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Thread: Lessons

  1. #1 Lessons 
    Moderator Moderator AlexP's Avatar
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    I think the lessons that some of you wicked smart mathematics types have been doing (algebraic number theory, category theory) are awesome. It's great that you're willing to take your time to teach others. I understand that it takes time and effort, and I understand if your current threads take precedence, but there are a few things I wouldn't mind seeing lessons on, if any of you are up to it. For now I'm mainly interested in set theory, group theory, and maybe some basic topology stuff. I have high hopes of really getting into QM even without the schooling for it, and I know that I need to work on my mathematics basis for it. Good quality free resources (about the only type I can afford) aren't easy to come by, so it would be awesome if I could learn a bit here. Much thanks to anyone who steps up to the challenge.


    "There is a kind of lazy pleasure in useless and out-of-the-way erudition." -Jorge Luis Borges
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  3. #2  
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    Yeah, Chemboy, I am up for it. I have tried to do some of the stuff on your list here, but the obvious place to start would be some elementary set theory. then groups, then vector spaces, then tensor spaces, then topological spaces then manifolds, then, ..

    Obviously, I know nothing about QM, but between us (we wicked math whizzos!?) could possibly cook up some sort of tutorial tread?

    I have no intention of doing it single handed, so ask, for example, serpicojr, river_rat, bit4bit and any others if they have an interest.

    It might be fun, given certain ground rules........


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    Forum Professor serpicojr's Avatar
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    I only know the most basic set theory, enough to get by, but I'd be happy to contribute to discussions on group theory and topology.
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  5. #4  
    Moderator Moderator AlexP's Avatar
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    Glad you're interested, Guitarist. The sequence you listed in your post looks good... I've had some experience with vector spaces, but nothing else you have there. I think I'll see who else would like to get in on this and then try to organize the first thread.

    Quote Originally Posted by serpicojr
    I only know the most basic set theory, enough to get by, but I'd be happy to contribute to discussions on group theory and topology.
    I was thinking one would need set theory before they learned group theory, though maybe it's not as essential as I thought. Glad you're interested.

    Maybe I'll work on compiling a list of the mathematics areas that are more essential to QM...though of course I don't mind learning anything unrelated either. I'm interested in the things I mentioned (set theory, group theory, topology) in any case, so it'll all be good.
    "There is a kind of lazy pleasure in useless and out-of-the-way erudition." -Jorge Luis Borges
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  6. #5  
    Forum Professor serpicojr's Avatar
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    Set theory means all sorts of different things to different people. There's the basic set theory that every working mathematician needs to know. There's axiomatic set theory which is the logical foundation for mathematics. And then there's set theory the discipline, which is studying and classifying sets according to various criteria: how big they are, how they're defined, how they behave under different axioms of set theory, etc.

    You certainly don't mean this third sense--you don't need this to do anything you're talking about. You may mean the second sense, but again, you don't really need this to do what you're talking about. Mostly, it's just good to be aware that people have sat down and made sure that everything we want to do with sets in the many branches of math is kosher.

    So this leaves the first sense which, yes, you really need to know. But the first sense only covers so much, and people often just pick this stuff up by studying other branches of math--group theory, vector spaces, and topology, for example. However, many textbooks start with an introductory section which covers the set theory you should know.

    What topics do you need to know? Basic set relations and operations--subsets, intersections, unions, complements, and the various operations that can be built up out of these. You also need to know how these things interact with one another, for example De Morgan's laws. You should know various constructions: power sets, set exponentiation, products, disjoint unions. A basic understanding of cardinality is important: What does it mean for two sets to have the same size? How big is the union of a bunch of sets? How big is the power set? Understanding ordinals is also important, but less so. At some point in algebra and topology, you'll be interested in using the Axiom of Choice, so knowing a little bit about this is important. And it's important to understand, on some level, the difference between actual sets and mere collections of objects.
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  7. #6  
    Moderator Moderator AlexP's Avatar
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    Quote Originally Posted by serpicojr
    What topics do you need to know? Basic set relations and operations--subsets, intersections, unions, complements, and the various operations that can be built up out of these. You also need to know how these things interact with one another, for example De Morgan's laws. You should know various constructions: power sets, set exponentiation, products, disjoint unions. A basic understanding of cardinality is important: What does it mean for two sets to have the same size? How big is the union of a bunch of sets? How big is the power set? Understanding ordinals is also important, but less so. At some point in algebra and topology, you'll be interested in using the Axiom of Choice, so knowing a little bit about this is important. And it's important to understand, on some level, the difference between actual sets and mere collections of objects.
    I'm interested in all three senses, but I guess the things above I should, and would like to, learn first. Is there someone who would be willing to start a thread on set theory in that first sense?
    "There is a kind of lazy pleasure in useless and out-of-the-way erudition." -Jorge Luis Borges
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  8. #7  
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    I think I can do a thread on group theory. In fact, not just groups but also rings and fields – that is to say, abstract algebra. If you’re ready for it, let me know.
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  9. #8  
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    For what it's worth, I have already tried to run tutorials here on group theory, vector spaces and topology, which I can try and dig out if anyone is interested.

    It might save a bit of effort......
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  10. #9  
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    Well, I have another idea. I’ve been reading up on algebraic topology recently – so why don’t I start a primer on introductory algebraic topology? :P It will also help me consolidate what I’m learning.
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  11. #10  
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    Well, it might be fun, but obviously not quite what Chemboy is looking for.

    I am willing to skate over the surface of set theory if anyone wants, but like most here, I suspect, most of what I know came by a process of "osmosis" when studying structured sets.

    We have an unwelcome house-guest tonight, it will be good to have something to retreat to.......
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  12. #11  
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    I’m still willing to do groups. Since you suggest having it straight after sets, I’ll wait for you to do your sets first. :P

    Once the basics of groups have been done, rings and fields (as well as more advanced group theory) can be done in tandem with vector spaces. That’s my suggestion.

    Meanwhile, I’d still like do my algebraic topology. If anyone feels it’s too heavy going, they can just ignore it. 8)
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  13. #12  
    Forum Masters Degree bit4bit's Avatar
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    From what I've heard, a proper mathematical understanding of quantum mechanics uses alot of linear algebra, and PDE's.
    Chance favours the prepared mind.
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  14. #13  
    Moderator Moderator AlexP's Avatar
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    Quote Originally Posted by bit4bit
    From what I've heard, a proper mathematical understanding of quantum mechanics uses alot of linear algebra, and PDE's.
    Yeah, I'm aware of that. I'll be taking Calculus III in the fall, so by then I'll probably be in a position to study differential equations on my own, and from there on to PDEs. Unfortunately differential equations isn't required for my course of study, and I don't know if I'd want to take it on in real course form if I don't really have to. I know of a website with good linear and differential equations notes, and I've gotten through the linear.
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  15. #14  
    Forum Masters Degree bit4bit's Avatar
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    Cool, I'm just working through 'Calculus III' (which is the vector calculus right?), and also want to start learning differential equations afterwards (mainly cause I want to go into some kind of analysis). I recently got a book on them, since I should be finishing this calculus book soon.
    Chance favours the prepared mind.
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  16. #15  
    Moderator Moderator AlexP's Avatar
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    Quote Originally Posted by bit4bit
    Cool, I'm just working through 'Calculus III' (which is the vector calculus right?), and also want to start learning differential equations afterwards (mainly cause I want to go into some kind of analysis). I recently got a book on them, since I should be finishing this calculus book soon.
    I know Calc III as multivariable calculus, but basically, yeah.
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