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Rooney
Registrace 10. 08. 2020
My goal is to better understand concepts in higher mathematics by teaching them. Hopefully, I can help a few people along the way!
Galois twisted forms and nonabelian H1
A good reference for this material is Gille and Szamuely's "Central Simple Algebras and Galois Cohomology".
Animation is done with "manim" community edition.
Animation is done with "manim" community edition.
zhlédnutí: 926
Video
the cartesian product: type theory vs. set theory
zhlédnutí 1,4KPřed 2 lety
When I was reading through Section 1.5 on the Homotopy Type Theory book, I was initially very confused as to what was going one, particularly regarding the uniqueness principle. I don't go into detail about the constructions they provide in the chapter, but hopefully this will be a helpful overview of what is going on intuitively. Credits 3b1b's animation library "manim": github.com/3b1b/manim ...
E-local is equivalently A-null
zhlédnutí 236Před 3 lety
We are trying to build towards the result that E-localizations always exist. A crucial part of this is the second lemma in this video, which is lemma 2.4 in the nLab. The nLab doesn't provide a proof, but references two other papers that do (one being the original source). Credits References: ncatlab.org/nlab/show/Bousfield localization of spectra ncatlab.org/nlab/files/VanKoughnettLocalization...
E-localization: a first definition
zhlédnutí 331Před 3 lety
credits Reference: ncatlab.org/nlab/show/Bousfield localization of spectra Animation: A modified version of 3b1b's manim library: github.com/treemcgee42/youtube
Adjunctions as Kan Extensions
zhlédnutí 884Před 3 lety
Credits: I use a modified version of 3Blue1Brown's animation library "manim": github.com/treemcgee42/youtube Riehl, Categorical Homotopy Theory- math.jhu.edu/~eriehl/cathtpy.pdf Lehner, “All Concepts are Kan Extensions”: Kan Extensions as the Most Universal of the Universal Constructions www.math.harvard.edu/media/lehner.pdf
The Yoneda Perspective
zhlédnutí 10KPřed 3 lety
Credits: I use a (modified version of) 3Blue1Brown's animation library "manim". The main inspiration was this series of blogs on math3ma (an AMAZING website) www.math3ma.com/blog/the-yoneda-perspective www.math3ma.com/blog/the-yoneda-embedding www.math3ma.com/blog/the-yoneda-lemma
ITHT: Part 12- Model Structure on Topological Spaces
zhlédnutí 401Před 3 lety
Credits: nLab: ncatlab.org/nlab/show/Introduction to Homotopy Theory#TheClassicalModelStructureOfTopologicalSpaces Animation library: github.com/3b1b/manim My own code/modified library: github.com/treemcgee42/youtub...
ITHT: Part 11- Quillen Adjunctions
zhlédnutí 318Před 3 lety
Credits: nLab: ncatlab.org/nlab/show/Introduction to Homotopy Theory#QuillenAdjunctions Animation library: github.com/3b1b/manim My own code/modified library: github.com/treemcgee42/youtub... Music: ► Artist Attribution • Music By: "KaizanBlu" • Track Name: "Remember (Extended Mix)" • CZcams Track Link: bit.ly/31Ma5s0 • Spotify Track Link: spoti.fi/2NUH3xZ
ITHT: Part 10- Derived Functors
zhlédnutí 577Před 3 lety
Credits: nLab: ncatlab.org/nlab/show/Introduction to Homotopy Theory#DerivedFunctors Animation library: github.com/3b1b/manim My own code/modified library: github.com/treemcgee42/youtub... Music: ► Artist Attribution • Music By: "KaizanBlu" • Track Name: "Remember (Extended Mix)" • CZcams Track Link: bit.ly/31Ma5s0 • Spotify Track Link: spoti.fi/2NUH3xZ
ITHT: Part 9- The Homotopy Category
zhlédnutí 420Před 3 lety
Credits: nLab: ncatlab.org/nlab/show/Introduction to Homotopy Theory#TheHomotopyCategory Animation library: github.com/3b1b/manim My own code/modified library: github.com/treemcgee42/youtube Music: ► Artist Attribution • Music By: "KaizanBlu" • Track Name: "Remember (Extended Mix)" • CZcams Track Link: bit.ly/31Ma5s0 • Spotify Track Link: spoti.fi/2NUH3xZ
Introduction to Homotopy Theory: Part 8- Homotopy in Model Categories
zhlédnutí 446Před 3 lety
Credits: nLab: ncatlab.org/nlab/show/Introduction to Homotopy Theory#homotopy_2 Animation library: github.com/3b1b/manim My own code/modified library: github.com/treemcgee42/youtube Music: ► Artist Attribution • Music By: "KaizanBlu" • Track Name: "Remember (Extended Mix)" • CZcams Track Link: bit.ly/31Ma5s0 • Spotify Track Link: spoti.fi/2NUH3xZ
