Mathematics General: Atiyah Edition

why is grothedieck bald

>>8449671
To keep the government out

>>8449662
do we have confirmation that Atiyahs proof is accepted or not?

The dude is getting old and his abstract seems to suggest he didn't need any new ideas to solve it. Seems strange that the proof would of been missed for this long.

Holy shit, he proved it in one page. I hope this is legit, but there are very few premises in his argument. It's really straightforward. Goddamn!

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Atiyah's a cool guy. He solves problems and doesn't afraid of anything!

>What is /sci/ reading?
Cohomology of Sheaves by Birger Iversen

>What problems are you working/stuck on?
I'm trying to construct a way to classify abelian categories using categories of modules as my tool. How to generalize this from small subcategories to cover general abelian categories is still a mystery.

>Progress on research/theses?
I have yet to recheck my arguments for the classification of small abelian categories, but it seems to be done.

>>8449662
Thank you OP.
This is inspiring.

>>8449779
how do you classify abelian categories use cats of modules?

Is this some sort of coherent sheaves type business?

>>8449662
>What is /sci/ reading?
Introduction to Representation Theory by Etingof
Complex Analysis by Dolbeault
Begriffsschrift by Frege

>What problems are you working/stuck on?
Finding good problems to work on... I'm pretty bored

>Progress on research/theses?
Not working on anything specific atm

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>>8449903
The small ones I managed to classify using the Mitchell embedding. As I mentioned, I need to recheck that everything is correct up to this point, but I do believe I have been able to use equivalences of module categories between their subcategories, and, later, between the embedded abelian categories. The problem is that it is still very restrictive, as there is the limiting assumption of the categories being small.

>>8449919
Ok so the Mitchell embedding says, there is an equivalence of cats between any small abelian cat and a full subcat of some cat of modules (that is compatible with kernals and cokernals i guess? )

Is this what you mean by classify?

I wasn't totally aware of this result but i guess it makes sense now because when ever someone is working with an abelian cat they seem to be implicitly using module theory.

I wish I could concentrate on the maths that I'm supposed to be doing for class. I keep fucking checking out books that are not related to my classes at all and spending hours and hours pouring over numb theory/group theory/ ab algebra stuff instead of doing my insanely boring calc/stats busy work.

UGGGGh, why are these classes so mind numbingly horrible. Why do I have to take classes, why can't I just live on a library sofa and read books!!! REEEEEEEEEE#E

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>tfw no math gf

I'm a freshman at University of Maryland College Park. Currently I'm taking MATH340 and reconsidering my decision in majoring in mathematics. The book we are using is Advanced Calculus of Several Variables. We are also supposed to "self-learn" a lot of stuff. I've taken linear before so I understand it but the pace is too fast for me to remember everything. It's been 2 months and we are already in Chapter 8. Is it normal for undergrads learn this shit this fast?

Chapter 1 is a brief incursion in some topological aspects. Chapter 2 directional derivatives, differentials. Ch3. Chain rule. Ch.4 Critical points. Ch. 5 MANIFOLDS (patches ?! ) and Lagrange multipliers (and this is around a bit over page 100!). Ch 6 Taylor's in one and Ch. 7 several variables. Ch 8 Classification of critical points. Part III begins with Newton's method and contraction mappings. Then goes to Multivariable mean theorem, Inverse and Implicit Mapping Theorem. Ch 4 (III) is Manifolds in Rn and finishes with higher derivatives. Part IV is Multiple Integrals, n-dimensional integrals, Riemman sums, Fubini's theorem, Change of Variables, Improper Integrals, Path Lenght and Line Integrals, Green's theorem, some applied problems, Line and Surface Integrals. Book end with Differential Forms, Stoke;s theorem, Classical Theorems of Vector Analysis, Closed and Exact Forms, Normed Vectors Spaces, Variational Calculus the Isoperimetric problem.

