/sci/ - Science & Math

[Anonymous No.9767847]

>Rudin Analysis
>Lang Basic Math
>Artin Algebra
>Axler Linear Agebra
>Spivak Calculus

[Anonymous No.9767852]

>>9767847
All meme textbooks which are only good AFTER you've studied from a brainlet textbook.

Beginners: STAY AWAY FROM THESE BOOKS.

[Anonymous No.9767855]

>>9767847
>>9767852
what are the based high-test alpha chad versions of these books

[Anonymous No.9767862]

>>9767852
why does everyone recommend these sorry books for BEGINNERS instead of recommending them AFTER the brainlet intro books the person actually needs?

[Anonymous No.9767887]

>>9767855
Folland’s Real Analysis, Dummit/Foote’s Abstract Algebra, and Serre’s A Course in Arithmetic.

Standard reading for a fourth year undergrad at a reputable university

[Anonymous No.9767894]

>>9767852
Is Calculus by Adams/Essex a brainlet textbook?

[Anonymous No.9767903]

>>9767852
why do people recommend these books to beginners?

[Anonymous No.9767994]

>>9767903
pseud cred

[Anonymous No.9768027]

>>9767855
The manga guide to calculus.

[Anonymous No.9768033]

>>9767903
It's a meme. I fell for this once and it's not pretty. Don't do it.

[Anonymous No.9768044]

>>9767852
Lang and Spivak, sure. What's wrong with Rudin and Axler for beginners? Rudin is hard, but still perfectly reasonable for an undergraduate course in analysis. Axler reads pretty easily. I'm not familiar with Artin.

[Anonymous No.9768064]

>>9767847
All these books are good. This should be the actual /sci/ starter kit.

[Anonymous No.9768066]

>>9768064
most of /sci/ is filled with amerisharts so they can't read anything more than Stewarts calc books.

[Anonymous No.9768070]

>>9768066
This is why we should have math captcha with abstract algebra problems. It's the best brainlet filter.

[Anonymous No.9768092]

>Pinter's abstract algebra

[Anonymous No.9768114]

>>9767887
>Dummit/Foote
Or, how about Lang's Algebra (revised 3rd edition), the superior Algebra text-book.

[Anonymous No.9768126]

>topology without tears

[Anonymous No.9768154]

>Stewart calculus

[Anonymous No.9768155]

>SGA

[Anonymous No.9768505]

>>9767847
Kobayashi & Nomizu

[Anonymous No.9768530]

>>9768044
Axlers assumes you know computational linear algebra. Then motivates and proves the theoretical basis of it. If you don’t know how to even do rref I’d recommend holding off on axler until you learn the computational aspects first to build intuition. It’s okay to mix proofs in the computational parts of linear algebra.

[Anonymous No.9768548]

>>9767847
Russell & Whitehead

[Anonymous No.9768603]

>>9767847
>anything but Divine Proportions: Rational Trigonometry to Universal Geometry

[Anonymous No.9768617]

>>9767847
>>9768044
what about apostol's calculus as an introduction? Is it also a meme for learning calculus for the first time?

[Anonymous No.9768733]

>>9768617
Yes, it is a meme. If you have never done calculus in your life, you should be starting with a high school level book. Spivak, Apostol, Stewart and all the other memebooks are the /sci/ equivalent of /lit/ recommending Pynchon and James Joyce: totally worthless to someone not already chin deep in academia.

[Anonymous No.9768818]

>>9767852
>>9767847
Axler is easy. Way easier than any other LA theory book like shilov or hoffman and kunze. In part because the theoretical machinery he builds up to avoid relying on determinants is just more intuitive, and in part because he asks less mathematical maturity of the reader. His exposition is careful and he explicitly asks you to justify some obvious facts he uses in the exposition.

>>9768530
What would you recommend instead as a linear algebra theory book for someone who never had computational linear algebra, then?

[Anonymous No.9768887]

>>9768818
Because they'd probably fail the machine learning courses that actually use applied linear algebra vs what axler teaches?

They might know about bilinear forms but if they can't do simple computations with matrices it's not good.

I suggest they take an elementary linear algebra class that has proofs + computations, then move to theoretical linear algebra that is more graduate level material

[Anonymous No.9769205]

>>9767903
Stewart, Khan Academy, Professor Leonard, 3blue1brown, Paul's Online Math Notes.

[Anonymous No.9769206]

>>9768733
>>9768033
>>9767903
>>9767994
>>9767852
Do you guys actually have problems with these books? The meme books have been a godsend for me, personally.

Go slower and try harder, maybe?

[Anonymous No.9769216]

>>9769206
I'm sure they are good if you already know calculus but not for learning it for the first time.

[Anonymous No.9769217]

>Spivak Calculus on Manifolds

[Anonymous No.9769382]

>Strang Linear algebra

[Anonymous No.9769392]

>stanley enumerative combinatorics

[Anonymous No.9769397]

>>9769206
I'm a tutor and believe me: even the advanced students doing AP Calculus BC have a hard time understanding even the simplest of rigorous mathematics.

Rigorous math is a great way to organize information, but it's a terrible way to teach, in particular to teach beginners (aka brainlets)

[Anonymous No.9769749]

>>9769216
I used Spivak for learning calculus for the first time and I'm learning Axler to learn LA for the first time. Of course I supplement responsibly and do computational problems as well, but I have not found a problem with my method as of yet.