Introduction to Homotopy Theory: Part 7- Small Object, Retract Arguments
zhlédnutí 403Před 3 lety
Credits: nLab: ncatlab.org/nlab/show/Introdu... Animation library: github.com/3b1b/manim My own code/modified library: github.com/treemcgee42/youtube Music: ► Artist Attribution • Music By: "KaizanBlu" • Track Name: "Remember (Extended Mix)" • CZcams Track Link: bit.ly/31Ma5s0 • Spotify Track Link: spoti.fi/2NUH3xZ
Introduction to Homotopy Theory: Part 6- Projective and Injective Morphisms
zhlédnutí 460Před 3 lety
Credits: nLab: ncatlab.org/nlab/show/Introdu... Animation library: github.com/3b1b/manim My own code/modified library: github.com/treemcgee42/youtube Music: ► Artist Attribution • Music By: "KaizanBlu" • Track Name: "Remember (Extended Mix)" • CZcams Track Link: bit.ly/31Ma5s0 • Spotify Track Link: spoti.fi/2NUH3xZ
Introduction to Homotopy Theory- Part 5- Transition to Abstract Homotopy Theory
zhlédnutí 574Před 3 lety
Credits: nLab: ncatlab.org/nlab/show/Introdu... Animation library: github.com/3b1b/manim Music: ► Artist Attribution • Music By: "KaizanBlu" • Track Name: "Remember (Extended Mix)" • CZcams Track Link: bit.ly/31Ma5s0 • Spotify Track Link: spoti.fi/2NUH3xZ
Sequential Spectra- Part 5: Spectrification
zhlédnutí 197Před 3 lety
The second part of the Omega spectra section on nLab. Credits: nLab: ncatlab.org/nlab/show/Introdu... Animation library: github.com/3b1b/manim Music: ► Artist Attribution • Music By: "KaizanBlu" • Track Name: "Remember (Extended Mix)" • CZcams Track Link: bit.ly/31Ma5s0 • Spotify Track Link: spoti.fi/2NUH3xZ
Introduction to Homotopy Theory- Part 4: Fibrations
zhlédnutí 1,7KPřed 3 lety
Introduction to Homotopy Theory- Part 4: Fibrations
Sequential Spectra- Part 4: Omega spectra
zhlédnutí 170Před 3 lety
Sequential Spectra- Part 4: Omega spectra
Sequential Spectra- Part 3: Stable Homotopy Groups
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Sequential Spectra- Part 3: Stable Homotopy Groups
Sequential Spectra- PART 2: Preliminary Definitions
zhlédnutí 333Před 3 lety
Sequential Spectra- PART 2: Preliminary Definitions
Sequential Spectra- PART 1: Introduction / Motivating Spectra
zhlédnutí 679Před 3 lety
Sequential Spectra- PART 1: Introduction / Motivating Spectra
Introduction to Homotopy Theory- PART 3: CELL COMPLEXES
zhlédnutí 1,6KPřed 3 lety
Introduction to Homotopy Theory- PART 3: CELL COMPLEXES
Introduction to Homotopy Theory- PART 2: (TOPOLOGICAL) HOMOTOPY
zhlédnutí 2,9KPřed 3 lety
Introduction to Homotopy Theory- PART 2: (TOPOLOGICAL) HOMOTOPY
Introduction to Homotopy Theory- PART 1: UNIVERSAL CONSTRUCTIONS
zhlédnutí 10KPřed 3 lety
Introduction to Homotopy Theory- PART 1: UNIVERSAL CONSTRUCTIONS
Differential Forms: PART 2- COVECTORS AND ONE FORMS
zhlédnutí 7KPřed 3 lety
Differential Forms: PART 2- COVECTORS AND ONE FORMS
Differential Forms: PART 1A: TANGENT SPACES (INTUITIVELY)
zhlédnutí 15KPřed 3 lety
Differential Forms: PART 1A: TANGENT SPACES (INTUITIVELY)
Differential Forms: PART 1- TANGENT AND COTANGENT SPACES
zhlédnutí 6KPřed 3 lety
Differential Forms: PART 1- TANGENT AND COTANGENT SPACES
Your voice ♡♡♡♡♡
Sleepy ahh vibe 😂
Is \gamma really from M to TM? I'm not sure that makes sense.
Yoneda’s lemma and its dual allow any small category C to be described from morphisms into and out of it also known as its hom-set. Given an object X ∈ C there exists a functor Hom(-,X): C^op -> Set describing morphisms into C. For morphisms out of C there are also exists a functor Hom(X,-): C -> Set. This provides a concrete way to implement Grothendieck’s relative point of view of considering morphisms of a category instead of objects of that category in order to understand a category. It is important to note that the functor Hom(-,X): C^op -> Set is a presheaf of the category C. The presheaves are the probing questions or morphisms into C as the maps f: - -> C you mentioned in your examples of a deck of cards and topological space.
Can this twisted form be thought of as analogous/related to a sesquilinear form?
loved it!
never ever play music while talking like this.
2:02 3B1B character?
why only one video :)) loved the explanation
Can't watch because of music. Pls remove.