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>>8449977
By classifying the small ones I mean essentially this:
>take any two small abelian categories
>embed them in suitable module categories
>see if there is an equivalence between the module categories
Now, suppose such an equivalence exists.
>have the equivalence induce an equivalence between the "image" subcategories
>have that equivalence induce an equivalence between the categories we started with
I'll redo what I have done so far now. It should point out possible mistakes or give me certainty I'm not a total crank.

>>8449919
what if, given a large abelian cat you move to a small abelian sub cat and use the embedding there? Does some sort process where you work locally work?

REEEEEEEEEEEEEEE

If i want to learn MIT 6.00 what pre req math should i do?

>>8449999
ok so you are looking for A eq to B iff their embeddings are eq?

>>8449996
My third semester Multivariable calculus course covered pretty much what you listed except for vector calculus theorems (Green's, Stokes' and Divergence theorems). No differential forms and we skipped some stuff here and there such as Taylor's theorem, but plenty of normed vector spaces. I'd say it's alright as long as your professor doesn't expect you to be proficient at all of those at the same time, it depends on what kind of tests you get really.

>>8449996
would certainly be a bit much for a second year calc 3 class but 340 makes it sound like a third year class where that wouldn't be unreasonable

>>8449662
I think about Cartan lemma and Poincare lemma from homological point of view.

Poincare lemma seems to be a homotopy between id and zero "maps", and Cartan lemma... between Lie derivative and zero?

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>>8450003
I've been thinking about "covering" the category with these small categories, and maybe even have them all embedded into the same module category. It would be cool to have that done, but I haven't yet had time to check if it can be done. I also tried to define this equivalence by constructing a category of rings with different conditions, and say two abelian categories are equivalent if these categories are, but this became very messy. An example would be to have a ring of endomorphisms for every projective object, and then have the objects be sequences of rings indexed by the collection all such objects, with each ring being Morita equivalent to the ring of endomorphisms of its index P. This I abandoned pretty quickly because it seemed too complex to be the optimal solution to this problem. I may dig it up from its grave, though, if all other methods fail me.

>>8450011
I constructed a "retraction functor" to give me an eq for the embeddings if there is an eq for the whole categories. I would like it to be such that the whole categories of modules are equivalent to have possibly less equivalences between the original ones.

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Is there a better first introduction to algebraic topology than Matveev?

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>>8450196
I like Rotman's book. It is good for self studying.

>>8450196
hatcher bro

why do faggots feel the need to make a general for everything under the sun

you don't need a "math general" on a fucking math board

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>>8450223
>>8450196
thanks, i've looked at these but they're much longer than matveev (~80 pages), i think i'll stick with this for now as a real 'intro' among intro texts

>>8450253
have you seen what the other threads look like? this board is a shithole

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>>8450260
meant to reply to >>8450241

>>8449980
Don't let your classes quench that thirst. I'm telling you, if you focus on what fascinates you at the cost of grades, and REALLY commit to chasing curiosity, you will be leaps and bounds ahead of most others you meet. Get to the point where you can start your own research and it will set you apart far more than good grades. That kind of initiative speaks a lot to your character. You go, dude!

>>8449697
>>To keep the government out
kek, it is not false actually

Does someone have good references to understand Zernike polynomials ?
I'm not a mathematician, but I would like to understand it for microscopy applications.

Please never say mathematics is for autists/autistic.

When they begin to talk and joke, I feel alienated every time. I just can't. I can appreciate what they say and relate to it but that doesn't matter. I feel very uncomfortable since most autists there gradually stopped acting autistic, and I didn't. I mean I don't want to talk to them, I just wanted to fit in while not talking. That worked in the beginning. But that doesn't work any more. No one looked weird for being autistic before and now it seems that I do. Do I really need to talk to people. I don't want to. that's so sad.

>>8451269

Thanks for the encouragement, it actually means a lot to me.