>>9769397
well then they simply need more practice with rigorous mathematics, and we both know the only way to do that is to do read more rigorous mathematics. After your mature enough, shit gets ezpz and I get annoyed with conversational books like Lang's Calc, or Stewarts totally non rigorous approach

[Anonymous No.9770761]

>>9767852
>All meme textbooks which are only good AFTER you've studied from a brainlet textbook.
>Beginners: STAY AWAY FROM THESE BOOKS.
How the fuck is Lang's Basic Mathematics unsuitable for beginners? It's one of a very few math books that doesn't require any prerequisite knowledge and explains everything from the beginning.

[Anonymous No.9770920]

>>9770761
You’re right. It’s only a meme because everyone recommends it. axler’s precalculus book is a good substitute and even better axler has answers to all odd problems (standard exercises not the more challenging ones) in the text. Lang has answers to random problems. So on this alone axler is better for self-study where you can check your work. Both books provide good coverage of precalculus.

[Anonymous No.9770938]

>>9769749
Those are fine books. The problem lies in the fact the education system is so poor in the US many students aren’t ready for those books. They need a bridge book like book of proof, how to prove it, or some discrete math book.

If you don’t know linear algebra whatsoever and look at axler for an upper division computational statistics course that assumes mastery of computational techniques from linear algebra, then axler probably isn’t a good place to start. A book on computational linear algebra with proofs would be probably what those statistics courses are expecting. Axler himself says his book is a second course on linear algebra. The computational approach leads to intuition for the theoretical approach. It is a pre-rigorous, rigorous, post-rigorous approach Tao lays out.

Axler can be used as a first book, but I’d recommend people do what you are which is supplement it with computations.

Strang is a horrible book. There are better beginner books. The one thing I see with axler is it’s a good pre-categorical approach to linear algebra. But I’m looking for something more 21st century mathematics with a category theory emphasis.

[Anonymous No.9770985]

>>9769749 #
Those are fine books. The problem lies in the fact the education system is so poor in the US many students aren’t ready for those books. They need a bridge book like book of proof, how to prove it, or some discrete math book. But I also think if you skip the intro to proofs book should and dive right in, then you you can learn as you go. So on that basis I think Spivak and Axler are ok for a motivated high school student that doesn’t know calculus or linear algebra. You get better at proofs by doing them and both axler and Spivak provide a lot of practice.

I think strangs book is trash and there are better linear algebra books out there

[Anonymous No.9770993]

>>9769749
Those are fine books. The problem is many students come by out of high school aren’t ready for those books. They need a bridge book like book of proof, how to prove it, or some discrete math book. But I also think you can skip the intro to proofs book and dive right in. But if that is too much of a jump then you can go through the basics of proofs techniques through an introductory math class or intro to proofs book.

So on that basis I think Spivak and Axler are ok for a motivated high school student that doesn’t know calculus or linear algebra. You get better at proofs by doing them and both Axler and Spivak provide a lot of practice.

I think strangs book is bad and there are better linear algebra books out there

[Anonymous No.9770998]

>>9768733
Not to be "that high school guy," but I recently graduated after taking AP Calculus BC and self-studying the more advanced concepts of single-variable calculus. Would Apostol be a good series to brush up on a more proofs-based form of single-variable in addition to an apt introduction to multi-variable and linear algebra?

[Anonymous No.9771209]

>>9767903
>Ask friend to help me get prepped for calc
>Gives me their Spivak Calc book and tells me to do it so I'll be ahead
>Struggle through it but I have a month before classes start
>Some questions take a day or two from just chapters 1 and 2 because my algebra knowledge is shit
>Finally finish the textbook
>Friend congratulates me and says I'm ready
>Class starts
>Using fucking Stewart's

The only thing I even bothered studying after that was the applications portions. Only person who got an A from the prof. I'd recommend it but it's not going to be easy.

[Anonymous No.9771359]

>>9771209
How long did it take you to finish it?
Does Spivak prepare you to do the computational problems in Stewart? My concern is id know the theory deep but couldn’t solve an integration by parts problem

[Anonymous No.9771457]

>>9771209
>Gives me their
>using non-gendered pronouns unironically

[Anonymous No.9771616]

>>9770998
Read this, real math is proof based.
https://www.people.vcu.edu/~rhammack/BookOfProof/BookOfProof.pdf

Apostol mathematical analysis not apostol calculus over rudin.

[Anonymous No.9771626]

>>9767903
I would assume most people learn some form of calc in high school long before buying some textbook of their own volition they heard from an anime board.

Also Axler's Linear Algebra Done Right has always struck me as beginner? It's really easy to read, and lin. alg. is usually introduced in a sort of proofs-based way anyways in university coursework, correct? You can of course start first with more applicative lin alg but its quite normal to just start with the proofs.

[Anonymous No.9771707]

>>9771616
>Read this, real math is proof based.
What most so-called "mathematicians" do can hardly be called "proofs", at least not until the usage of Coq becomes more widespread.

[Anonymous No.9771744]

>>9771209
you literally learned a 6-month class in a month. If I did that, I'd be upset at the tremendous waste of time.

[Anonymous No.9771756]

>>9771626
If one does linear algebra done right would they learn enough computational linear algebra succeed in an applied machine learning course that uses linear algebra?

