wonderful explanation! inverse limits came up in some stuff on symmetric polynomials I was reading and I've never come across the concept before, thanks!
Dear Dr. Rambow, I have enjoyed your RUclips videos. I hold a PhD in electrical engineering, and I am a mathematics autodidact. As such, my principal mathematical learning is through reading. One thing that seems to appear “in passing” in many contexts is the topic(s) of Inverse Limit and Direct Limit. I understand the basic definitions, but I have no intuitive (or other) practical understanding of the concept. I would greatly appreciate any suggestions of material (papers, books, tutorials, etc.) that I can study from. The use of category theory/language is also inconvenient as I am not fully versed in CT. I’d love to understand the construction of simple concrete cases. Even the Vietoris Solenoid, which seems simple to define, lacks any numerical example. Materials that I have found seem quite cryptic or assume the reader already understands. For example, (I believe) a point in the Solenoid is a sequence (net?) {x_n} such that x_n=k_(n+1) x_(n+1) mod 1 The Solenoid is the set of such sequences. Is this correct? Thank you in advance, Gary Centerport, NY
Hi Gary. You're very right that some textbooks can be very cryptic or terse. There are many good Abstract Algebra books that give a more detailed description of these constructions, such as Dummit and Foote's "Abstract Algebra" and Jacobson's "Basic Algebra". If you want to learn more category theory, Saunders Mac Lane's "Categories for the Working Mathematician" isn't bad. One of the main reasons mathematicians use this construction is to help demystify complicated or unwieldy structures. Some of the objects I mentioned in the video are unwieldy just because they are uncountably infinite. In the case of your solenoid they are unwieldy because it is analytically intricate. By imagining these objects as similar objects that progressively change they can be better understood. Also the description of a complicated object via the use of these limits allows one to apply some of the techniques of category theory to help understand the object. As for your question about the solenoid, I haven't studied solenoids much, but I can say the "mod 1" you wrote doesn't make sense. Based on the link below, a point in the solenoid would be more like this. Let's go with the "classical" solenoid where the wrapping numbers are always 2; n_i = 2 for all i. Consider the unit circle, and it'll help to imagine it inside the complex plane. Pick a point z_1 on the unit circle. A point in the solenoid is the sequence created by repeatedly multiplying z_1 by 2; z_i = (z_1)^{2^{i-1}}. Naturally there would be an argument why this description is isomorphic as a topological group to the solenoid as described by wrapping tori inside of other tori. encyclopediaofmath.org/wiki/Solenoid I hope this helps.
wonderful explanation! inverse limits came up in some stuff on symmetric polynomials I was reading and I've never come across the concept before, thanks!
Please do more videos! This is great :)
Thank you so much for this video, really helps! :)
I'm glad.
@@bygradforgrad5787 Hopefully you could do a video on some basic topological properties of p-adic numbers or p-adic integers.
@@juanpaolosantos3816 I might just do that in the near future.
@@bygradforgrad5787 Alright, looking forward to it :)
Dear Dr. Rambow,
I have enjoyed your RUclips videos. I hold a PhD in electrical engineering, and I am a mathematics autodidact. As such, my principal mathematical learning is through reading. One thing that seems to appear “in passing” in many contexts is the topic(s) of Inverse Limit and Direct Limit. I understand the basic definitions, but I have no intuitive (or other) practical understanding of the concept. I would greatly appreciate any suggestions of material (papers, books, tutorials, etc.) that I can study from. The use of category theory/language is also inconvenient as I am not fully versed in CT. I’d love to understand the construction of simple concrete cases. Even the Vietoris Solenoid, which seems simple to define, lacks any numerical example. Materials that I have found seem quite cryptic or assume the reader already understands.
For example, (I believe) a point in the Solenoid is a sequence (net?) {x_n} such that x_n=k_(n+1) x_(n+1) mod 1 The Solenoid is the set of such sequences. Is this correct?
Thank you in advance,
Gary
Centerport, NY
Hi Gary.
You're very right that some textbooks can be very cryptic or terse. There are many good Abstract Algebra books that give a more detailed description of these constructions, such as Dummit and Foote's "Abstract Algebra" and Jacobson's "Basic Algebra". If you want to learn more category theory, Saunders Mac Lane's "Categories for the Working Mathematician" isn't bad.
One of the main reasons mathematicians use this construction is to help demystify complicated or unwieldy structures. Some of the objects I mentioned in the video are unwieldy just because they are uncountably infinite. In the case of your solenoid they are unwieldy because it is analytically intricate. By imagining these objects as similar objects that progressively change they can be better understood. Also the description of a complicated object via the use of these limits allows one to apply some of the techniques of category theory to help understand the object.
As for your question about the solenoid, I haven't studied solenoids much, but I can say the "mod 1" you wrote doesn't make sense. Based on the link below, a point in the solenoid would be more like this. Let's go with the "classical" solenoid where the wrapping numbers are always 2; n_i = 2 for all i. Consider the unit circle, and it'll help to imagine it inside the complex plane. Pick a point z_1 on the unit circle. A point in the solenoid is the sequence created by repeatedly multiplying z_1 by 2; z_i = (z_1)^{2^{i-1}}. Naturally there would be an argument why this description is isomorphic as a topological group to the solenoid as described by wrapping tori inside of other tori.
encyclopediaofmath.org/wiki/Solenoid
I hope this helps.