Grassmann algebra and deRham cohomology - Lec 12 - Frederic Schuller

[@taylankargin7474]

Prof. Schuller is a legend with an icecream on hand!

[@tobiaspower381]

I didn't see where the ice cream was derived from... it just appeared out of nowhere. Magic theoretical physics :)

[@gianlucarusso8475]

I wish I had a Prof like him when studying Physics...a real master !

[@OhadAsor]

Thanks Prof. Schuller, I was just rewatching your General Relativity course and was so happy to find you have more here!

[@J.Rahman]

Can you post a link to the General Relativity course?

[@david52875]

Jillur Rahman search W.E Heraus international winter school on gravity and light

[@tim-701cca]

2:24 diff form 11:43 tensor product of form is not form 14:12 wedge products 22:10 pullback of form 25:12 Grassmann algebra 38:49 exterior derivative 1:02:46 formulas involving d, wedge and pullback 1:11:27 de Rham cohomology

[@abcdef-ys1sb]

I am a big fan of you. Your lecture is simply amazing

[@jackozeehakkjuz]

At 52:25 he expands the commutator as "X acting on fY minus fY acting on X", which is not completely correct, since he has not defined the action of a vector field on another. What he should have said is "X composed with fY minus fY composed with X". I.e. (X∘(fY) - (fY)∘X). Then at 54:00 he goes on using the Leibniz rule of omega over the composition (X∘(fY)). Althought this is ultimately true, it is not so trivial. Besides these details, the proof is correct.

[@theleastcreative]

Are the problem sets for this course available anywhere?

[@burakcopur3841]

I wish he had made more lectures available, whatever he was teaching at the time :)

[@socialglitch2663]

My favorite lecturer!

[@joelcurtis7447]

Taking a page out of Susskind's playbook, eating sweets while lecturing.

[@mrnarason]

Was that a Häagen-Dazs?

[@synaestheziac]

And yet in the very same lecture he makes a jab at (physicists like) Susskind who describe the Grassman numbers as “numbers which don’t commute”, when of course it is only an operation that does or does not commute. That said, Susskind’s supersymmetry lectures are pretty great: kzbin.info/aero/PL_IkS0viawhpBbVkmwRLiysdV5eXegvd_

[@vivalibertasergovivitelibe4111]

This was really great. I knew the definitions but had little intuition about the de rham cohomology group. You have helped me a lot there. Thank you!

[@PrachiGarella]

Is there a way I can get a hold of the problem sheets?

[@Adivasilover10]

Your lectures are very important. From India

[@jeffreyelawani3022]

Legendary class!

[@WayneKimRecords]

Is that a Häagen-Dazs? i lol'd after realizing Frederic was actually holding an icecream

[@mrnarason]

I thought it was an eraser because why would he be eating a ice cream while lecturing lol.

[@tomkite1933]

I am quite interested in the approach to classical dynamics using the cotangent bundle over a manifold. Does anyone know any good sources to learn about this?

[@MagufoBoy]

Proffesor Schuller has a course in classical mechanics using differential geometry. Just write "Frederic Schuller Theoretische Mechanik" on the KZbin browser and you will find it

[@tomkite1933]

@@MagufoBoy Fantastic, thank you! I will watch it and hope the captions can translate all the mathematical terms properly

[@rounak5106]

@@MagufoBoy That course has a language barrier. Can you suggest some course which is in english or preferably some book (apart from Flanders) or lecture notes?

[@thomasbissinger2235]

​@@rounak5106 If you're still looking, the book by Marián Fecko (Differential Geometry and Lie Groups for Physicists) devotes a chapter to the Hamiltonian and Legendre formalism from a differential geometrist's perspective. It's generally a good read to accompany Frederic Schuller's lectures, there's some similarities in the presentation, although Fecko doesn's start with axioms of set theory. Apart from Fecko, Marsden and Ratiu (Introduction to Mechanics and Symmetry) have a readable intro to Lagrangian and Hamiltonian systems, although they are sometimes a bit vague on the math. I must admit that it's been a bit since I last looked into their book, so I can't guarantee you will find exactly what you're looking for there. Oh, and of course, I think the final lecture of this lecture series (28 - Application: Kinematical and dynamical systems) does a fine job in exploring the connection, too. If you haven't reached it yet. It's great. I mean, it's Schuller.