>>8452309
I'm glad to have had a positive impact on you. If you maintain a solid attitude, I'm sure I will be reading your work some day.

>>8452309

You're taking advice from a huge autist who's most probably much more intelligent than you are.

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>>8452837
You are being mean to this guy! >>8451716

>>8449980
Are you at least in ACalc, like real analysis? Cause number theory and algebra are for girls.

>>8452841

He's previously admitted to having been diagnosed with autism. It's reality that's being mean, not me.

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>>8452881
To be honest, my aspie radar beeps at him, but that other guy asked not to call mathematicians autists, so apologize to him (not OHP)! RUDE!

>>8452890

I didn't call mathematicians autists, just OHP.

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>>8452911
And I don't even care, to be frank. There is nothing wrong with autism if the autist is functional enough to do things. Otherwise it is sad.

>>8452918

I didn't say there was anything wrong with being an autist, it's just that I think OHP seems to think that anyone would have been able to replicate what he's done, and I'm doubtful about that.

It looks to me like he's very optimistic and idealistic, and I want to give that anon (and others) proper context for what he has to say, and a large part of that context can be conveyed by saying he's an actual autist.

>>8450020
Your description is pretty much spot on, and I have seen formulations of both in terms of homotopies (well, chain homotopies, but it's just an infinitesimal homotopy in synthetic differential geometry anyways, so the intuition is "right.")

>>8452837
>>8452841
>>8452881
>>8452890
>>8452911
>>8452918
>>8452922

Mathematicians are not all autistic, and I understand that you weren't saying that. I do think there is a higher proportion of spectres ("people on the spectrum", laughing my ass off at this slang right now). I also admit to being pretty romantic about the potential I see in everyone, but I sincerely believe that the biggest barrier in accomplishing anything is your mindset. If you are constantly saying to yourself, "I can't do this, I am just not special," then you are right and wrong. Nobody is really all too special, but nearly anyone can accomplish goals if they have the determination. It's okay if you disagree, and it's probably healthy for that anon to get both sides of things and form his own opinion on his potential.

Have a good day guys! Gotta run to class.

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>>8452922
Nor did I claim you said so. It was to solidify a point that I think of autists as equals, so that any aitistic person reading my posts doesn't have a mental breakdown. You are right about that optimism, he has a lot of it. It is inspiring, too. That's one reason why I enjoy having a chat with him.

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>>8452926
The mindset really is a grand barrier. I've only recently got to the point that I consider it possible for me to actually reach some results on my own. Guess what, I have been actually making some progress now that I have decided to believe the nice words people have said to me about being good (in a layman's or a student's eyes). One reason has been that a talented guy has said to be interested in my projects, you. This is getting too sappy for me, so I'll leave it to this.

Have a nice class.

>>8452929

>It is inspiring, too.

I find it debilitating, mainly because I don't have it, but I won't put it against him and I wouldn't ask him to change.

Also

>aitistic

Ayy

>>8452926

I don't know how you think, and you don't know how I think, and neither of us knows how anyone else thinks, but when I try to I strongly feel (and in some sense, I can tell) that I just can't, either because I truly can't or because I don't want to, and either of those options leaves me hopeless.

I find it hard to think at all, it's all so confusing to me, and I get immensely distracted by anything too easily. I have no idea if there's any effective method on how to think in general and how to concentrate in particular, but anything I've tried until now (which I'll admit is not much) has basically been useless and I don't know what to do.

How am I supposed to go into research (what I'd like to do) if I don't know how to think effectively and I can't even ask questions to myself? It's not just that I won't be able to answer any of them so I don't bother, it's that even when I do I just get completely lost on my thoughts until I forget I was even trying to solve a problem.

And the worst thing is that probably nobody can help me, I'm on my fucking own like always. Sorry for the blog anons, but what can I do?