[Anonymous No.9771770]

>>9771756
>would they learn enough computational linear algebra succeed in an applied machine learning course that uses linear algebra?
I would assume the "applied" machine learning course is akin to a computer science course (focused on programming) with some cursory coverage of theory. In that case, you should be fine if you have decent knowledge of algorithms. In any case, you should succeed just fine because even if the numerical/comp lin alg stuff seems foreign at first it shouldn't take much side-study to come up to speed. It's much harder for the "applied-only" folks to backlearn the theory in my experience.

[Anonymous No.9771797]

>>9771770
Okay, thanks!

[Anonymous No.9771821]

Can someone please give me some concrete examples as to why abstract algebra is a useful tool for science? I love algebra but I'm confused as to why it's a prereq for this board. I see studying it as more of a luxury.

[Anonymous No.9771824]

>>9768548
BASED and redpilled

[Anonymous No.9771832]

>>9771821
There is no simple example. People usually don't study abstract algebra because of the applications but because it has a beautiful theory that lets you reformulate known facts in a new nicer language.

[Anonymous No.9771843]

>>9771832
That's fair but I don't think it would be very convincing to someone who doesn't love math. It seems like you could make that argument for any field, but it's all algebra has going for it.

[Anonymous No.9771879]

>>9771843
tbqh
there's no need to learn abstract algebra if you don't love math

[Anonymous No.9771946]

>>9771457
The gender isn't relevant to the greentext though.

>>9771744
Maybe. I spent maybe 6-8 hours a day at most and took a break every few days. So the amount of time spent on it is about the same as most people put in. Just more condensed. Then the class is essentially one big review. I don't feel like I missed much or left with poor knowledge in a subject.

[Anonymous No.9771960]

>>9767847
These textbooks are all perfectly fine.

[Anonymous No.9771969]

>>9771946
didnt ask

[Anonymous No.9772105]

>>9771946
How did you check your solutions were correct from Spivak?

[Anonymous No.9772108]

>>9771209
Does Spivak teach teach the plug and chug methods you NEED for Stewart in addition to the theory or do you have to go back and fill that in?

[Anonymous No.9772117]

>>9767903
Because they're either pretentious cucks, or they just didn't think how it would actually be to start with one of those books straight away.

[Anonymous No.9772282]

ITT: Brainlets who get ass-blasted that Spivak teaches people the underlying reasoning and logic as to why you can do all those things you can in calculus.

[Anonymous No.9772511]

>>9772282

[Anonymous No.9772544]

>>9768733

Are you talking about James Stewart's calc book? If so, why lump that in with Spivak or Rudin? Those two are in a different galaxy of rigor.

[Anonymous No.9773444]

>>9772105
I was given the answer book with it as well as some in the back of the text. If you need to see or check the work more specifically you can Google "Spivaks Calc Problem xx" and you'll usually find someone else asking about it or asking if their way about it is correct.

[Anonymous No.9773460]

>>9772108
In regards to calculus its mostly the derivatives and integral work so there will be some plug-in-chug but not much. Stewart's has large sections about "Applications of the Derivative" which is mostly stuff like compound interest, speed, velocity, optimization, etc. That Spivaks doesn't have since he's not prepping engineers to take engineering classes early.
In regards to the derivative and integral stuff Stewart's just sticks with what he teaches you, you'll never get a derivative that requires more than just doing the product rule inside the quotient rule and by the time you get to anything that would be difficult that way you get taught substitution to simplify it. Spivak makes some jokes and gives you slightly more difficult problems through the chapter exercises but nothing you shouldn't be able to do if you understand it, it's mostly tedious if you do but I suppose it's to check to make sure you know the rules he's throwing at you. (ég one of the first larger derivatives being Cos(Cos(Cos(Cos(x)))) with a few more cosines thrown in). He does have lengthy explanations and it's not as dry as some people say but it's information dense.

[Anonymous No.9774191]

Best book for applied maths

[Anonymous No.9774212]

>>9767847

[Anonymous No.9774266]

>>9772511
He is right though

[Anonymous No.9774267]

What's the meme book for Complex Analysis?

[Anonymous No.9774279]

>>9767894
just finishing it up, it's decent

[Anonymous No.9774287]

>>9774267
needham

[Anonymous No.9774386]

>>9773460
Thanks. But does Spivak each you enough to do derivatives and integral computations from Stewart?

[Anonymous No.9774387]

>>9774266
Right about what? I just posted an interesting question. It has nothing to do with the comment I replied to other than it’s a fun problem from the book the comment references

[Anonymous No.9774521]

>>9774267
>What's the meme book for Complex Analysis?
Lang

[Anonymous No.9774536]

>>9774386
>Thanks. But does Spivak each you enough to do derivatives and integral computations from Stewart?
computations are for computers

[Anonymous No.9774612]

>>9769206
>Do you guys actually have problems with these books?
Pseud spotted

[Anonymous No.9774690]

>>9768114
This.

You'll find professional Algebraists that still don't care for everything in that book. It's your standard reference.

But real talk, it'd be dumb to go into that book for a first time look at algebra.

[Anonymous No.9774696]

>>9767855
>Papa and Grandpa Rudin
>Mathematical Logic Schoenfield, Euclids Elements, and Dummit/Foote up to fields in order to deduce laws of basic math.
>Lang Algebra
>Baby Rudin

[Anonymous No.9774701]

>>9774696
Oops forgot
>Axler Linear --> Greub Linear

[Anonymous No.9774748]

>>9774696
>Euclids Elements
>just gives axioms
kys

[Anonymous No.9774769]

>>9767847
Those textbooks are great.