[@gchtrivs7897]

Top-notch quality content! Smashed that like button right on the middle

[@dpr265]

Finally I’m back, hopefully I can finish the rest of lectures.

[@rishabhkumar9587]

At 32:40 , What if I take the wedge product of two forms of dimensions n1 < dim(M) and n2 < dim(M) such that n1+ n2 > dim(M) ?

[@rishabhkumar9587]

I think it will be 0 in the set of n1+n2 dimensional forms. Please correct me if I am wrong

[@UnforsakenXII]

I think you're right because once you shuffle it around a little, you will see that two of the basis forms in the new wedge are linearally dependent and then the anti communitivity kills it off.

[@kockarthur7976]

@Rishabh Kumar You are right.

[@TurtleTube123]

Thank you for asking and answering this, I had exactly the same question!

[@8gigabytes]

Thank you very much, what a great lecture!

[@josephgarretsullivan4762]

Near the end when he covers deRahm cohomology he refers to the domain of the exterior derivative maps as vector spaces. But since the domain are differential forms of the previous order (that is, (r,0) tensor fields on the manifold), are these domains usually vector spaces or rather are they usually modules? In other words, do we in physics prefer for some reason to equip a generic grade of (r,0) tensors with the ring of C^infty (M) (0-form), over the field F that underlies the fiber bundle structure of TM?

[@millerfour2071]

8:26, 15:00, 24:00, 26:18, 30:07, 36:06, 43:12, 48:30, 55:42, 58:08, 1:01:28, 1:16:25, 1:27:20, 1:36:33, 1:41:49

[@netrapture]

1:01:50 it "commutes" "if you know what you are doing" :)

[@nihaljalal5693]

18:00, Are those two the same? Isn't the set of n-forms a subset of the set of all covector fields?

[@kvazau8444]

covector fields are 1-forms

[@paulfischer3271]

I don't understand. At 43:40 it looks like a vector field is eating a smooth function. Isn't the X_i a vector field, that is a section of a smooth tangent bundle? Then it should map a point on the manifold to one of it's tangent vectors. But omega is a 0-n tensor field, that is a function that takes n vector fields and outputs a smooth function. So how can you give the X_i a smooth function when it should eat a point of the manifold? What am I missing?

[@falconcyril]

A vector field X of a manifold M acts on a function f from M to R in the following way X(f)(x):=T_xf(X(x)), where x is a point of M and Tf is the tangent map of f, X(f) is again a map from M to R. In this definition, X(x) is a point of T_xM and T_xf is a linear map from T_xM to T_{f(x)}M, so every things makes sense.

[@danideboe]

The Lie derivative L_X is identified with X such that L_X(f) =: X(f)

[@mathgeek43]

This reply is really late, but I hope it helps. ω is an n-form field. This means that at each point p∈M, ω assigns an n-form ωₚ which takes n vectors at p and outputs a real number. The expression ω(X₁,...,Xₙ) is a function f:M→ℝ because you can think of f as taking the point p ∈M and outputting the value that ωₚ gives you.

[@paulfischer3271]

How exactly is the push forward of a vector field used at 20:23 defined?

[@creillyucla]

good catch. it is defined pointwise. remember there is never any issue pushing a tangent vector forward, though this is not the case for vectors fields.

[@jackozeehakkjuz]

Yeah, he missed that one. It's not defined. Actually you need to evaluate omega at h(p) and apply it to (the pushforward of h at p, applied to each X_i evaluated at p). Later in the course he clarifies this, although he never corrects himself about this lecture.