>>8452953
Interesting points my friend, I don't judge the "blogginess." As far as finding a way to clearly think and let your thoughts cohere, I am not sure what to say. I know that I have always had a propensity for producing developed thoughts, but I have been practicing metacognition lately and have come to realize that a big part of my thinking style involves how my brain produces a strong signal when I read something. To clarify what I mean, suppose I read something like "the derivative of f at x." My brain, quite rapidly, sends a few clear signals: the point (x,f(x)) on the curve, a tangent hyperplane at that point, but also perturbations in that tangent as it "vibrates" between different secants close to it. I also think about curvature, which sends other signals, and this little network of ideas proliferates in that split second. I think this sort of action is what allows me to connect new ideas to old ones, and really understand what is happening. I do it while I learn a new topic, and I do it when I think about old topics.

Maybe you can practice that sort of thing. Read some math you are comfortable with, but instead of just reading it, try to manually think about all of the things that relate to that idea for you. Maybe if you practice this sort of exercise, you will find it easier to keep clear lines of thought and produce those ever-important connections.

As you said, we don't know how the other thinks, but I don't want to just roll over if I can offer anything useful to you.

>>8452938
It makes me really happy to know that I have done a little bit of good for the mathematics community, even if only for one person. Have a nice class!

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>>8452953
Oops, I'm channeling ayylien messages! This seems to be where our opinions differ, though. I know I'm limited, but I don't those limitations and more talented people crush my spirit. Instead I use them for inspiration, they too are only humans.

>>8452968
If my work leads to anything, I will make sure you hear of it.

>>8452953
>I just can't, either because I truly can't or because I don't want to, and either of those options leaves me hopeless.
That's what you get when you have to rely on a shitty uni. I've been through that. That will be hard and painful no matter what. But you have to start working on what good unis do, read what that people say, disregard what's happening in your place and stop the bad influences as hard as you can. This will be really hard. But that's what you have to do to save the real scientist and human being in you.

I used to study in a real ill atmosphere (uni that was once good but not any more, and hence simulates everything for the sake of saving its status, profs that do bullshit, don't know their subject and fool students into going in the same bullshit, hatefulness around, genuine interest being SUPPRESSED, so on). I was disclosing one lie after another. The greatest lie they all shared was that what's happening is ok and usual for good places, and that they are a good place. I started investing on other places, reading blogs of people I knew, and what they said about this uni. I was shocked they were telling the same thing I felt all the time. Soon after that I quitted regretting nothing. I felt so good after that. Now I visit unofficially a very good place. It's been a year already and I still value every other person here, their attitude. I just know the price of a good education and a healthy academic atmosphere now. That was a tough lesson.

>>8453163
*really
*investigating

>>8453163

I'm not sure what you're trying to tell me. It's not like my uni doesn't have support for this kind of stuff, I could probably talk about it with my advisor and my professors, I just don't like talking about myself with people, much less so someone who's as busy as a prof. I'd rather not waste their time with my sob stories about how hard I have it considering basically fucking everyone around has it at least as hard.

>>8453176
he's not trying to tell you anything
it's just a faggy, whiny blogpost about his own life

>>8453176
Yes, maybe I just project my case on you, I just felt similar way because I didn't see the whole picture.

I just started my STEM degree and I want to understand math more deeply and develop my mathematical intuition, are there any lectures I can watch to help with that?
Right now I'm watching TTC's The Art and Craft of Mathematical Problem Solving (with Paul Zeitz, who wrote a fairly well-regarded book on problem solving) and I'm really enjoying it, I wondered if there's some more material similar to that.

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>>8449662
Can someone PLEASE give me an explanation on filtrations and adapted processes?

From my understanding we take a stochastic process [math] X_t [/math] and then every sigma algebra [math] \mathcal{F}_t [/math] we just add the sets [math]\{ \omega \in \Omega: X_t(\omega) = X_t, \ \forall s < t \} [/math]. In words; we add a set with all the [math]\omega[/math]'s such that, from our observations, one of the [math]\omega[/math]'s in that set is the real path of the process, and of course we generate a new sigma algebra out of that set.