Look, I'm 19 years old.
I am handsome, smart, athletic and virile.
I have a research paper that is in it's final editing stage, and a professor at my college has read the first draft and thinks it's publishable.
I have a girlfriend who is confident, articulate, playful and spontaneous.
I have a small group of interesting friends from different social and academic backgrounds, and I also have many other acquaintances who see me as a reliable source of humour and good company.
Both my parents are alive and in good health.
I have no regrets.
I have already experienced three existential crises, the latter of which was described as having the depth and profundity of a man twice my age.
I am a passionate lover, a sharp thinker, and a trader of witty repartee.
I am not self-pitying, meek or needlessly humble.
I will live a good life at your expense you fucking brainlet.

[Anonymous No.9774771]

>>9774748
>Thinks Euclid was just axioms.

[Anonymous No.9774772]

>>9774769
>Look, I'm 19 years old.
>I am handsome, smart, athletic and virile.
>I have a research paper that is in it's final editing stage, and a professor at my college has read the first draft and thinks it's publishable.
>I have a girlfriend who is confident, articulate, playful and spontaneous.
>I have a small group of interesting friends from different social and academic backgrounds, and I also have many other acquaintances who see me as a reliable source of humour and good company.
>Both my parents are alive and in good health.
>I have no regrets.
>I have already experienced three existential crises, the latter of which was described as having the depth and profundity of a man twice my age.
>I am a passionate lover, a sharp thinker, and a trader of witty repartee.
>I am not self-pitying, meek or needlessly humble.
>I will live a good life at your expense you fucking brainlet.

[Anonymous No.9774786]

>>9774696
>Mathematical Logic Schoenfield
Such a dense book.

[Anonymous No.9774794]

>>9774786
It's this book and baby rudin that made me realize that the brain has limited processing power. Up until my second year of college or so, I could just read and understand. But once I reached these books, a real perspective shift happened inside. When you can read a sentence, understand what it meant, but still have to sit and let it sink for longer than a moment. Maybe even a day, two days, or a week.

[Anonymous No.9774839]

>>9774769
Back to le Reddit with thee.

[Anonymous No.9775877]

>>9774386
He does. The only difference would be he will give you a couple variable forms such as the integral from a to b instead of a number. The cases he'll give you actual numbers might be to test your knowledge with a number to an integral or an integral of an integral. Not as difficult as it sounds.

So... They're derivatives and integrals. They're the same in Stewart's or Spivaks.

[Anonymous No.9775971]

Any non-meme charts?

[Anonymous No.9775986]

>>9775971
>Any non-meme charts?
Yep

High School:
• Euclidean geometry, complex numbers, scalar multiplication, Cauchy-Bunyakovskii inequality. Introduction to quantum mechanics (Kostrikin-Manin). Groups of transformations of a plane and space. Derivation of trigonometric identities. Geometry on the upper half-plane (Lobachevsky). Properties of inversion. The action of fractional-linear transformations.
• Rings, fields. Linear algebra, finite groups, Galois theory. Proof of Abel's theorem. Basis, rank, determinants, classical Lie groups. Dedekind cuts. Construction of real and complex numbers. Definition of the tensor product of vector spaces.
• Set theory. Zorn's lemma. Completely ordered sets. Cauchy-Hamel basis. Cantor-Bernstein theorem.
• Metric spaces. Set-theoretic topology (definition of continuous mappings, compactness, proper mappings). Definition of compactness in terms of convergent sequences for spaces with a countable base. Homotopy, fundamental group, homotopy equivalence.
• p-adic numbers, Ostrovsky's theorem, multiplication and division of p-adic numbers by hand.
• Differentiation, integration, Newton-Leibniz formula. Delta-epsilon formalism.

[Anonymous No.9775988]

>>9775986
Freshman:
• Analysis in R^n. Differential of a mapping. Contraction mapping lemma. Implicit function theorem. The Riemann-Lebesgue integral. ("Analysis" by Laurent Schwartz, "Analysis" by Zorich, "Theorems and Problems in Functional Analysis" by Kirillov-Gvishiani)
• Hilbert spaces, Banach spaces (definition). The existence of a basis in a Hilbert space. Continuous and discontinuous linear operators. Continuity criteria. Examples of compact operators. ("Analysis" by Laurent Schwartz, "Analysis" by Zorich, "Theorems and Problems in Functional Analysis" by Kirillov-Gvishiani)
• Smooth manifolds, submersions, immersions, Sard's theorem. The partition of unity. Differential topology (Milnor-Wallace). Transversality. Degree of mapping as a topological invariant.
• Differential forms, the de Rham operator, the Stokes theorem, the Maxwell equation of the electromagnetic field. The Gauss-Ostrogradsky theorem as a particular example.
• Complex analysis of one variable (according to the book of Henri Cartan or the first volume of Shabat). Contour integrals, Cauchy's formula, Riemann's theorem on mappings from any simply-connected subset C to a circle, the extension theorem, Little Picard Theorem. Multivalued functions (for example, the logarithm).
• The theory of categories, definition, functors, equivalences, adjoint functors (Mac Lane, Categories for the working mathematician, Gelfand-Manin, first chapter).
• Groups and Lie algebras. Lie groups. Lie algebras as their linearizations. Universal enveloping algebra, Poincaré-Birkhoff-Witt theorem. Free Lie algebras. The Campbell-Hausdorff series and the construction of a Lie group by its algebra (yellow Serre, first half).

