7 ms·

What Is Abstract Algebra? [video]

192 points by espeed 10y ago

gragas 10y ago
Here are fantastic Algebra lectures (Abstract Algebra, Fall 2003, Harvard) by my favorite lecturer Benedict Gross: https://www.youtube.com/playlist?list=PLelIK3uylPMGzHBuR3hLMHrYfMqWWsmx5 https://www.youtube.com/playlist?list=PLelIK3uylPMGzHBuR3hLM...
- stablemap 10y ago
  There's a link on the sidebar of Gross's page [1] to an old version of the course site; this lets you download video, audio, notes, and homework. Even if you know everything in Artin's book the lectures are rarely boring. [1] http://www.math.harvard.edu/~gross/ http://www.math.harvard.edu/~gross/
  - joebergeron 10y ago
    Do you mean Michael Artin's Algebra? Just took his Algebra I (18.701) at MIT last semester. An incredibly challenging class, but he's a really excellent lecturer and a very likeable guy. Still really sharp.
    - stablemap 10y ago
      Yes, thanks -- I should have made clear that this is what Gross used. Wonderful book.
- tptacek 10y ago
  I'm one lecture in and this is already pretty fantastic. Thanks!
adamnemecek 10y ago
If you are curious how does Abstract Algebra relate to software engineering, this book might be of your interest https://smile.amazon.com/Elements-Programming-Alexander-Stepanov/dp/032163537X https://smile.amazon.com/Elements-Programming-Alexander-Step... The main selling point of this approach is that it can cut out a lot of code In this talk, Sean Parent, at that time working on Adobe Photoshop, estimated that the PS codebase could be reduced from 3,000,000 LOC to 30,000 LOC (=100x!!) if they followed ideas from the book https://www.youtube.com/watch?v=4moyKUHApq4&t=10s https://www.youtube.com/watch?v=4moyKUHApq4&t=10s Another point of his is that the explosion of written code we are seeing isn't sustainable and that so much of this code is algorithms or data structures with overlapping functionalities. As the codebases grow, and these functionalities diverge even further, pulling the reigns in on the chaos becomes gradually impossible. Bjarne Stroustrup (aka the C++ OG) gave this book five stars on Amazon (in what is his one and only Amazon product review lol). https://smile.amazon.com/review/R1MG7U1LR7FK6/ https://smile.amazon.com/review/R1MG7U1LR7FK6/
- RossBencina 10y ago
  I have not yet read "Elements of Programming," but there is another book "From Mathematics to Generic Programming," also co-authored by Alexander Stepanov, which links Abstract Algebra to program design. I found the latter very approachable: https://smile.amazon.com/Mathematics-Generic-Programming-Alexander-Stepanov/dp/0321942043 https://smile.amazon.com/Mathematics-Generic-Programming-Ale...
  - stillworks 10y ago
    I always had a strong belief that principles of Group Theory must correlate into writing type systems. Just looked at the ToC of this book and there is an entire chapter on Groups in that book ! Not only Groups but also goes into Rings as well... Must read this book.
    - RossBencina 10y ago
      In a Group, every element has an inverse element. If you drop this property, you get a Monoid.[0] Functions under function composition form a Monoid (or Monoidal Category). Brian Beckman did a nice intro video to this at MS Channel 9 a few years back.[1] [0] https://en.wikipedia.org/wiki/Monoid https://en.wikipedia.org/wiki/Monoid [1] "Brian Beckman: Don't fear the Monad" https://www.youtube.com/watch?v=ZhuHCtR3xq8 https://www.youtube.com/watch?v=ZhuHCtR3xq8
    - tel 10y ago
      Groupoids, a generalization of groups, factor heavily into type theory!
    - empath75 10y ago
      I think what you want is Category Theory.
  - charlieflowers 10y ago
    Question -- clearly, when you look at Haskell, you see abstract algebra. Were Stepanov's ideas big influencers of Haskell, or vice versa, or neither?
netvarun 10y ago
This was previously posted slightly over a year ago and had some good discussion - https://news.ycombinator.com/item?id=10686610 https://news.ycombinator.com/item?id=10686610
joe563323 10y ago
The common trend in educational videos is very interesting. The videos that are made approximately from 2-10 minute long with animation and pictures are very easy to follow than to watch hour long stanford or mit live class room recordings.
- andrepd 10y ago
  Often, though, hour long lectures is the only way to go. You just can't learn much in 10 minutes with an animated video. Sometimes you just got to delve in.
  - empath75 10y ago
    I think watching something like this before watching a deep dive or reading a text book could be useful, just to give you kind of a lay of the land and some intuition.
  - dantheman 10y ago
    I disagree very rarely does one explanation take that long. It can normally be built up by lots of little videos. Once someone has completely 10 videos on the basics you can do slightly longer ones that span the concepts and explains tradeoffs between techniques.
- xemdetia 10y ago
  I feel that the hidden trick to the live classroom recordings is to watch at 2x-4x speed and rewind and go back to 1x the moment you don't understand something or if you are explicitly following a worked example (even then I usually cruise at 2x). If you have pause you can just work at a different pace.
  - amelius 10y ago
    If you don't mind the chipmunk-voices, that could work :)
    - soberhoff 10y ago
deleted 10y ago
[deleted]
espeed 10y ago
Also see the "Modern Algebra - Visual Group Theory" online course and video lectures by Prof Matthew Macaule at Clemson: Course material, videos, and slides: http://www.math.clemson.edu/~macaule/classes/m16_math4120/index.html http://www.math.clemson.edu/~macaule/classes/m16_math4120/in... YouTube Playlist: https://www.youtube.com/watch?v=UwTQdOop-nU&list=PLCc9vhgj7wo7ghkzJCUaeAw1ceB1BOaXG https://www.youtube.com/watch?v=UwTQdOop-nU&list=PLCc9vhgj7w...
mgalka 10y ago
Great explainer. Though I'm still not sure what the relation is between abstract algebra and regular algebra, other than the application of higher dimensional equations.
- number-sequence 10y ago
  Abstract algebra is often about studying structure on sets. That usually takes the form of studying operator(s) with certain properties and the elements that they work with. Some early examples that students see are things like rotations/reflections of n-gons, modular arithmetic, polynomials, matrix spaces, and fields. It is different from regular algebra in the sense that often one starts with a set and operation one supposes to have a particular kind of property (perhaps it's a group, or perhaps it's a field, or perhaps it's a certain kind of field (maybe separable, or local, or ...)). Then from these assumptions one proves facts about the assumed algebraic structure. It's different in that often particular elements aren't of much interest, and it's not as tied down to a particular context.
nishs 10y ago
Évariste Galois, a major contributor to abstract algebra, had an interesting life. He had participated in the French Revolution of 1830; worked on mathematics while in jail; and died when he was 20 years old as a result of being shot in the abdomen in a duel. https://www.youtube.com/watch?v=pdYe4BKcm74 https://www.youtube.com/watch?v=pdYe4BKcm74 (Galois Theory - Math History) https://en.wikipedia.org/wiki/%C3%89variste_Galois https://en.wikipedia.org/wiki/%C3%89variste_Galois