I'm looking for an intuitive explanation that isn't just "filtration is all the information known at time t".

>>8449982
>Tfw you picked Microelectronic engineering as a major and 80% of it so far has been com sci tier computet theory classes and programming in C++, not even Assembly.

Kill me.

Working on a model rocket with a SAS I made to see if you can get away with only one gyroscope.

There has been a lot of doubt expressed by experts on whether Atiyah's proof holds any merit. Besides being a big name, it doesn't seem like there is much to suggest Atiyah finally proved it. Keep in mind this is one of the open problems that has a new proof of existence or non-existence every year. Not quite as bad as the Jacobian conjecture though.

Although, I am no expert. I only just understand a bit of KR-theory, only the basics.

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>>8453234
>There has been a lot of doubt expressed by experts on whether Atiyah's proof holds any merit.
where? the only discussion i've seen so far is on http://mathoverflow.net/questions/253577/atiyahs-paper-on-complex-structures-on-s6 which isn't much

>>8453348
>implying I meant online discussion
Not everything said happens online.

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>>8453506
?...
where did i imply you meant online?

>>8453510
Mathematicians tend to have conversations in real life, so it is not surprising that you cannot see a discussion online of what professionals are saying about it.

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>>8453513
i don't see why you keep mentioning the online

i only asked where

>>8453517
irl obviously

>>8452923
Yep, but I have no geometrical understanding of these and of de Rham cohomology itself.

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>>8453524
very helpful bud

>>8453525
One thing to consider is that, in synthetic differential geometry, the de Rham complex is basically the Moore complex of infinitesimal simplices in the space. It's a lot like infinitesimal homotopy type, which motivates the simple synthetic proof of de Rham's theorem.

>>8453527
Linked to your link was http://mathoverflow.net/questions/1973/is-there-a-complex-structure-on-the-6-sphere?rq=1 which supports exactly what that fellow was suggesting---skepticism. Didn't look very far did you?

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>>8453553
>http://mathoverflow.net/questions/253577/atiyahs-paper-on-complex-structures-on-s6
everything there is speculation

it's not even first-hand skepticism, just people reporting on other supposed experts being skeptical

there's not even a single remark about any actual mathematical issue

how far am i supposed to look into this literally nothing post?

>>8453565
Are you upset? Do you like KR-theory or something? How much do you know about it?

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>>8453568
>Are you upset?
no? what's with your weird attitude?

>Do you like KR-theory or something? How much do you know about it?
sure? you might as well be asking if i 'like' screwdrivers

>>8453574
How much do you know about KR-theory?

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>>8453576
not much, i understand what it is but have (relatively) more experience on the side of algebraic k-theory (pic related) than topological k-theory

>>8453574
KR-theory is a tool used to remove screws that any fool with half a brain can use, and has used in her life ???

>>8453590
its a tool, it doesn't matter if you 'like' it or not, it'll still solve the same problems

>>8453588
Well I leave an interesting and relevant exercise for you:
a) what is KR of a flag manifold?
b) a grassmannian?

>>8453594
i dunno about you, but i like math, so KR theory is very different from a screw driver.

>>8453634
how so?

>>8453530
Ah, not sure if that's what you mean, but seems like differential forms just act on "simplexes" by integration, and vice versa any linear form on simplexes is a kind of integration of diff form, diff form at each point is derived by substituting infinitesimal simplexes. Now it seems to me that not only de Rham cohomologies coincide with simplicial but their complexes are simply the same for good enough cases.

>>8453852
i don't like using screwdrivers.