[Anonymous No.9775993]

>>9775988
Sophomore:
• Algebraic topology (Fuchs-Fomenko). Cohomology (simplicial, singular, de Rham), their equivalence, Poincaré duality, homotopy groups. Dimension. Fibrations (in the sense of Serre), spectral sequences (Mishchenko, "Vector bundles ...").
• Computation of the cohomology of classical Lie groups and projective spaces.
• Vector bundles, connectivity, Gauss-Bonnet formula, Euler, Chern, Pontryagin, Stiefel-Whitney classes. Multiplicativity of Chern characteristic. Classifying spaces ("Characteristic Classes", Milnor and Stasheff).
• Differential geometry. The Levi-Civita connection, curvature, algebraic and differential identities of Bianchi. Killing fields. Gaussian curvature of a two-dimensional Riemannian manifold. Cellular decomposition of loop space in terms of geodesics. The Morse theory on loop space (Milnor's Morse Theory and Arthur Besse's Einstein Manifolds). Principal bundles and connections on them.
• Commutative algebra (Atiyah-MacDonald). Noetherian rings, Krull dimension, Nakayama lemma, adic completion, integrally closed, discrete valuation rings. Flat modules, local criterion of flatness.
• The Beginning of Algebraic Geometry. (The first chapter of Hartshorne or Shafarevich or green Mumford). Affine varieties, projective varieties, projective morphisms, the image of a projective variety is projective (via resultants). Sheaves. Zariski topology. Algebraic manifold as a ringed space. Hilbert's Nullstellensatz. Spectrum of a ring.
• Introduction to homological algebra. Ext, Tor groups for modules over a ring, resolvents, projective and injective modules (Atiyah-MacDonald). Construction of injective modules. Grothendieck Duality (from the book Springer Lecture Notes in Math, Grothendieck Duality, numbers 21 and 40).
• Number theory; Local and global fields, discriminant, norm, group of ideal classes (blue book of Cassels and Frohlich).

[Anonymous No.9775995]

>>9775993
Sophomore (cont):
• Reductive groups, root systems, representations of semisimple groups, weights, Killing form. Groups generated by reflections, their classification. Cohomology of Lie algebras. Computing cohomology in terms of invariant forms. Singular cohomology of a compact Lie group and the cohomology of its algebra. Invariants of classical Lie groups. (Yellow Serre, the second half, Hermann Weyl, "The Classical Groups: Their Invariants and Representations"). Constructions of special Lie groups. Hopf algebras. Quantum groups (definition).

Junior:
• K-theory as a cohomology functor, Bott periodicity, Clifford algebras. Spinors (Atiyah's book "K-Theory" or AS Mishchenko "Vector bundles and their applications"). Spectra. Eilenberg-MacLane Spaces. Infinite loop spaces (according to the book of Switzer or the yellow book of Adams or Adams "Lectures on generalized cohomology", 1972).
• Differential operators, pseudodifferential operators, symbol, elliptic operators. Properties of the Laplace operator. Self-adjoint operators with discrete spectrum. The Green's operator and applications to the Hodge theory on Riemannian manifolds. Quantum mechanics. (R. Wells's book on analysis or Mishchenko "Vector bundles and their application").
• The index formula (Atiyah-Bott-Patodi, Mishchenko), the Riemann-Roch formula. The zeta function of an operator with a discrete spectrum and its asymptotics.
• Homological algebra (Gel'fand-Manin, all chapters except the last chapter). Cohomology of sheaves, derived categories, triangulated categories, derived functor, spectral sequence of a double complex. The composition of triangulated functors and the corresponding spectral sequence. Verdier's duality. The formalism of the six functors and the perverse sheaves.

[Anonymous No.9775997]

>>9775995
Junior (cont):
• Algebraic geometry of schemes, schemes over a ring, projective spectra, derivatives of a function, Serre duality, coherent sheaves, base change. Proper and separable schemes, a valuation criterion for properness and separability (Hartshorne). Functors, representability, moduli spaces. Direct and inverse images of sheaves, higher direct images. With proper mapping, higher direct images are coherent.
• Cohomological methods in algebraic geometry, semicontinuity of cohomology, Zariski's connectedness theorem, Stein factorization.
• Kähler manifolds, Lefschetz's theorem, Hodge theory, Kodaira's relations, properties of the Laplace operator (chapter zero of Griffiths-Harris, is clearly presented in the book by André Weil, "Kähler manifolds"). Hermitian bundles. Line bundles and their curvature. Line bundles with positive curvature. Kodaira-Nakano's theorem on the vanishing of cohomology (Griffiths-Harris).
• Holonomy, the Ambrose-Singer theorem, special holonomies, the classification of holonomies, Calabi-Yau manifolds, Hyperkähler manifolds, the Calabi-Yau theorem.
• Spinors on manifolds, Dirac operator, Ricci curvature, Weizenbeck-Lichnerovich formula, Bochner's theorem. Bogomolov's theorem on the decomposition of manifolds with zero canonical class (Arthur Besse, "Einstein varieties").
• Tate cohomology and class field theory (Cassels-Fröhlich, blue book). Calculation of the quotient group of a Galois group of a number field by the commutator. The Brauer Group and its applications.
• Ergodic theory. Ergodicity of billiards.
• Complex curves, pseudoconformal mappings, Teichmüller spaces, Ahlfors-Bers theory (according to Ahlfors's thin book).

