After all the political rabble rousing and talking about talking about talking about vitalism, I for one am thoroughly refreshed learning a bit of actually vital and beautiful piece of abstract math.
I found group isomorphisms so intriguing because they come so close to being an actual mathematical construct of empathy. No wonder that I had to learn about them from you, Rajeev.
Are they similar in concept to organic isomers? This is all beyond me. It's fun to realize that people you chat with have access to a world of symbolic logic that you will never the have access to. I barely made it through 300lvl differential equations....20 some yrs and several concussions ago. I know just enough to have some decent intuitions about complex systems.
A better example from organic chemistry might be the different structural diagrams of organic molecules tbh. If you are looking at Lewis structure, skeletal structure and Fischer projection of the same molecule, say, then the conceptual frameworks of each type of diagram are homomorphic groups. The molecule itself is just the “function” that can then be translated between those groups without losing fundamental structure.
(I am still new to this concept so @Rajeev, correct me if this isn’t a good example)
The concept of group (& ring & field) isomorphisms from algebra probably applies most directly to stereoisomers but not structural isomers from organic chemistry.
Stereoisomers different in how the molecule is arranged in space due to things like electric charge, light, heat in the environment – but the bond structure between elements is preserved. Structural isomers actually differ in the bond structures between constituent molecules; the only exceptions are enantiomers (mirrors) that have the exact same structure but opposite chirality.
In all of these examples, I talked about Abelian groups, which means all the elements are commutative. There are plenty of example of non-Abelian groups: in fact the dihedral group mentioned above (D3) about rotations/reflection about the equilateral triangle is one such example.
One could construct an homomorphisms among different non-Abelian groups that are kind of algebraic enantiomers. Or else, for example, certain groups contain subgroups that are 'mirror images' of each other within the larger group.
Riffing off of your Bezout's identify point, I think people would find it cool if you picked a pair of numbers which were not co-prime in order to show how the cosets end up matching up with multiples of the associated gcd. If you factor this map through a suitably chosen co-prime map, you can show how homomorphisms get chained and the isomorphism theorems that come up there. Also, you didn't need to show that the homomorphisms preserved the identity since that comes for free by showing that the map respects the operations.
Yup! You've probably heard of Smith normal form to decompose these types of integer quotient maps in higher-dimensional spaces. At some point, it becomes nearly impossible to get all the factors to be co-prime, so one must pull in the machinery you're referring to, which converts the group into a chain of cyclic groups.
I feel like I'd heard about it in passing but never actually looked into it before. It's amazing all the ways in which deeply theoretical questions (how to conceive of the structure of this quotient of a free module) can be answered by these computational matrix methods. The algorithms themselves are tedious, but when you read into the theory about why you take each step, it's like glimpsing heaven.
Quotients within mathematics as a whole are beautiful when they can be visualized. I love thinking about topological quotient constructions for some spaces like the torus and Klein bottle, but what's mind blowing is how there are theorems allowing you to *compute* the number of "holes" of different dimensions in spaces you can't even visualize as long as you know the space you're starting with and keep track of the identifications which end up producing the quotient space.
Super true, had not ever thought about it in the context of 'gaps' and 'discontinuities'. Remarkable how you can show that topologies that appear drastically dissimilar are essentially identical.
The quotient group from the originam homomorphism is identical to the image (which is Z in this case). That's the identification I'm referring to, though it may not be as clear as it could be in my wording. Each integer corresponds to a different 'line' in the 2D plane, and all the lines together make up the integer lattice.
There's no "usual" way, but here's an example that usually makes sense to people. Imagine a square and "identify" the opposing edges with each other, which is to say we want to make them the same as each other (glue them together). So you'd roll the paper until you get two sides to touch, at which point you have a cylinder with a tunnel going through it. From there, if you imagine this is really stretchy, you can actually unite the two end holes so that they feed into each other. From this, you get a donut shaped thing with a tunnel running through the donut's interior.
One of the reasons isomorphisms are so wonderful is that they allow you to look at the exact same thing in completely different ways. Same architecture, same operations, same scope – completely new context. Then, it becomes a game to dance between the different views and let each of them bring you closer to the elemental truths beneath them all.
Noooooo, not synecdoche!!! That movie still pops up in my head when I deal with the mind numbing drivel that you get from midwit leftists masquerading as geniuses. Transdisciplinarity my ass!
After all the political rabble rousing and talking about talking about talking about vitalism, I for one am thoroughly refreshed learning a bit of actually vital and beautiful piece of abstract math.
I found group isomorphisms so intriguing because they come so close to being an actual mathematical construct of empathy. No wonder that I had to learn about them from you, Rajeev.
