I might be a little late but this is exactly what I was looking for. Homological algebra and its connections to algebraic topology are so interesting to me and these videos are very helpful for making the ideas more concrete. Thanks for posting them😊
This is fantastic! Thanks! Although it might be a chicken and egg thing, I always thought the finite order elements are called torsion because they appear in the homology groups of objects that get twisted. Things like klein bottle or mobius strip contain torsion components whereas sphere, torus, etc. do not.
@Desiderato Productions Hello! I didn't seriously study math until college. I took calculus in high school but that was it. If you have access to college courses, then I would just focus and do the best I could in those. If you are studying on your own, then I would just pursue the things that interest me the most. College level math is mostly proof-based. If you haven't done it yet, then a good intro to that would be discrete math. That opens the door to study a bunch of other things. A first course in abstract algebra or group theory. It's hard to recommend anything specific without knowing your background more.
I think that the name of torsion element comes from the work of Noether in which that elements correspond with an actual twist of, for example, a Mobius band.
To throw my hat in the "why call it torsion" ring: if we think of abelian groups as Z-modules, then we can kind of pretend that we're doing linear algebra. Taking some a in A, the "span" of a "should" go off forever and be a big line (if we want to pretend that we're doing linear algebra) but sometimes the line "twists around" and comes back, if a is finite order. Torsion meaning twisting, we get that the span of a is "twisted", so a has "torsion." That's my thought at least.
That’s what I’ve always imagined too. If you think about Cayley graphs, a torsion element will yield a cycle in the graph, and free groups, for example, are torsion free and so their Cayley graphs are trees - no cycles! So torsion really is like a ‘twisting’ or at least ‘cycling’ motion. (I guess the converse isn’t true: if you pick a Z linearly dependent generating set you can get cycles even without torsion because there won’t be a unique path from the identity to any element that admits more than one expression as a word in the generators.)
Conjecture: It is okay to love another human being you've never met. Let us assume that we are allowed to employ the axiom of unconditional love choice. Although this may seem to render the statement a tautology, this is not so obvious. Hint: Recall that we take the definition of "okay" to mean whatever is advantageous for the group as opposed to a specific individual in that group.
You are yeeting these algebra lectures like nobody's business, thank you so much for this Professor Borcherds.
I might be a little late but this is exactly what I was looking for. Homological algebra and its connections to algebraic topology are so interesting to me and these videos are very helpful for making the ideas more concrete. Thanks for posting them😊
Thank you so much for wonderful lecture! I always wanted to learn what Tor is, and this lecture is a wonderful start for me!
This is fantastic! Thanks! Although it might be a chicken and egg thing, I always thought the finite order elements are called torsion because they appear in the homology groups of objects that get twisted. Things like klein bottle or mobius strip contain torsion components whereas sphere, torus, etc. do not.
there is a footnote in rotman's homological algebra where he says that free groups are free because they lack "entangling relations"
@Desiderato Productions Hello! I didn't seriously study math until college. I took calculus in high school but that was it. If you have access to college courses, then I would just focus and do the best I could in those. If you are studying on your own, then I would just pursue the things that interest me the most. College level math is mostly proof-based. If you haven't done it yet, then a good intro to that would be discrete math. That opens the door to study a bunch of other things. A first course in abstract algebra or group theory. It's hard to recommend anything specific without knowing your background more.
At 6:30, shouldn't A be tensored with B? Or am I missing something?
I also think so but fortunately it does not change the definition of Tor
@@robalo7155 yes, it should be A tensored with B. But as Marques said, it does not change the definition.
At 13:22, shouldn't it be ds+sd = f-g ?
yes
I think that the name of torsion element comes from the work of Noether in which that elements correspond with an actual twist of, for example, a Mobius band.
The 1st homology groups of the mobius strip is actually just Z, the Klein bottle or projective space are better examples
To throw my hat in the "why call it torsion" ring: if we think of abelian groups as Z-modules, then we can kind of pretend that we're doing linear algebra. Taking some a in A, the "span" of a "should" go off forever and be a big line (if we want to pretend that we're doing linear algebra) but sometimes the line "twists around" and comes back, if a is finite order. Torsion meaning twisting, we get that the span of a is "twisted", so a has "torsion." That's my thought at least.
That’s what I’ve always imagined too. If you think about Cayley graphs, a torsion element will yield a cycle in the graph, and free groups, for example, are torsion free and so their Cayley graphs are trees - no cycles! So torsion really is like a ‘twisting’ or at least ‘cycling’ motion.
(I guess the converse isn’t true: if you pick a Z linearly dependent generating set you can get cycles even without torsion because there won’t be a unique path from the identity to any element that admits more than one expression as a word in the generators.)
At 3:52 there seems to be a typo in the exact sequence which is being written.
But that turns out to be correct because, Z/2Z (x) Z/2Z is isomorphic to Z/2Z
Thank you, king! Great lectures.
Very nice lecture
Thank you, sir.
Wow just yesterday learn what is homologycal algebra
Thanks!!!
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Conjecture: It is okay to love another human being you've never met. Let us assume that we are allowed to employ the axiom of unconditional love choice. Although this may seem to render the statement a tautology, this is not so obvious. Hint: Recall that we take the definition of "okay" to mean whatever is advantageous for the group as opposed to a specific individual in that group.