[Anonymous No.9775999]

>>9775997
Senior:
• Rational and profinite homotopy type. The nerve of the etale covering of the cellular space is homotopically equivalent to its profinite type. Topological definition of etale cohomology. Action of the Galois group on the profinite homotopy type (Sullivan, "Geometric topology").
• Etale cohomology in algebraic geometry, comparison functor, Henselian rings, geometric points. Base change. Any smooth manifold over a field locally in the etale topology is isomorphic to A^n. The etale fundamental group (Milne, Danilov's review from VINITI and SGA 4 1/2, Deligne's first article).
• Elliptic curves, j-invariant, automorphic forms, Taniyama-Weil conjecture and its applications to number theory (Fermat's theorem).
• Rational homotopies (according to the last chapter of Gel'fand-Manin's book or Griffiths-Morgan-Long-Sullivan's article). Massey operations and rational homotopy type. Vanishing Massey operations on a Kahler manifold.
• Chevalley groups, their generators and relations (according to Steinberg's book). Calculation of the group K_2 from the field (Milnor, Algebraic K-Theory).
• Quillen's algebraic K-theory, BGL^+ and Q-construction (Suslin's review in the 25th volume of VINITI, Quillen's lectures - Lecture Notes in Math. 341).
• Complex analytic manifolds, coherent sheaves, Oka's coherence theorem, Hilbert's nullstellensatz for ideals in a sheaf of holomorphic functions. Noetherian ring of germs of holomorphic functions, Weierstrass's theorem on division, Weierstrass's preparation theorem. The Branched Cover Theorem. The Grauert-Remmert theorem (the image of a compact analytic space under a holomorphic morphism is analytic). Hartogs' theorem on the extension of an analytic function. The multidimensional Cauchy formula and its applications (the uniform limit of holomorphic functions is holomorphic).

[Anonymous No.9776000]

>>9775999
Specialist: (Fifth year of College):
• The Kodaira-Spencer theory. Deformations of the manifold and solutions of the Maurer-Cartan equation. Maurer-Cartan solvability and Massey operations on the DG-Lie algebra of the cohomology of vector fields. The moduli spaces and their finite dimensionality (see Kontsevich's lectures, or Kodaira's collected works). Bogomolov-Tian-Todorov theorem on deformations of Calabi-Yau.
• Symplectic reduction. The momentum map. The Kempf-Ness theorem.
• Deformations of coherent sheaves and fiber bundles in algebraic geometry. Geometric theory of invariants. The moduli space of bundles on a curve. Stability. The compactifications of Uhlenbeck, Gieseker and Maruyama. The geometric theory of invariants is symplectic reduction (the third edition of Mumford's Geometric Invariant Theory, applications of Francis Kirwan).
• Instantons in four-dimensional geometry. Donaldson's theory. Donaldson's Invariants. Instantons on Kähler surfaces.
• Geometry of complex surfaces. Classification of Kodaira, Kähler and non-Kähler surfaces, Hilbert scheme of points on a surface. The criterion of Castelnuovo-Enriques, the Riemann-Roch formula, the Bogomolov-Miyaoka-Yau inequality. Relations between the numerical invariants of the surface. Elliptic surfaces, Kummer surface, surfaces of type K3 and Enriques.
• Elements of the Mori program: the Kawamata-Viehweg vanishing theorem, theorems on base point freeness, Mori's Cone Theorem (Clemens-Kollar-Mori, "Higher dimensional complex geometry" plus the not translated Kollar-Mori and Kawamata-Matsuki-Masuda).
• Stable bundles as instantons. Yang-Mills equation on a Kahler manifold. The Donaldson-Uhlenbeck-Yau theorem on Yang-Mills metrics on a stable bundle. Its interpretation in terms of symplectic reduction. Stable bundles and instantons on hyper-Kähler manifolds; An explicit solution of the Maurer-Cartan equation in terms of the Green operator.

[Anonymous No.9776003]

>>9776000
Specialist (cont):
• Pseudoholomorphic curves on a symplectic manifold. Gromov-Witten invariants. Quantum cohomology. Mirror hypothesis and its interpretation. The structure of the symplectomorphism group (according to the article of Kontsevich-Manin, Polterovich's book "Symplectic geometry", the green book on pseudoholomorphic curves and lecture notes by McDuff and Salamon)
• Complex spinors, the Seiberg-Witten equation, Seiberg-Witten invariants. Why the Seiberg-Witten invariants are equal to the Gromov-Witten invariants.
• Hyperkähler reduction. Flat bundles and the Yang-Mills equation. Hyperkähler structure on the moduli space of flat bundles (Hitchin-Simpson).
• Mixed Hodge structures. Mixed Hodge structures on the cohomology of an algebraic variety. Mixed Hodge structures on the Maltsev completion of the fundamental group. Variations of mixed Hodge structures. The nilpotent orbit theorem. The SL(2)-orbit theorem. Closed and vanishing cycles. The exact sequence of Clemens-Schmid (Griffiths red book "Transcendental methods in algebraic geometry").
• Non-Abelian Hodge theory. Variations of Hodge structures as fixed points of C^*-actions on the moduli space of Higgs bundles (Simpson's thesis).
• Weil conjectures and their proof. l-adic sheaves, perverse sheaves, Frobenius automorphism, weights, the purity theorem (Beilinson, Bernstein, Deligne, plus Deligne, Weil conjectures II)
• The quantitative algebraic topology of Gromov, (Gromov "Metric structures for Riemannian and non-Riemannian spaces"). Gromov-Hausdorff metric, the precompactness of a set of metric spaces, hyperbolic manifolds and hyperbolic groups, harmonic mappings into hyperbolic spaces, the proof of Mostow's rigidity theorem (two compact Kählerian manifolds covered by the same symmetric space X of negative curvature are isometric if their fundamental groups are isomorphic, and dim X> 1).
• Varieties of general type, Kobayashi and Bergman metrics, analytic rigidity (Siu)

[Anonymous No.9776035]

>i posted it again mom!