Incredible, I had never even made the connection to empathy: same substance traced through differing frames.
Are they similar in concept to organic isomers? This is all beyond me. It's fun to realize that people you chat with have access to a world of symbolic logic that you will never the have access to. I barely made it through 300lvl differential equations....20 some yrs and several concussions ago. I know just enough to have some decent intuitions about complex systems.
A better example from organic chemistry might be the different structural diagrams of organic molecules tbh. If you are looking at Lewis structure, skeletal structure and Fischer projection of the same molecule, say, then the conceptual frameworks of each type of diagram are homomorphic groups. The molecule itself is just the “function” that can then be translated between those groups without losing fundamental structure.
(I am still new to this concept so @Rajeev, correct me if this isn’t a good example)
I like this.
The concept of group (& ring & field) isomorphisms from algebra probably applies most directly to stereoisomers but not structural isomers from organic chemistry.
Stereoisomers different in how the molecule is arranged in space due to things like electric charge, light, heat in the environment – but the bond structure between elements is preserved. Structural isomers actually differ in the bond structures between constituent molecules; the only exceptions are enantiomers (mirrors) that have the exact same structure but opposite chirality.
In all of these examples, I talked about Abelian groups, which means all the elements are commutative. There are plenty of example of non-Abelian groups: in fact the dihedral group mentioned above (D3) about rotations/reflection about the equilateral triangle is one such example.
One could construct an homomorphisms among different non-Abelian groups that are kind of algebraic enantiomers. Or else, for example, certain groups contain subgroups that are 'mirror images' of each other within the larger group.
I was moved by your vision to teach this to your children someday.
This is amazing 🥲
Riffing off of your Bezout's identify point, I think people would find it cool if you picked a pair of numbers which were not co-prime in order to show how the cosets end up matching up with multiples of the associated gcd. If you factor this map through a suitably chosen co-prime map, you can show how homomorphisms get chained and the isomorphism theorems that come up there. Also, you didn't need to show that the homomorphisms preserved the identity since that comes for free by showing that the map respects the operations.
Yup! You've probably heard of Smith normal form to decompose these types of integer quotient maps in higher-dimensional spaces. At some point, it becomes nearly impossible to get all the factors to be co-prime, so one must pull in the machinery you're referring to, which converts the group into a chain of cyclic groups.
https://math.stackexchange.com/questions/2760082/calculate-smith-normal-form-cyclic-group-decomposition
I never got far enough in my studies to do this stuff on a regular basis.
I feel like I'd heard about it in passing but never actually looked into it before. It's amazing all the ways in which deeply theoretical questions (how to conceive of the structure of this quotient of a free module) can be answered by these computational matrix methods. The algorithms themselves are tedious, but when you read into the theory about why you take each step, it's like glimpsing heaven.
Quotients within mathematics as a whole are beautiful when they can be visualized. I love thinking about topological quotient constructions for some spaces like the torus and Klein bottle, but what's mind blowing is how there are theorems allowing you to *compute* the number of "holes" of different dimensions in spaces you can't even visualize as long as you know the space you're starting with and keep track of the identifications which end up producing the quotient space.
Super true, had not ever thought about it in the context of 'gaps' and 'discontinuities'. Remarkable how you can show that topologies that appear drastically dissimilar are essentially identical.
to be clear a homomorphism preserves structure but it doesn’t preserve *all* structure. When you map 2D to 1D, you do lose a lot.
The quotient group from the originam homomorphism is identical to the image (which is Z in this case). That's the identification I'm referring to, though it may not be as clear as it could be in my wording. Each integer corresponds to a different 'line' in the 2D plane, and all the lines together make up the integer lattice.
There's no "usual" way, but here's an example that usually makes sense to people. Imagine a square and "identify" the opposing edges with each other, which is to say we want to make them the same as each other (glue them together). So you'd roll the paper until you get two sides to touch, at which point you have a cylinder with a tunnel going through it. From there, if you imagine this is really stretchy, you can actually unite the two end holes so that they feed into each other. From this, you get a donut shaped thing with a tunnel running through the donut's interior.
One of the reasons isomorphisms are so wonderful is that they allow you to look at the exact same thing in completely different ways. Same architecture, same operations, same scope – completely new context. Then, it becomes a game to dance between the different views and let each of them bring you closer to the elemental truths beneath them all.
Don't think I'm qualified to make sense of your questions, but I bet you could make some nice and kooky art out of your curiosity here.
Noooooo, not synecdoche!!! That movie still pops up in my head when I deal with the mind numbing drivel that you get from midwit leftists masquerading as geniuses. Transdisciplinarity my ass!
I promise you're not like them. You have humility and an open mind.