>>8453891
*their cochain complexes

>>8449662

Hey /math/

I'm taking a course on discrete dynamical systems. It comes a point in which a Chaotic function f is defined by three properties:

1) f is transitive.
2) the set of periodic points of f is dense.
3) f is sensitive to initial conditions

It turns out that 1) and 2) => 3) the thing is that 1) and 2) can be seen as pure topological properties wich is not always the case for 3). So right now I'm looking for articles in wich 3) might be generalized a bit more in hopes to give a definition of chaotic function in a general topological space instead of a metric space.

Does anyone here have any insights or thoughts on this?

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>>8449662
Atiyah does it again the absolute madman

>>8453891
I'm not entirely certain about your last point, as the synthetic de Rham complex is the Moore complex of infinitesimal simplexes modded out by the simplexes with at least one contractible face (I think that was it), so there will be a difference for ugly spaces. It may be the case that they coincide for stuff like CW complexes. That sounds right.

I wish I could focus more on my assignments and what I need to do right now than the more interesting parts of math I want to study. For instance I'm taking analysis right now but I can't concentrate on the exercises because I just find it dull and uninteresting. There's no topological or even metric spaces, this just feels like proving convergence of sequences, one after another. I did a gap year where I learned all kind of interesting math and now that I have to go back to do the more basic stuff, i get stuck.

Everytime I try to focus and actually learn the material or do the exercises, I end up learning about something else. What do you guys do to keep yourself focused?

>>8454034
> I did a gap year where I learned all kind of interesting math and now that I have to go back to do the more basic stuff, i get stuck
You should talk to your profs and take exam prematurely or something. That's not a good situation for your development.

>>8454060
Unfortunately, the program is quite rigid. I'm doing a 3 year bachelor but the courses that you take in the first two years are set and there's no real way to get around it. I wished we had something like honors real analysis but this course doesn't exist in my university.

The worst is that I have proved harder stuff and I went through more difficult books, but since I have trouble staying focused during the lectures and I don't really enjoy the material, I take too much time doing simpler problems. When I did some analysis with Tao's book I found the material fun. When I went through the 'Analysis in normed vector spaces' section of Lang's undergraduate analysis it was the same. I had fun proving the lemmas, theorems and so on. Now this just feels like a disorganized mess of facts to memorize and "tricks" to find to solve a particular convergence problem.

>>8454034
>>8454091
Here's some motivation for you. You don't understand at a reasonable level what is by your own description material ripped straight from calculus II. If that isn't extremely disconcerting to you as a mathematics student there is a big problem.

>>8454348
I'm pretty sure ε-δ proofs of just about everything in Calculus isn't calc II, no. Also it's not that I don't understand, it's that I can't find motivation, and then I slack off and it all starts to pile up.

>>8454557
merely smart but lazy
I see

>>8454566
I'm not sure this has anything to do with being smart to be honest. And I'm not sure this is laziness either. When it comes to the kind of math I'm actually interested in, I can spend hours every day working through a book.

>>8454091
Learning to walk isn't hard but it is difficult to motivate if you already know how to drive. However, when one learns to drive they do not forget how to walk.

You have put yourself in the unfortunate position of having to learn the basics of a topic as if you don't already know its more advanced aspects. It makes sense that you would find it difficult to motivate and probably don't see the point. That said this is still material you're expected to know and if you have this gap it will raise some serious concerns about your ability just any normal person would when encountering someone who drives everywhere but apparently doesn't know how to walk.

>>8454626
This is a very valid point, indeed. I guess I should just find an isolated place where I can study only the material we're covering in class.

hi
how do I into Stokes generalised theorem, differentiable manifolds? i want to understand integration of differential forms and so on.

>>8454651
do you need a reference or someone to explain it to you?
Anyway, here's a reference:
Jeffrey M. Lee "Manifolds and Differential Geometry"

>>8454651
You'll need to learn calculus on manifolds. Spivak's book or Munkres' Analysis on Manifold is good for that. As for differentiable manifolds, Lee's introduction to smooth manifolds is a classic.

>>8453192
Someone please help me out here.

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I'm giving up on my algebra.