[@yogeeshreddy6373]

Jackozee Hakkiuz By X_i evaluated at p, he probably means this: If X_1=X_gamma,q. Then h_*(X_1)=(h_*)_q (X_gamma,q) belonging to T_h(q) N subset of TN. Notice that p and q can be different, but since the element I described above is subset of TN, omega(p) being a one-form can act on this

[@tysonche3321]

Hello everyone! I hope you're all doing well. I was hoping to get some clarification on the coefficient 1/(n!m!) in the definition of the wedge product. I've been considering the space of n-forms as a subspace of the space of (0,n)-tensors, which are anti-symmetric under the action of Sₙ. In this context, I was wondering if it's a common technique in group representations to take the 'average' of an object, like the (0,n)-tensor in this case, and map ω⊗σ to the 'averaged' version. If that's the case, I was thinking it might be more natural for the coefficient to be 1/(n+m)!, since (n+m)! represents the size of S_(n+m). Could someone please kindly shed some light on this and help me understand the reasoning behind the 1/(n!m!) coefficient? I would really appreciate it!

[@thisisapersonwhois]

The coefficient is chosen to be 1/(n!m!) since the sum involves n!m! copies of each unique term, thus cancelling out with the coefficient. To see that there are n!m! copies of each unique term note that the first n vector field arguments of ω⊗σ can be permuted amongst themselves in n! ways and the last m vector field arguments in m! ways. The sign accrued by such permutation will cancel out with the sgn(\pi) in front.

[@dpr265]

it’s getting difficult, plan to rewatch several times

[@jelmar35]

"We would like to extend the D" 40:08 Mentioning my insecurities.

[@songvasu5256]

The link is not downloadable for some reason?

[@mastershooter64]

schuller's lectures are like asmr lol

[@balasubramannyans9469]

Which text book is following ?

[@samwinnick4048]

I think you have a typo on the board at around 33:40. I think the second summand of the Grassmann algebra should be T*M not Gamma(T*M).

[@tim-701cca]

It is not a typo.

[@imrematajz1624]

On the one hand this is aha! On the other it is an ice cream🎉

[@danielroddy2032]

Dude eats an ice cream at 1:05:15… :0

[@trewq398]

amazing

[@Mgaak]

Is this lecture a part of PhD course ?

[@UnforsakenXII]

I think it's undergraduate. It's pretty reachable if you read it along with an introductory book on mfds that covers some aspects of topology, analysis and abstract algebra. But then again, i'm necrocommenting. It's been a year, whoops.

[@kockarthur7976]

No it's definitely aimed more at graduate students. Maybe some extremely advanced undergraduates could follow and benefit from these lectures, but they would certainly be the exceptions. From the program's main page: "Background knowledge minimally required of students is what one normally would have acquired after the second year of university courses in physics and/or mathematics. The course aims at graduate students (roughly year 4 to 5) at universities but will be accessible to advanced undergraduates and still very rewarding to beginning postgraduate students."

[@josef-namnguyen6884]

Is Magnum ice cream a thing in Germany? Because it really looks like a Magnum.

[@jelmar35]

It is!

[@mr.mxyzptlks8391]

Kinda late, but the ice really distracted me 🎉

[@mrnarason]

1:15:28 nice meme

[@bodeco5145]

O cara é um deus mesmo, sacou um picolé no meio da aula e ta nem aí

[@Igdrazil]

Ok I've got a Airbzb Theory for the "BZ-theorem". Indeed, since algebraic topologists travel to much to bring back Home the O-Logos, some became traumatise in cheep Airbnb, getting crazy in broken (non connex) BZ morning clic-clac folding. As these strange beast become connectedless manifolds when they get aging, their unfolding is as trivial as their folding back turns to be a morning nightmare. As for the zero Cohomology group, the first connected part is always trivial to handle. But as soon as one try's to connect it to unseaworthy free electron other parts, it gets much harder. And thrying a threefold cohomologic connexion, it leads to the famous morning universal nightmare. Thus they named the dreadfull obstruction to this pull back differential connectivness, the BZ Obstruction Theorem ! Morality : Airbzb makes maths many fold harder!!!

[@Karim-nq1be]

This is kindergarten algebra my friends 🙂

[@BigMoneyPauper]

yo its a bit sus to say the word n forms outloud bro

[@SadSocks]

Ladroh

[@johnchessant3012]

Z