192 points by espeed 10y ago

Here are fantastic Algebra lectures (Abstract Algebra, Fall 2003, Harvard) by my favorite lecturer Benedict Gross: https://www.youtube.com/playlist?list=PLelIK3uylPMGzHBuR3hLMHrYfMqWWsmx5 https://www.youtube.com/playlist?list=PLelIK3uylPMGzHBuR3hLM...

stablemap 10y ago
There's a link on the sidebar of Gross's page [1] to an old version of the course site; this lets you download video, audio, notes, and homework. Even if you know everything in Artin's book the lectures are rarely boring. [1] http://www.math.harvard.edu/~gross/ http://www.math.harvard.edu/~gross/
- joebergeron 10y ago
  Do you mean Michael Artin's Algebra? Just took his Algebra I (18.701) at MIT last semester. An incredibly challenging class, but he's a really excellent lecturer and a very likeable guy. Still really sharp.
  - stablemap 10y ago
    Yes, thanks -- I should have made clear that this is what Gross used. Wonderful book.
tptacek 10y ago
I'm one lecture in and this is already pretty fantastic. Thanks!

adamnemecek 10y ago

If you are curious how does Abstract Algebra relate to software engineering, this book might be of your interest https://smile.amazon.com/Elements-Programming-Alexander-Stepanov/dp/032163537X https://smile.amazon.com/Elements-Programming-Alexander-Step... The main selling point of this approach is that it can cut out a lot of code In this talk, Sean Parent, at that time working on Adobe Photoshop, estimated that the PS codebase could be reduced from 3,000,000 LOC to 30,000 LOC (=100x!!) if they followed ideas from the book https://www.youtube.com/watch?v=4moyKUHApq4&t=10s https://www.youtube.com/watch?v=4moyKUHApq4&t=10s Another point of his is that the explosion of written code we are seeing isn't sustainable and that so much of this code is algorithms or data structures with overlapping functionalities. As the codebases grow, and these functionalities diverge even further, pulling the reigns in on the chaos becomes gradually impossible. Bjarne Stroustrup (aka the C++ OG) gave this book five stars on Amazon (in what is his one and only Amazon product review lol). https://smile.amazon.com/review/R1MG7U1LR7FK6/ https://smile.amazon.com/review/R1MG7U1LR7FK6/

