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
🚨🚨 #MathTwitter Announcement! 🚨🚨 As I'm going through and finalizing chapters in my #VisualAlgebra book, I'm recording a new set of YouTube videos in tandem. This will be more than twice as long as my existing #VisualGroupTheory playlist, with MUCH more content. 1/3 👇
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Obligatory retweet
The Seahawks have literally never played in a normal game.
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What's the biggest heartbreak you've ever experienced?
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I've accepted that tonight is going to completely break me emotionally, or it'll be the best sports night of my life with nothing in between. I'm terrified.
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Seven hours away and I’m already an emotional wreck 😭😭😭😭
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Doctor: “do you have any pre-existing heart conditions?” Me: “yes, the Mariners are currently in the playoffs.”
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SOMEWHERE IN HEAVEN, A YOUNG DAVE NIEHAUS IS SMILING 🥹 THE @MARINERS ARE ONE GAME AWAY FROM THE WORLD SERIES! I DON'T BELIEVE IT! MY, OH MY! 🔱 #MLBPlayoffs
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Jorge Polanco with the 15th walk-off hit in a winner-take-all game in postseason history 17th walk-off win in a winner-take-all game overall INCREDIBLY … the only other ALDS winner-take-all walk-off win? Edgar’s double BASEBALL!!!
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I’m celebrating the 4th of July this year by working all day in the office and drinking tea with a cricket match in the background. ‘Merica! 🇺🇸🇺🇸🇺🇸
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Tucker Carlson is obsessed with defending Russia and the KGB thug that runs it.
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Even if you don't care one bit about scientific research, it's important to recognise the value of publicly funding people working on very hard problems and training bright young minds on how to solve them. Do you think it’s a coincidence that so many successful entrepreneurs
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The Vatican has the chance to do the funniest thing...
I was excited to hear that President Trump is open to the idea of being the next Pope. This would truly be a dark horse candidate, but I would ask the papal conclave and Catholic faithful to keep an open mind about this possibility! The first Pope-U.S. President combination has
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One more thing! You can find the entire playlist (currently have 45 lectures, am planning about 100) at the following webpage. I have the slides posted as well. https://t.co/EycH0QzR5Q 17/16
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Here's a summary of our 4 examples actions and the fundamental features. Please share, I think this will be helpful to algebra students and instructors alike! Lecture 5.4: Examples of actions https://t.co/jlzy50DGtI 16/16
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**Don't let someone tell you that you can only take the quotient by a subgroup if it's normal!** You can ALWAYS quotient by the right cosets and get a G-set. If it's normal, that G-set happens to be a group. [In Ch 3, we constructed G/H by collapsing by *left* cosets.] 15/16
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Our last action is new, but arguably the most important: G acting on cosets of some subgroup H≤G. Also, *every* transitive G-set is isomorphic to such a G-set! This is constructed by collapsing the *right* cosets in a Cayley graph. 14/16
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One of my favorite examples is A₄, and it's something we've seen since Chapter 3. These last 2 pictures are from an old lecture, way back when we introduced normal subgroups: Lecture 3.4: Normalizers and normal subgroups https://t.co/CbT1y8dBen 13/16
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Here's the action of G on its subgroups by conjugation. Once again, we'll characterize our "five fundamental features", and revisit our two theorems on orbits. It's helpful to superimpose the action graph on the subgroup lattice. 12/16
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Here's the action of G on its elements by conjugation. It's worth characterizing our "five fundamental features": 1. orbits 2. stabilizers 3. fixators 4. kernel 5. fixed point set 11/16
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Next, we'll see 4 examples: a group G acting on... 1. Its elements by multiplication 2. Its elements by conjugation 3. Its subgroups by conjugation 4. Its cosets by multiplication Here's the first one, and a 1-line proof of Cayley's theorem. 10/16
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The orbit-counting theorem says the avg # of checkmarks per row (i.e., the avg size of a fixator) is the number of orbits. It's worth comparing this to our running example of D₄ acting on binary squares. Lecture 5.3: Two theorems on orbits https://t.co/Ir4BAORAPB 9/16
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