Відео

Curiously enough, Michael Penn just posted an algebraic geometry video today where he says he isn't able to wrap his mind around the concept of sheaves

Sir, this video deserves an award

Great video. But, what I have learned about "solvable" group just requires the quotients to be Abelian, not prime cyclic ( named "supersolvable" ). What confuses me is that, why do we have the meaning of definition "solvable" other than "supersolvable"? It seems sufficient we just define "supersolvable" then solve the problem of radical solution. And what I have learned is through "solvable" groups... Is it just a result of generalization to some extent?

Hello again my friend. Just randomly stumbled across this video, and wanted to ask you to do a video on Sphere packing (in arbitrary dimensions), which is apparently a growing and blossoming field these days. Also, I wish to collect some ideas of it for work in particle theory on the physical side of things. In any case, some insight from you on this area would be useful and certainly entertaining. Thank you for all your work and contributions to mathematics education to a broader audience. Also, I love your new Algebraic Geometry series! Can't wait for more!

I always think of a sheaf as a formalization of partial functions. Partial functions with an intersection operation.

I'm an engineer and I always called that a graph

very nice!

This is beyond great, unlike other videos that are not straight to the main idea! Would you like to make some videos about rings of differential operators, particularly with polynomial coefficients? It is highly related to Gröbner Bases (and of course, Weyl Algebra). I am currently studying it for my thesis. Thank you! Also, I have already hit that subscribe and like button ;)

Great video. But when I saw the video at the end, I had doubts about the set S, shouldn't S be containing s(a) ≠ 0 (otherwise [1] can not be contained in S ) ? Sorry to bother.

I noticed the pattern F is iso if there exists a G: D -> C where GF equals id_C and FG equals id_D, also F is equiv if there exists a G: D -> C where GF iso id_C and FG iso id_D. Is there an even weaker notion where G: D -> C where GF equiv id_C and FG equiv id_D? And if said weaker notion exists then are there infinitely many of such notions each weaker than the last?

This is an amazing series, really well done. You get quite a kick from visualisation :))

@7:00 Functors are not vanilla arrows. They must be arrows between arrows *_and_* between objects, otherwise they make no sense. So in CAT you cannot ignore the objects. That's why you cannot get an element-free definition for a _full functor._ So Category Theory is definitely not "just about the arrows". It is only that an _emphasis_ is on the arrows.

Thank you 🎉nice explanation

Fantastic explanations and thought provoking material. Thanks a bunch!

@5:20 oh man, what a downer. I really like your series and relaxed delivery, but Mathematica™? Seriously? That prices out a lot of poor kids (and myself). Can't you bend a little to redo interactives in SAGE or Maxima or similar. In Jupyter you can use Sympy and Galgebra (the pypi library, not the gui Geogebra, although the latter is useful too) combined with Plotly. You have to support free-libre software dude. So much of the world runs on free-libre, we all should give back by refusing proprietary software. (I do realize the irony of posting this on youtube.)

Love the longer format videos like this. Thanks!

As a chemist I definitely liked the analogy. Thanks for the great explanations!

found your explanations in this video kinda confusing

have you seen the book sheaf theory through examples? it really focuses on sheaves outside of algebraic geometey

Classically, Gauss genus theory of quadratic forms for quadratic fields Q(-d^1/2) is derived from the general linear group GL(2, Z). The action the group SL(2, Z) on the upper half plane H is the starting point of modular functions and modular forms. See “Primes of the form x^2 + ny^2” (1989) which includes an account of the genus theory found in Gauss’s Disquisitiones Arithmeticae. Representations of both modular forms and quadratic fields can be equated by Serre’s modularity conjecture proven by Khare and Wintenberger. This logarithmic asymptotic staircase looks similar to the psi function ψ(x) = x - log(2π) + {zeros of ζ} determined by zeros of Riemann zeta function bounded by the prime number theorem. I wonder if the Cantor set may describe a similar psi function determined by the zeros of the L-functions of modular forms and their equivalent quadratic fields. The Cantor function and Riemann zeta function both described by Bernoulli numbers. Note "Integrals Related to the Cantor Function." (2004) by Gorin, E. A. and Kukushkin, B. N. The Cantor function is a map between the interval c: I -> I to itself with a ternary function of deleting the center 1/3. Following A^1-homotopy theory this the interval can be replaced by the affine line A^1. This equivalence is a result of the Thom-Pontryagin theorem between the algebraic cobordism groups A of a smooth quasi-projective scheme over field k (ie affine variety A^n_k such as affine line A^1_k) and some homotopy groups π_A given by classical homotopy. A commutative diagram can be constructed to compose both from the equivalence class of homotopy groups from the homotopic paths of the interval f: I -> X and the equivalence class of algebraic paths of the affine line g: A^1 -> Y. There is a quadratic integer ring Z[ω] = {a + ωb} where ω = (1 + D^1/2)/2 with discrimination D of associated to an quadratic field Q(D^1/2). The coordinate ring is equivalent to the affine line Z[ω] = A^1 by Hilbert’s Nullstellensatz. The automorphism a: A^1 -> A^1 described by the Cantor map descends to the ring of integers a quadratic integers. Quadratic integers form lattices which can be considered equivalent and furthermore automorphic up to homothety given by matrices in SL(2,Z).

very unexpected!

