Thanks for the great visual explanations! @15:43 Since the graphs change based on the generators used, does this effect "well-defined"-ness when studying groups through Cayley graphs? Because the group is the same, but the graphs are changing. Therefore, do you think we can only study the relation between the group and the generators, rather than the group itself?
I am flattered by your words, as usual ;-) And I also deserve praise, as usual, as the topic is just visually appealing by default ;-) The Cayley graph depends in an essential way on the choice of generators. So yes, the Cayley graph is a way to study a group via a fixed set of generators. That is often sufficient, you can learn cool things this ways, but maybe not perfect. Well, life is not perfect I guess ;-) But there is still something you can do if you are really interested in the group and not the group plus generators: The graph changes when the generators change, ok, not much we can do. But some properties of the Cayley graph do not depend on the generators, i.e. those are really properties of the group. Most famously, in geometric group theory you can associate a metric to the Cayley graph, making it a metric space. Different Cayley graphs give isometric spaces in an appropriate sense, so this metric is a property of the group and not of the presentation of it. That is, imho, quite cool! But was a bit too much for the video; the video is already long, and arguably too long ;-), anyway.
Thank you for the great explanation! Both categories and Cayley graphs are directed graphs; I wonder if there's any connection? Could this allow you to view groups as categories in another way than as a one object category?
Thanks for the feedback, and good observation! In this case I would say that its a coincidence: many things are encoded by directed graphs. In some sense too many thing ;-), they are not directly related in general. This however comes close to what you are looking for: en.wikipedia.org/wiki/Groupoid
Thanks man, you are truly a gifted educator and expositor
I am trying my best, but I am far from perfect. Your kind words are still a very welcome encouragement, thank you.
Thanks for the great visual explanations! @15:43 Since the graphs change based on the generators used, does this effect "well-defined"-ness when studying groups through Cayley graphs? Because the group is the same, but the graphs are changing. Therefore, do you think we can only study the relation between the group and the generators, rather than the group itself?
I am flattered by your words, as usual ;-) And I also deserve praise, as usual, as the topic is just visually appealing by default ;-)
The Cayley graph depends in an essential way on the choice of generators. So yes, the Cayley graph is a way to study a group via a fixed set of generators. That is often sufficient, you can learn cool things this ways, but maybe not perfect.
Well, life is not perfect I guess ;-) But there is still something you can do if you are really interested in the group and not the group plus generators: The graph changes when the generators change, ok, not much we can do. But some properties of the Cayley graph do not depend on the generators, i.e. those are really properties of the group.
Most famously, in geometric group theory you can associate a metric to the Cayley graph, making it a metric space. Different Cayley graphs give isometric spaces in an appropriate sense, so this metric is a property of the group and not of the presentation of it. That is, imho, quite cool! But was a bit too much for the video; the video is already long, and arguably too long ;-), anyway.
Great video. Thanks 😊
Glad that you liked it - the video and most importantly Cayley graphs, of course.
It is also a nice topic, which makes my life easy ;-)
@@VisualMath very cool stuff. I find I can intuit graphs way better than groups, so I'd much rather deal with graphs!
Same here ;-)
Thank you for the great explanation! Both categories and Cayley graphs are directed graphs; I wonder if there's any connection? Could this allow you to view groups as categories in another way than as a one object category?
Thanks for the feedback, and good observation!
In this case I would say that its a coincidence: many things are encoded by directed graphs. In some sense too many thing ;-), they are not directly related in general.
This however comes close to what you are looking for: en.wikipedia.org/wiki/Groupoid
very good video!
Thanks you for your kind feedback. Cayley graphs are very nice, so not sure whether I deserve your praise: the script for the video "wrote itself" ;-)
great video 👍👍
Thank you, I am glad that you liked the video. Cayley graphs are great, so I had an easy life ;-)