Invariants: How Mathematicians Distinguish Between Objects

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Mathematical Invariants are properties of class of mathematical objects that don't change under various transformations. Studying invariants are one of the big themes of mathematics. In this video we will see three examples of invariants, one from graph theory called the chromatic number, one from geometry called the Euler Characteristic, and one from Linear Algebra called the Determinant (geometrically and algebraically). Enjoy!
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0:00 Graph Theory: Chromatic Number
4:52 Big Idea of Invariants
5:19 Geometry: Euler Characteristic
9:34 Linear Algebra: Determinant
13:24 brilliant.org/TreforBazett
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[@tateallen9972]

I see the Peterson graph, I click

[@DrTrefor]

ha:D

[@johnchessant3012]

11:00 wow, that's a very interesting way to generate Pythagorean triples out of thin air!
e.g. [[1, -1], [1, 1]] * [[m/n, 0], [0, n/m]] * [[1/2, 1/2], [-1/2, 1/2]] = (1/2mn) [[m^2+n^2, m^2-n^2], [m^2-n^2, m^2+n^2]]
and the fact that this has determinant 1 gives us (m^2-n^2)^2 + (2mn)^2 = (m^2+n^2)^2.

[@briandwi2504]

After your last video explaining your intentions I thought I'd watch. I have watched several of your University series and benefited from them. I am not a huge fan of 'popular ' maths, but this wasn't dumbed down and didn't patronise the viewer. I thought it was very interesting so great start Dr Trefor and the best of luck with this project.

[@DrTrefor]

Thank you so much, that is exactly my goal!

[@northernlight1000]

This will be a legendary series! Thanks so much Dr.!

[@Bunnokazooie]

I love 3 coloring and isomorphisms, glad to see a spectacular introduction to it

[@mostafasaleh5594]

Very nice idea for a series Doc! Please do some abstract algebra in the future

[@DrTrefor]

Definitely some coming!

[@Itsmisterunknown9]

Huge love from India

[@DrTrefor]

Thank you!

[@PRIYANSH_SUTHAR]

The study leave is totally going to be worth it.

[@ToolA10]

Clear explanation of invariance with easy to follow examples. Your excellent animations really help too! And thanks for defining terms like isomorphism. Many videos use it without doing so. Same thing for what the Determinant of a matrix means geometrically. Your depiction makes the linear algebra make more sense.

[@user-gs6lp9ko1c]

Great start! Looking forward to the rest of the series.

[@DrTrefor]

Thanks!

[@_P_a_o_l_o_]

Can't wait to see the next invariants!

[@giornoGiovanna964]

I would love if you would do a number theory course I would love it

[@andrewharrison8436]

Hmm: Invariants, bet Euler gets a mention (actually Maths, bet Euler gets a mention - the odds are good).
My memory of topology is that it is entirely about Invariants - knot theory particularly.
Anyway, not dissapointed, makes me realise that I need to revise matricies - that was elegant.

[@putin_navsegda6487]

thank you, always pleasure to watch your videos !

[@DrTrefor]

Glad you like them!

[@kmjohnny]

The more videos I watch on yt, the more I see how everything in math is connected. I wonder how all math is connected to Trefor's favorite T-shirt.

[@user-wk8ci7ek8r]

Beautiful shirt and interesting series..

[@jimmelton]

Would love to see some in-depth linear algebra stuff!

[@DrTrefor]

I have a playlist on linear algebra - check it out!

[@briandwi2504]

@@DrTrefor It is brilliant!

[@siamtalukdar-sx7uj]

It will be really heplful for a mechanical engineering student for Continuum mechanics if you make a course on Tensor Algebra And Tensor Analysis., as there is no reliable course on youtube or net.

[@MegaMONI45]

Just finished my complex analysis exam and am a little drunk but came to say...Dr Bazett full-time youtuber LET'S GOOOOOOO!!!!!!

[@MegaMONI45]

ps. If you're doing full-time youtube for a bit, Dr Bazett, could you have a crack at turning your wonderfully creative explanations to some hectics stuff? E.g. Curry-Howard Correspondence? Perhaps live stream you trying to solve the Collatz conjecture? Maybe run through the proof of the Poincare conjecture that won the millennium prize?! Just some ideas...

[@SumanYadav-wr3cn]

Sir please provide lectures on sieve theory

[@ArdenChew271]

Will you do PDEs in the future

[@Grassmpl]

Good old Petersons graph

[@guruji75]

Is there any link between invariants and conservation laws in physics? Im thinking Noether's theorem...

[@ValkyRiver]

4:40 caveat - assuming no exclaves

[@fdileo]

Could you recommend a book about graphs?

[@EpicMethGaming]

WHOOOOOAAAAAA SKIBID TILET??

[@Stellectis2014]

Two graphs have different chromatic numbers, there cannot be a graph isomorphism between them. Then how do you explain subgroups?

[@DrTrefor]

The chromatic number of a subgraph is less than or equal to the chromatic number of the original graph.

[@Stellectis2014]

@@DrTrefor Thank you Dr. Trefor.

[@Itsmisterunknown9]

Hi sir

[@me-tubemath7927]

How to create videos plz make a video how to make video for videos like you?

[@DrTrefor]

I have this! ua-cam.com/video/hmQd_P_qj1w/v-deo.html

[@me-tubemath7927]

@@DrTrefor shifu, any source to learn matlab for graphing step by step.

[@uwuowo7775]

I like geese

[@DrTrefor]

who doesn't??

[@imnimbusy2885]

Do geese see God?

[@taradavies3073]

me too.

[@uwuowo7775]

@@DrTrefor people who don't like geese

[@logielleEntiopya]

​@@uwuowo7775 r/technicallythetruth

[@sgiri2012]

Topologists can't distinguish between the two diametrically opposite objects present in your v necked shirt. Expecting topology videos from you