Problems like these take a hell lot of time to solve + solutions are ridiculously long.

Please look at these and say: are they really this hard or maybe I'm missing something? Please answer, I need to know:

1. For which [math]d \in \{2, 3, \ldots 10\}[/math] ring [math]\mathbb{Z}+\mathbb{Z}[\sqrt{d}][/math] is integrally closed?
2. Prove that the quotient ring of [math]\frac{\mathbb{C}[x, y, z]}{\left< x^3 + y^3 +z^3 - 1 \right>}[/math] is isomorphic to the field of rational functions of two variables.
3. How many solutions does [math]x^2+y^2=1[/math] have in [math]\mathrm{F}_q[/math] (q is prime)?

>>8455424
You probably are just not clear on how to approach the problems. A lot of students are hand-fed proofs which make them think that "proofs" are really just a direct chain of applications of lemmas and theorems from class.

In reality, the proofs required be these exercises require you to apply yourself and use lemmas and theorems in an indirect fashion to get close enough to a solution to make the necessary logical leap.

>>8455424

It would be helpful if you could tell us what they've taught you in the course, maybe give us the textbook you're using.

>>8455424
>3. How many solutions does x2+y2=1 have in Fq (q is prime)?
do it for Fp first obviously

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>>8451277
Falsely factual

I have to finish writing some chapters for me EE thesis "Integrated resistor matching algorithm" and I'm a procrastinating son of a bitch.

fucking hate maths desu

>>8455684
quality contribution

>>8455688
hows yours any better than mine...

>>8455684
why?

>>8455700
just joking. I hate what im learning right now which made me frustrated enough to come here and write a mean spirited comment at maths

>>8455598
We barely had any theorems, except for little ones that were put in problem sets. However, I see no interconnections and reasoning behind the problems, seems to me like a bunch of random facts that use definitions from the lectures.

The lectures were about what spec is, localizations, zeroes of an ideal, integral extensions, closures, how can some types of singularities on the zeroes' graph of a polynomial ideal tell that corresponding factor ring isn't integrally closed, nullstellensatz (no proof), any finite generated ring over field k with degtr_k = 1 is isomorphic to a quotient field of k[x,y]/<f> (no proof, only a probabilistic argument, lol), and what is a model of a functional field.

I barely understand what these lectures are about.
This prof is doing his lectures based on some shafarevich's book, but I don't see things like these in shafarevich's books I've downloaded.

Yeah... i'm fucked.

If complex variables is so important to applied math, then why is it only an elective for my school's applied math program?

>>8455900
Because your school is shit rofl

I know you guys get this question alot, but it wasn't in the sticky, so I'll keep it short:

What is a great textbook for rigorously teaching myself algebra(in high school terms, Alg 1-preCalc)

>>8455424
1. Depending on what you know about the rings of integers of quadratic fields, it could be easy or long and boring (try d=2 and d=5 to see what's going on). Either way, not super complicated.
2. Tricky
3. Not very long but requires a clever idea

>>8455746
skipping proofs in lectures isn't the worst thing. Your first port of call is to get comfortable with actually doing calculations with affine varieties, and seeing the geometric ideas and picture. This is the proof he didn't want to do https://en.wikipedia.org/wiki/Noether_normalization_lemma

If you want a good reference for algebraic geometry, my favorite is ravi vakils free book http://math.stanford.edu/~vakil/216blog/FOAGdec2915public.pdf which is a much more modern approach.

Hartshorne is good too, and probably more in line with the approach your prof is taking, but be warned, key concepts are introduced as exercises and some exercises themselves can be extremely difficult with no warning.

