wrote this chapter over the summer, and unlike what I've posted during the first 6 weeks, most of this contains never-before-seen visuals. Let's dive in!
Mon 1/10 π§΅π
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 π§΅π
We started #Week5 of #GraduateVisualAlgebra with group actions. Imagine a "group switchboard". Every element has a button, that permutes a set S. The only rule is:
"pressing the a-button, then the b-button, is the same as pressing the (ab)-button": Ο(a)Ο(b)=Ο(ab).
1/13 Mon π§΅π
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 π§΅π
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 π§΅βοΈπ
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?? π€
1/9 Mon π§΅π
#Week1 of #GraduateVisualAlgebra in the books! As promised in my ugrad #VisualAlgebra meta-thread (https://t.co/ymz8DObloI), I'll post summaries for each day -- LOTS of never-before-seen materials!
We'll return to Felix & his topological πΌ later in the class.
1/8 Wed π§΅π
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!ππ§΅
I'll teach #GraduateVisualAlgebra in #Fall2022! Yes, our 1st semester grad algebra course can be visual. I'm not 100% sure how it'll go, but LOTS of basic proofs will be left as exercises (show __ is a subgroup, etc). And, I will be tweeting this course as well. Stay tuned!
3/6
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