I have a silly question. For the graph + vector space example, is the (pre)sheaf just the assignment of vector spaces and the restriction maps between them? Or, is it somehow a valid assignment of elements in those vector spaces? I guess this last question can be phrased purely in terms of images of compositions of the restriction maps. I am asking since, from what I understand, sheaves are useful to show obstructions to certain things existing (which, from what I understand, is a very different motivation than in AG). As an example (which I will phrase informally as to maybe also help other people), let's say we want to show that there does not exist a line that I can draw on the mobius strip (where I don't distinguish the two sides) that is non-zero and locally constant. From what I understand, sheaf theory could be used to show that such a function does not exist. Is this done by showing that such a function is not a valid assignment according to the sheaf that one can construct on the mobius strip? If not, then what is done? Hopefully that makes sense. Thanks for your videos, they've always been very helpful and my go-to resource if I encounter something new.

8:41 😂😂

graphs are so simple yet so complicated

I think that any presheaf F: C^op -> Set for small category C and category of sets Set are functors that categorify any set into small categories. Since it is a functor, it also maps morphisms by sending functors of small categories to functions of sets. Proper classes are collections of objections collection of morphisms of large categories. This is given by definition. Proper classes are in some sense “larger”than sets. This makes wonder if cardinality (size of sets) and size of categories are equivalent. This largeness of either also suggests to me a way to quantify Gödel incomplete theorem for either set theory or category theory.

Thank you

All I ever say is silly, and I jump right in. Thx

非常感谢您，比起书上的公式，您的讲解更显而易懂！ Thank you, your explanation is easier to understand than the textbook！

Sorry about the floods in Bavaria (I think you're German?)

I think of sheaves as presheaves following Yoneda’s lemma which also satisfy a covering condition. This covering condition describes which open sets are local and how to glue them together. This additional covering condition makes the presheaf into a sheaf. From the categorical perspective and following Yoneda’s lemma a presheaf describes the homomorphism into an object X ∈ C as Hom(-,X): C^op -> Set of an opposite small category C^op and category of sets Set. In this way the object X has a map X -> Hom(-,X) and can be understood by the morphisms (presheaves) into itself. See “Isabell duality” (2023) by Baez. The category of locally constant sheaves is equivalent to the category of covering spaces. See Example 1.2 in “Sheaves, covering spaces, monodromy and applications” (2016) by Calabrese.

nice!

In complex analysis, Liouville's Theorem states that a bounded holomorphic function on the entire complex plane must be constant. The boring case :)

Thank you!

been looking forwards to this one!

Goodstein not Goodsteen.

im 11 yr old and im interested

Great video!!!! Thank you so muchhh

very nice video!

Oh no! It was all a trick to teach us topos theory secretly!

Here's another two: the graphs of the n-ary versions of the AND and OR logic operators form 'inverse fractals.'

12:03 I didn´t really get the term either, until I accidentally found out that it actually is "faisceau" in French. So a bundle basically, but the term is already in use for other mathematical objects.

I think what you’re trying to get at is describing the stability of a system through some iterative process where the system is some dynamical system and the iterative process can be described in combinatorics. Since this happens in general on different objects stable categories should be considered. One way to approach this is to also consider coalgebras. See “The Seirpinski Carpet as a Final Coalgebra” (2021) by Noquez and Moss and “Fractal Sets as Final Coalgebras Obtained by Completing an Initial Algebra” by Bhattacharya and Moss.

Oh yeah!! Your videos are excellent! I suggest them to all of my friends.

Thank you!!! This is such a hard and sparse topic. Hard to find NON-COMPLEX information on it over the internet.. You just get the definitions with no draws or intuition.. And here you are FOR THE SAVE!! thanks man!

Thanks again for the video :) I loved the example of the hemisphere, in my syllabus we only discussed the gluing in general and it was hard to imagine about it, but now I see where the term comes from. I've watched a few of the other videos too and they were helpful too, I love your enthusiasm.

At 4:36 the diagonal arrows suggest a spectral sequence. There is a Künneth spectral sequence. I think your idea of computing cohomology by each piece suggests the locality of sheaves and their sheaf cohomology which is derived using a spectral sequence.

Hi, this might be a stupid question but why exactly isn’t the outside of Alexander’s sphere not simply connected

I really need to read that stanford book abt geometric algebra 😭😂 is it the sea of wisdom ??

CW complexes retract nicely onto a sub complex….!! Yay Xournal! Ctrl+shift+f

One day in the hopeful future I would love to sit and chat Yang-Mills with you bc ive been digging into it for a few months now as a step along the knot theory rabbit hole, but I want to leave you with this link! ua-cam.com/video/sVS2sVBQYO8/v-deo.htmlsi=QAD07U-i0VfF7mI3 the glueball is apparently one of the big markers of mathematical consistency in YM theories, i don't pretend to understand why yet, but i *do* know it's significant and this discovery is like empirical verification of the yet-to-be-discovered mathematical truth. how exciting!