[Anonymous No.9776118]

>>9776035
>>i posted it again mom!
Who are you quoting?

[Anonymous No.9776147]

>>9775988
You do know that not once was this curriculum classroom tested right? I emailed the creator of the curriculum about it.

[Anonymous No.9776176]

>>9776147
>You do know that not once was this curriculum classroom tested right?
I didn't, but I don't see why it would matter. No researchers rely on classroom tested curricula.

[Anonymous No.9776179]

>>9774769
>The existential crises at age 19.
Yeah, you have some serious problems going on right now, regardless of your research status. And combinatorics is not mathematics, so don't act like you are doing mathematical research.

[Anonymous No.9776182]

>>9776179
bro can you reply i am all intrigued and sqt is dead

[Anonymous No.9776185]

>>9776182
>faggots
Why the homophobia?

[Anonymous No.9776189]

>>9776185
ree you are in 4chan. Reply plz
>>9776176 you too

[Anonymous No.9776194]

>>9776179
>combinatorics is not mathematics
Explain

[Anonymous No.9776196]

>>9776194
>mathematics
This is not well-defined.

[Anonymous No.9776209]

>>9776176
Because you are giving out a curriculum that may not be the best for anyone, even the top researchers of the world. The man behind this teaches at the top undergraduate school for mathematics at Russia, and even the most time crunched curriculum there teaches maybe the first two or so years during the four year curriculum.
I am not saying that it is a bad idea to study these subjects at this pace under guidance of a proper instructor, but even so you are missing out on more thourough material and specializing quite fast by this curriculum. You cover the bare minimum of analysis required and many other fields are just as lacking.

[Anonymous No.9776221]

>>9776194
Well, at the graduate level and above it certainly does, but if a young undergraduate is doing research chances are they are doing it in elementary combinatorics, which is a discipline based entirely on problem solving and without theory. Many of the methods of elementary combinatorics are ad hoc, which is characteristic of an engineer.

[Anonymous No.9776222]

>>9775986
thx meme bro

[Anonymous No.9776228]

>>9776209
>Because you are giving out a curriculum that may not be the best for anyone
They have free will, they can choose not to follow it.

>The man behind this teaches at the top undergraduate school for mathematics at Russia, and even the most time crunched curriculum there teaches maybe the first two or so years during the four year curriculum.
No researchers rely on classroom tested curricula.

>you are missing out on more thourough material
What do you mean by thorough material?

>specializing quite fast by this curriculum.
Of course you're free to set your own pace.

>You cover the bare minimum of analysis required and many other fields are just as lacking.
What is missing?

[Anonymous No.9776230]

>>9771821
Representations of clifford algebras can be used to describe spin, in non relativistic quantum mechanics these are the pauli matrices, in quantum field theory they are the gamma matrices. They are used as the spin operators and to compute scattering cross sections respectively.

[Anonymous No.9776258]

>>9776228
>this is actually based off of something

[Anonymous No.9776265]

>>9776258
>>this is actually based off of something
What are you trying to say?

[Anonymous No.9776274]

>>9776147
>>9776265
I'm just interested where this is from. Can anyone give the name of the guy or a link?

[Anonymous No.9776279]

>>9776274
>Can anyone give the name of the guy or a link?
https://en.wikipedia.org/wiki/Misha_Verbitsky
http://imperium.lenin.ru/~verbit/MATH/programma.html

[Anonymous No.9776288]

>>9776228
I suppose I should be more explicit. I followed this curriculum myself for the first couple of years in undergraduate, though my interests diverged soon after. But even then I am on the algebraic side of things, and so if you are in any way not of this form then this curriculum would certainly not be the ideal for you to learn from. The analysis covered in the entire curriculum amounts to only a half-year's worth at best, and if you cannot see that then clearly you should not be blasting the curriculum out everywhere if you are not qualified yourself to use it.

[Anonymous No.9776318]

>>9776288
>I suppose I should be more explicit. I followed this curriculum myself for the first couple of years in undergraduate, though my interests diverged soon after. But even then I am on the algebraic side of things, and so if you are in any way not of this form then this curriculum would certainly not be the ideal for you to learn from. The analysis covered in the entire curriculum amounts to only a half-year's worth at best, and if you cannot see that then clearly you should not be blasting the curriculum out everywhere if you are not qualified yourself to use it.
I asked what was missing.

[Anonymous No.9776403]

>>9776318
I do not know too much analysis personally, but if I had to give references, I would say
*Abstract Measure Theory
*Advanced Functional Analysis (as in Functional Analysis by P. Lax)
*Several Complex Variables
*Partial Differential Equations
*Harmonic Analysis/Distributions
*Operator Algebras
*Probability Theory
I could not give too much detail as I myself have not studied much analysis, but even with a cursory look I can see how little is covered by his curriculum. Meanwhile, he goes into what is standard for third-year graduate students and above with algebraic geometry.