RossBencina 10y ago
I have not yet read "Elements of Programming," but there is another book "From Mathematics to Generic Programming," also co-authored by Alexander Stepanov, which links Abstract Algebra to program design. I found the latter very approachable: https://smile.amazon.com/Mathematics-Generic-Programming-Alexander-Stepanov/dp/0321942043 https://smile.amazon.com/Mathematics-Generic-Programming-Ale...
- stillworks 10y ago
  I always had a strong belief that principles of Group Theory must correlate into writing type systems. Just looked at the ToC of this book and there is an entire chapter on Groups in that book ! Not only Groups but also goes into Rings as well... Must read this book.
  - RossBencina 10y ago
    In a Group, every element has an inverse element. If you drop this property, you get a Monoid.[0] Functions under function composition form a Monoid (or Monoidal Category). Brian Beckman did a nice intro video to this at MS Channel 9 a few years back.[1] [0] https://en.wikipedia.org/wiki/Monoid https://en.wikipedia.org/wiki/Monoid [1] "Brian Beckman: Don't fear the Monad" https://www.youtube.com/watch?v=ZhuHCtR3xq8 https://www.youtube.com/watch?v=ZhuHCtR3xq8
  - tel 10y ago
    Groupoids, a generalization of groups, factor heavily into type theory!
  - empath75 10y ago
    I think what you want is Category Theory.
- charlieflowers 10y ago
  Question -- clearly, when you look at Haskell, you see abstract algebra. Were Stepanov's ideas big influencers of Haskell, or vice versa, or neither?

netvarun 10y ago

This was previously posted slightly over a year ago and had some good discussion - https://news.ycombinator.com/item?id=10686610 https://news.ycombinator.com/item?id=10686610

joe563323 10y ago

The common trend in educational videos is very interesting. The videos that are made approximately from 2-10 minute long with animation and pictures are very easy to follow than to watch hour long stanford or mit live class room recordings.

andrepd 10y ago
Often, though, hour long lectures is the only way to go. You just can't learn much in 10 minutes with an animated video. Sometimes you just got to delve in.
- empath75 10y ago
  I think watching something like this before watching a deep dive or reading a text book could be useful, just to give you kind of a lay of the land and some intuition.
- dantheman 10y ago
  I disagree very rarely does one explanation take that long. It can normally be built up by lots of little videos. Once someone has completely 10 videos on the basics you can do slightly longer ones that span the concepts and explains tradeoffs between techniques.
xemdetia 10y ago
I feel that the hidden trick to the live classroom recordings is to watch at 2x-4x speed and rewind and go back to 1x the moment you don't understand something or if you are explicitly following a worked example (even then I usually cruise at 2x). If you have pause you can just work at a different pace.
- amelius 10y ago
  If you don't mind the chipmunk-voices, that could work :)
  - soberhoff 10y ago

deleted 10y ago

[deleted]

espeed 10y ago

Also see the "Modern Algebra - Visual Group Theory" online course and video lectures by Prof Matthew Macaule at Clemson: Course material, videos, and slides: http://www.math.clemson.edu/~macaule/classes/m16_math4120/index.html http://www.math.clemson.edu/~macaule/classes/m16_math4120/in... YouTube Playlist: https://www.youtube.com/watch?v=UwTQdOop-nU&list=PLCc9vhgj7wo7ghkzJCUaeAw1ceB1BOaXG https://www.youtube.com/watch?v=UwTQdOop-nU&list=PLCc9vhgj7w...

mgalka 10y ago

Great explainer. Though I'm still not sure what the relation is between abstract algebra and regular algebra, other than the application of higher dimensional equations.

number-sequence 10y ago
Abstract algebra is often about studying structure on sets. That usually takes the form of studying operator(s) with certain properties and the elements that they work with. Some early examples that students see are things like rotations/reflections of n-gons, modular arithmetic, polynomials, matrix spaces, and fields. It is different from regular algebra in the sense that often one starts with a set and operation one supposes to have a particular kind of property (perhaps it's a group, or perhaps it's a field, or perhaps it's a certain kind of field (maybe separable, or local, or ...)). Then from these assumptions one proves facts about the assumed algebraic structure. It's different in that often particular elements aren't of much interest, and it's not as tied down to a particular context.

nishs 10y ago

Évariste Galois, a major contributor to abstract algebra, had an interesting life. He had participated in the French Revolution of 1830; worked on mathematics while in jail; and died when he was 20 years old as a result of being shot in the abdomen in a duel. https://www.youtube.com/watch?v=pdYe4BKcm74 https://www.youtube.com/watch?v=pdYe4BKcm74 (Galois Theory - Math History) https://en.wikipedia.org/wiki/%C3%89variste_Galois https://en.wikipedia.org/wiki/%C3%89variste_Galois