Abstract Algebra: The Definition of a Group [video] | Hacker News Reader

Abstract Algebra: The Definition of a Group [video]

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74 pointsespeed31 comments

These sorts of videos take a lot of time to prepare and someone has gone to some trouble here. Abstract Algebra is a massive subject though, with thorough treatments running into the many hundreds of pages. So it's too much to expect to get to grips with the subject from a few short videos such as these.

Recently I've been watching Benedict Gross' Abstract Algebra lectures, to see if I can learn anything about how he presents material (he's a famous number theorist and known for his expository style).

http://wayback.archive-it.org/3671/20150528171650/https://ww...

One neat trick there is that groups are introduced via symmetric groups and via orthogonal groups, which allows a lot of geometric (visual) intuition, in a subject which is otherwise just symbolic (by definition).

What amazes me quite a lot is the total dearth of video lectures on commutative algebra online. This is taught in many undergrad courses, and there are plenty of written lecture notes online, but almost nothing when it comes to video presentation.

I think a lot of people who set out with the best intentions just give up when they find out how enormous the subject is, and how hard it is to keep errors out of presentations.

I started thisismath.org as a naive college kid in 2011 because there weren't any good video lectures on advanced maths. I knew it was going to be difficult but I couldn't appreciate how much work would go into actually editing the videos at the quality that I wanted until I felt the true weight of the commitment. I was in over my head and after a few videos I confronted the reality that I didn't have the extra time to commit and I just trailed off.

I still think there's an important need for advanced math video resources, and we should commend anyone who puts in the effort.

Anything that can help make this subject more approachable is alright in my book. In my experience, the hardest part of Abstract Algebra isn't the abstract part of it. It's the lack of examples and clarity around how different named concepts behave. The proofs aren't terrible even except for the fact that they require a familiarity with number theory that many (at least I myself) don't have. I love the subject though. Really interesting.

For me, it has always been the terminology. The basic theorems are relatively easy to understand... once you can remember the vast number of algebra-specific definitions. The field also has a proclivity for umm "colorful" choices of names.

It's not unlike say Haskell, which IMO is pretty straightforward once you have a feel for all the terminology.

Like this from wikipedia on Monoid?

Suppose that S is a set and • is some binary operation S × S → S, then S with • is a monoid if it satisfies the following two axioms:

Associativity For all a, b and c in S, the equation (a • b) • c = a • (b • c) holds. Identity element There exists an element e in S such that for every element a in S, the equations e • a = a • e = a hold.

In other words, a monoid is a semigroup with an identity element. It can also be thought of as a magma with associativity and identity. The identity element of a monoid is unique.[1] A monoid in which each element has an inverse is a group.

It seems to me that the definition presented there is relatively free of jargon, except the "in other words" section which links it to other (linked) structures.

> except the "in other words" section which links it to other (linked) structures.

That's what I was getting at. I took a few classes in AA and its sibling topology, including at graduate level. Most of the time was spent defining things, because there's so many terms with very specific definitions. Only after maybe 2/3rds of a course do you really start getting into deeper material. In my brain, reading definitions was always a process like: read through, substitute definitions for jargon terms, read through, substitute... iterate until you hit a fixed point.

That's standard for math, but AA is particularly well known for its colorful choice of names. Some people enjoy it, but I was always more attracted to the analysis-family areas of math.

I've always found the other name for category theory amusing though: https://en.wikipedia.org/wiki/Abstract_nonsense

Fair enough! My biggest struggle taking AA was also remembering the definitions of everything. There are so many things all tied together.

I think I only finally remembered what a monoid was when I realized it was a category with one object ("mono-"). Or that a groupoid is a category where every arrow is an isomorphism, so then a category which is both a monoid and a groupoid is a "group."

> Anything that can help make this subject more approachable is alright in my book.

Alright? Yeah, why wouldn't it be alright?

Lack of examples? What? Algebra is literally overflowing with examples. Algebraic structures generalize concrete structures you've probably already seen elsewhere in mathematics: from arithmetic, addition and multiplication, as the video explained, from discrete math, you can write down lots of interesting, finite examples, symmetry groups of geometric figures generate lots of interesting finite and infinite (Lie) groups, 1- and 2-dimensional wallpaper patterns are classic examples, etc, etc. In comparison, for example, it's quite non-trivial to describe a single interesting 4-manifold.

I think the issue is that people try to jump in abstract algebra too early, before they acquaint themselves with a variety of concrete examples.

