This is so fascinating and beautiful! Hypergraphs also fall under this definition and possibly many other useful structures. Can't wait to see how algebraic topology ties into all these
Answer to question at 9:30: a full k-simplical complex has k+1 over r+1 simplices of dimension r. Prove uses Pascal's triangle / binomial coefficients. Hope, this is correct.
Professor, i was waiting this for long time. Awesome explanation.When are you planning to finish uploading all other video lectures which explains other parts of your text book?. Thank you very much for these initial videos.
thank you, but i think this project is on hold for awhile. it did not get as much play as i hoped. am going to pivot back to calculus materials for the next year i think...
@@prof-g maybe it takes a little while for word to spread? Given the quality of the videos and the engaging presentation, I'm sure these will permeate. Build it, they will come..
What a shame! I understand, but I just found this yesterday and it's already been so helpful for my thesis prep. I'll be on the lookout in case you pick this up again, Dr. G!
Many thanks for your informative and perfectly designed contributions to maths. Only one formal question remains: Why is the full/hollow k-simplex at 10:12 not named full/hollow simplical complex? Or did I misunderstood something? Thanks for your reply.
Why are complexes defined to have all the n-1 dimensional faces of each n-simplex? Wouldn't it be computationally simpler just to put the largest n-simplex in the set, and not worry about the n-1 dimensional faces of that n-simplex (and their n-2 dimensional faces, and so on)? The result would be the same, but you no longer have to store as much data.
storage is a very interesting issue. as you note, if you have a full n-simplex, all you need to store is the set of vertices & you can reconstruct the rest. for more interesting complexes, storage is not so simple.
I find the graph transitioning in 10:00 quite beautiful however didn't you forget to add the triangle faces on the final graph? I think this could be just a graphing approach because there would be lots of simplices that aren't the full k+1 one that weren't graphed there
not 100% sure i'm understanding your question, but let me take a guess. i rendered out the big hollow k-simplex with a lot of transparency, so it might look like the outermost faces are missing; they are not, but merely have some translucency, so you can see that there is a lot more on the inside. there is no "outermost" faces -- all faces are on the boundary. it's problematic to illustrate that in 3-d for a higher dimensional structure.
@@prof-g As a PhD Theoretical Physicist (retired), I can say your videos, if viewed by college students- especially graduate students in applied mathematics (including physicists), would accelerate their understanding of critical mathemtics. And, you are welcome.
Great video, beautifully illustrated. Looking forward to the next ones.
This is so fascinating and beautiful! Hypergraphs also fall under this definition and possibly many other useful structures. Can't wait to see how algebraic topology ties into all these
Answer to question at 9:30: a full k-simplical complex has k+1 over r+1 simplices of dimension r. Prove uses Pascal's triangle / binomial coefficients. Hope, this is correct.
Professor, i was waiting this for long time. Awesome explanation.When are you planning to finish uploading all other video lectures which explains other parts of your text book?. Thank you very much for these initial videos.
thank you, but i think this project is on hold for awhile. it did not get as much play as i hoped. am going to pivot back to calculus materials for the next year i think...
@@prof-g maybe it takes a little while for word to spread? Given the quality of the videos and the engaging presentation, I'm sure these will permeate. Build it, they will come..
What a shame! I understand, but I just found this yesterday and it's already been so helpful for my thesis prep. I'll be on the lookout in case you pick this up again, Dr. G!
Great video with a great explanation ! Is this illustraded with after effects ?
no, i use 3-d tools, including c4d
Many thanks for your informative and perfectly designed contributions to maths. Only one formal question remains: Why is the full/hollow k-simplex at 10:12 not named full/hollow simplical complex? Or did I misunderstood something? Thanks for your reply.
Why are complexes defined to have all the n-1 dimensional faces of each n-simplex? Wouldn't it be computationally simpler just to put the largest n-simplex in the set, and not worry about the n-1 dimensional faces of that n-simplex (and their n-2 dimensional faces, and so on)? The result would be the same, but you no longer have to store as much data.
I think you can but they are included in the definition for completeness (or rather, closeness).
storage is a very interesting issue. as you note, if you have a full n-simplex, all you need to store is the set of vertices & you can reconstruct the rest. for more interesting complexes, storage is not so simple.
I find the graph transitioning in 10:00 quite beautiful however didn't you forget to add the triangle faces on the final graph? I think this could be just a graphing approach because there would be lots of simplices that aren't the full k+1 one that weren't graphed there
not 100% sure i'm understanding your question, but let me take a guess. i rendered out the big hollow k-simplex with a lot of transparency, so it might look like the outermost faces are missing; they are not, but merely have some translucency, so you can see that there is a lot more on the inside. there is no "outermost" faces -- all faces are on the boundary. it's problematic to illustrate that in 3-d for a higher dimensional structure.
Very interesting!
A first rate video!
thank you so much!
@@prof-g As a PhD Theoretical Physicist (retired), I can say your videos, if viewed by college students- especially graduate students in applied mathematics (including physicists), would accelerate their understanding of critical mathemtics. And, you are welcome.