So far i've watched 3-4 videos on homology/cohomology and they all had creepy music playing in the back and just an eerie vibe. From now on it will forever be creepy math for me
Nice! But why the background music?
Too much music, too little information.
Why are you doing everything to put one to sleep?! Is this some weird ASMR math channel?! Also, why steal from other creators?
Wow
Beautiful video.
The concept of Derivations, and the generalization of the derivative is really interesting to me.
Very cool! I have been very curious about the Yoneda lemma, and this was illuminating.
Why do you say that the tangent plane image is not a good way to think of the tangent space?
Dude turn up the mic.
"a section to the composition functor" what a testament to nLabs tendency to overcomplicate some things.... As if higher math isn't already complicated enough.. No wonder nLab is unusable for many...so unecessary
Please turn the volume up during editing
I would argue that a "nicer" example of a category enriched over itself is the category of k-vector spaces (or more generally, R-modules). Indeed, the set of linear maps between two vector spaces is a vector space, and composition is bilinear (so induces a canonical linear map from the tensor product of hom-spaces). Great video!
good explanation but the music is distracting.
I hate when people record videos forgetting about audio quality
First time that I ever managed to comprehend anything related to this topic. Wikipedia and ncatlab are very good websites, and yet they are bad for introduction to this topic. This is the first video that I watched from this channel about this topic, since I believe in general case first, special case later if ever, and still managed to comprehend the definitions, even if not the motivations or the reasons for the names.
baza yobanaya
It's pronounced "tah-pology" not "tope-ology". It's a small error, but it reveals you've never spoken to a mathematician.
boy what
Or that he's not studying math in the UK or the US Girl/boy what's your damn problem? Not all mathematicians are English language masters
@@imperfect_analysis I know, just providing context so his video can improve. I had the same problems starting out, you know "Yoo-ler", "Ho-mo-to-py", etc. The comment is not mean spirited in intent.
@@annaclarafenyo8185 alright:) sorry if I sounded mean but you're right
@@annaclarafenyo8185you cant just claim afterwards that it wasnt mean spirited when it clearly is. claiming that he „never spoke to a mathematician“ is dumb and hurtful. try better
Fantastic explanation! I have not seen Yoneda's Lemma introduced so delicately before and it has been much needed.
Do you have a discord server?
very good explanation! thank you very much; ❤❤❤
6:49 I think colimits are called direct limits while limits are inverse limits. Since it's confusing, most people just don't call them that anymore
Very well done , best explanation on CZcams.
For the (y, 0) vector field (3:14) it was unclear exactly what was being integrated and with respect to what? I assume we are integrating "y" with respect to theta along the circle? OK, later (7:19) I noticed that you are integrating y dx + 0 dy which is the line integral of the tangential component of the vector field with respect to the arc length in the language of vector calculus. I don't know if it is the same thing in the language of differential forms---I am still confused about whether "dx" in differential forms refers to an infinitesimal or a basis of a co-vector. Thanks.
Thank you for sharing your thoughts about such advanced and profound math problems. I think your videos are quite illuminating and I would like to express my respect for your worthwhile work!
Excellent cursory introductory look at a complex subject. It's all starting to take shape in my mind.
Man, you need to improve your diction.
CLARIFICATIONS: 1. At around 0:40 I say something to the effect of "if V_K is isomorphic to W_K as K-vector spaces, it may not be the case that V is isomorphic to W as F-vector spaces". I'm wrong there, that will always be true, since dim_K(V_K)=dim_F(V)=dim_K(W_K)=dim_F(W). I should have brought this point up later on when I had introduced taking G-invariants. The point I should be making is "if X and Y are isomorphic as K-vector spaces, it may not be true that X^G and Y^G are isomorphic as F-vector spaces". The intuition is that, in this latter example, the G-actions on X and Y may not be related, but the G-actions on V_K and W_K are. 2. The clarification (1) was necessary because dimension completely characterizes finite-dimensional vector spaces up to isomorphism. But one can show that the whole theory discussed in the video applies to vector spaces equipped with "(p,q)-tensor structure"s. The structure of an algebra is an example of such a structure. Dimension does not completely characterize algebras up to isomorphism, so it is possible for V_K and W_K to be isomorphic as K-algebras but V and W are not isomophic as F-algebras
Stumbled upon this when looking for a more intuitive explanation of what tangent spaces are. Pretty good video, would be even nicer if you spoke louder tho ;)
So succinct and great amk
Hole at infinity?
thanks! I liked the part of the cards; ❤❤🦊
excellent video
Platinium end
This is great. Is there a playlist / series this is part of ?
Wake up man !!
3:39 Note that ANY ELEMENT in π_•((QX)_k) is the image of a finite stage, not necessarily the whole group. Only this is needed for the coming proof.
Holy shit I finally understand the difference between a type and a set, I love you <3
brilliant!! can’t really find any other words. thank you for making this!
Galois Theory has always just went over my head, I'll have to learn some more first lol
It's definitely a theory that gets deeper each time I revisit it!