It's even worse with category theory -- how one is supposed to understand why functors are useful before encountering them in a natural context is beyond me.

I like to generate examples by trying to add random things together, and then adding constraints. Forget, for a moment, words like "group" and "ring". At the risk of fatally oversimplifying things, I'll try to help.

What happens if I add water and... oil? Well, first of all, let's specify that if I add water to water, I still get water; and if I add oil to oil, I still get oil. Now, I could have water on top of the oil, oil on top of the water, water in the oil, and oil in the water. Let's collapse this so that now we have a collection of combinations that are water, oil, water and oil mixed, water on top of oil, and oil on top of water. If we can agree that we can "add" things in this collection together and get other things in the collection, then we have created an abstract algebra. The math is just a formalization that gives us extra power to reason about this collection of things.

In a more computer sciencey case, what might be the outcome of "adding" two Twitter users? Does one follow the other? Do they both follow each other? Is there something else that happens? Monad was a buzzword for a while because it made questions like this solvable with some goodies like easy parallelization. What if 200 million accounts suddenly needed to be added together? It sure would be nice to have an idempotent, parallelizable way to do so. Monads were one way of doing that, and abstract algebra gives us mathematical methods to rigorously approach problems like this (and many that aren't like this one).

It might be worth it to check out algebird

Socratica was started by two Caltech grads: Michael Harrison (ex-Googler) and Kimberly Hatch Harrison (http://www.socratica.com/about.html).

The actress in the video is Liliana Castro (https://en.wikipedia.org/wiki/Liliana_Castro).

The videos are good but something is just off ( uncanny valley ?) about the way the person presents. It seems really fake :(

I wouldn't place it in the uncanny valley. It just sounds highly scripted and acted.

Another comment mentioned that she's an actress.

The presenter is an actress. It's effectively a monologue.

I saw the same thing... it was like she was trying to give sexy looks while also teaching me math. Rather distracting. Did you also notice that?

I suspect that's the point -- evoking a motivating and memory-enhancing dopamine shot -- a la the Van Wilder tutors.

Oh huh... I don't know if I like that, but if it works then I guess it works

Sounds like bogus. Being primed for math and being primed for sex feels like two completely different modes to me.

Thinking about sex primes the brain for analytical thinking: http://tedxtalks.ted.com/video/Mathematics-and-sex-Clio-Cres...

Too bad that's TEDx, and not a real TED talk.

I think the videos serve as an introduction to some basic concepts intended to bring some clarity for further in-depth study. The video is much like the "front matter" to a chapter in a book.

Of course, the _REAL_ learning work starts when the student suffers through countless problems and proofs. This is no replacement for that. One doesn't learn math by passively watching videos or listening to lectures. That's the way it is and that is the way it will always be. This just provides a clear 30,000 foot view of the topic, a little motivation and background and, yes, some eye candy (and that's OK too).

I've probably taken too many courses on this and related topics during my undergrad, going from basic group theory to category theory, representation theory, and algebraic geometry.

For the uninitiated, a great first book is Fraleigh's text - very down-to-earth and conversational. When you're done with that you can read Dummit/Foote and for those with serious gumption, Lang's classic text. Have to say, it was my favorite subject - clean, elegant, and abstract.

For some reason this reminds me of Twin Peaks. I was expecting a surreal, supernatural twist midway through.

Isn't is a bit sexist that they use a good looking actress instead of one of the creators doing it herself? I wonder why that is

I don't know. Why would it be or not be?

Or might it only "sort of but not exactly" be?

I've heard of things being said that any individual instance wouldn't necessarily be sexist, but the fact that it is particularly common is. Is that a relevant idea? I am not sure.

However, perhaps the question might not be best considered only in terms of sexism, but in terms of general views on attractiveness, and then relating that to sexism?

But then again, that could also not apply. I don't know. I'm not really making any claims here.

I figure that this probably isn't a particularly harmful thing in this instance, if it is at all, though it might be a useful thing as a starting point for thinking about a topic. But also, it could be useless for that also.

I don't know.

To me it feels forced. Idk, maybe we can just feel it when the presenter is reading a script. This feels a little... unnatural. Some youtube commentor said she's an "expression queen". I tend to agree, though I don't think they meant it in a negative way whereas I do. I guess that would be the appeal for some people but I am not used to learning stuff in such an expressive way.

This was good fun to watch. I didn't know much about group theory before this.

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