I also dislike Hartshorne chapter 1

>>8455424
2) is not even a field, how can it be isomorphic to [math] \mathbb{C}(s,t) [/math]

>>8456300
Khan Academy
Simmons' Precalculus Mathematics in a Nutshell: Geometry, Algebra, Trigonometry

>>8456420
>... that the quotient ring of ...
Its quotient field is considered

>>8456420
oh nvm you mean the fraction field, well in that case i think you are looking for a birational map

>>8456430
ya i really dislike the quotient ring terminology, ambiguous considering what we are taking the quotient ring of is a quotient ring :/

>>8456433
if you are still stuck, read the section on rational maps in ravi vakils notes i linked, in particular the examples/exercises starting on page 191

Why is it that every Optimization lecturer on the planet is garbage?

>>8450007
None. It's an introductory CS class.

>>8449996
Are you stupid? That's the hardest class we offer at UMD. Research your classes before you take them it's invaluable in college.

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I want to learn about mathematical treatment of non-classical logic, but all I ever find is some unrigorous vague philosobabble.

>>8456936
>non-classical logic
That sounds way too broad, what are you looking for infinitary stuff? continous logic? something else?

>practical formulations of QFT yields testable and accurate predictions of experiments but are not mathematically sound
>mathematically sound QFTs are either so weak that they only ever predict what was already discovered decades ago, or so esoteric that no physical system could ever be described by it

What are some Entry level books for someone interested in self-studying Higher Math? I'm talking about introductions to proof etc. Hard to describe what I mean, but I'm very interested in the More formalized approach to Math, which basically explains in detail the logic behind all the things we take for granted as far as basic calculations or even derivations go.

>>8456936
That's because mathematicians tend to work with their intuition and often times rigorous proofs and formalisms are viewed as something that just get in the way of doing mathematics.
>we're just using whichever assumptions make things work
>just do the natural thing
>it's true but we're not going to prove it
>handwaving

If you want to deal with mathematics in different logics or with a large focus on rigorous formalisms, proofs, and foundations then you should be looking at computer science instead. In particular programming language theory, because there's a correspondence between category theory, type theory, and logic where:
a logic = a programming language
a theorem = a type
a proof = a program
etc..

In this context a computer science researchers actually do a lot of work in how to formalize and intuit mathematics in non-classical logics (classical logic yields uninteresting programming languages). This is also the basis for theorem provers which are studied a subset of programming language theory.

>>8457672
Because QM is wrong at its core.

>>8457689
Look for some lectures on higher mathematics by Bill Shillito on YouTube. For proofs, Book of Proofs is quite good. As for the "more formalized approach" this is a mindset, not a specific field of study. Pick whichever subject you want, and read a textbook/watch lectures/whatever. There's no 'University Physics' type of books for mathematics, I reckon.

>>8457689
Try Gowers' Princeton Companion to Mathematics

please someone help this seems like such simple math but I literally cannot figure out how my professor reached the conclusion highlighted in red

>>8458107
Check your pic m8

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>>8458107
wait wrong file

>>8458112
if cb+ad=2 and c=-a then 2=cb+ad=-ab+ad=a(d-b)

>>8458124
yeah I get that part i meant the implication that d = b+1 or d = b +2 >.<

>>8458127
do you know anything else about these coefficients? are they supposed to be natural numbers or something?

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>>8458135
integer coefficients

after thinking about it it seems when a = 1 then (b-d) has to be 2 so d = b+2

and vice versa a = 2

the thing is I dont understand whats limiting a to be either 1 or 2

>>8458160
There seems to be a step skipped here. The first line of your proof limits a to be either +-1 or +-2, but the line you're confused about assumes a is either 1 or 2.

I don't see why your proof is throwing out the negative options

>>8458180
ohhh I see it. Man mathematical proof feels like grasping at straws for answers

>>8458192
What he did in the proof is still valid, but there is a step skipped.

The reason is that your expression is symmetric; if you swap a <-> c and also b <-> d, the answer stays the same.

Since one of a,c is positive it's okay to just assume that a is the positive one. If it wasn't you could swap and make it the positive one without changing the answer.

>>8457672
lel yes. but you yourself cannot verify anything that scientists claim