[Anonymous No.9776460]

>>9776403
>*Several Complex Variables
>*Operator Algebras
>*Harmonic Analysis/Distributions
>*Abstract Measure Theory
Each of these are discussed in at least one of the books he lists.

>*Advanced Functional Analysis (as in Functional Analysis by P. Lax)
>*Probability Theory
What place do these have in his curriculum?

>*Partial Differential Equations
These certainly fit, but irregardless I don't think the list was ever said to be exhaustive. I'm surprised this is how you think of it after claiming to have spent years following it

[Anonymous No.9776607]

>>9768044
Are you joking? how is Spivak calculus harder than Rudin?

[Anonymous No.9776673]

>>9776460
Sure those topics are in the books that he recommended, but they are not in his curriculum, which is the object of discussion. And I am not saying that his curriculum is bad (why would I have used it?), more-so that it is not useful for most mathematics majors and that the curriculum gets too much publicity. Certainly is not helpful for someone who does not even know the undergraduate calculus of the real line.

[Anonymous No.9776679]

>>9774769
This pasta is awesome

[Anonymous No.9776685]

>>9767852
It's hit and miss. I picked up a copy of Rudin's babby analysis before taking my University's intro to analysis course and I fucking annihilated it and wrecked the curve for everyone else. I tried to work through Artin's Algebra before I had any exposure to abstract algebra and I got pwned. Gave up on reading it and just read Pinter when I took an intro course on the subject.

[Anonymous No.9776718]

>>9776685
I had the opposite experience; Artin's algebra seemed nice and easy, but Rudin was harder to follow for me, coming from pleb linalg and calc background before each read.

[Anonymous No.9776723]

>>9776718
Yeah, I think at this point it comes down to preference and natural ability. Algebra is not always intuitive for me, and I typically need to study extra hard in my algebra courses. The idea of open balls and metric spaces is a concept that I have had a hard-on for since basic HS calculus.

[Anonymous No.9776991]

>>9769392
i took that class taught by stanley himself. 'twas tough, to say the least. mad respect for those which combinatorics comes easy

[Anonymous No.9776998]

>>9775877
Okay. Since Spivak has both computations and theory I’ll go through it

What coverage of calculus is it? Does it cover integration by parts?

[Anonymous No.9777032]

>>9767903
because some people actually want to learn proofs
who cares if some high school dropout wants an easier textbook because he thinks induction is hard

[Anonymous No.9777048]

>>9768733
>>9772544
afaik
Stewart/Thomas are HS level books

[Anonymous No.9777050]

>>9772544
But I think that post you're replying to is a copypasta that includes Stewart as bait

[Anonymous No.9777052]

>>9775877
>Not as difficult as it sounds.
What do you mean by that? It sounds like he wants you to evaluate the integral in the bound then evaluate the integral. Is there some theorem provided that expedites the computation?

[Anonymous No.9777053]

>>9776182
integrate and any parabola is sim. to ax^2

[Anonymous No.9777602]

bump

[Anonymous No.9778320]

>>9767847
What's a good brainlet Algebra textbook?

[Anonymous No.9778328]

>>9778320
>What's a good brainlet Algebra textbook?
Hungerford - Abstract Algebra: An Introduction

[Anonymous No.9779208]

>>9776998
Yeah it has that. It pretty much has all the basic derivatives and integrals. Just none of the applications Stewart has. It should cover you up to calc 2 at the least.

>>9777052
No expidition. I think it's just a way to show integrals are not as scary or intimidating as they look and are just another way to express a sum.

[Anonymous No.9779209]

>>9779208
Up THROUGH calc 2 I mean to say.

[Anonymous No.9779354]

>>9767847
math is gay

[Anonymous No.9779411]

>>9779354
no u

[Anonymous No.9779836]

>>9778320
Pinter or Gallian
Even Artin is brainlet friendly because it spends a lot of time on concrete examples, namely matrix computations and the like

[Anonymous No.9779841]

>>9775986
>>9775988
>>9775993
>>9775995
>>9775997
>>9775999
>>9776000
>>9776003
Nobody thinks this is funny you ass hat

[Anonymous No.9779846]

>>9779841
>Nobody thinks this is funny you ass hat
Why would a list of mathematical topics be funny?

[Anonymous No.9780166]

>>9779846

[Anonymous No.9780696]

>>9767847

[Anonymous No.9781286]

>>9774191
That's not Boas, so you're wrong.

[Anonymous No.9781304]

No love for hoffman and kunze linear algebra?

[Anonymous No.9781312]

So this image was wrong all along?

[Anonymous No.9781320]

>>9781312
>So this image was wrong all along?
Of course, memelists are always wrong.

[Anonymous No.9781323]

>>9781320

Ok so... besides the anon that provided a whole classified list of math books, can you provide books to said topics? what do you recommend?

OR

Do you have a better flowchart?

sincerely,

A brainlet that wishes to expand his math knowledge

[Anonymous No.9781326]

>>9781323

>classified list of math books

I meant list of math topics.

[Anonymous No.9781330]

>>9781323
>wishes to expand his math knowledge
end goal?

[Anonymous No.9781332]

>>9781323
>besides the anon that provided a whole classified list of math books, can you provide books to said topics?
There are books in that list (look in the parentheses), for the topics that don't have one listed, pretty much any one will do since the treatments have become standard

[Anonymous No.9781333]

>>9781330

I'm just interested in learning math... but I don't know what book should I use. Hell, I don't even know if logic should be one of the starting points.

[Anonymous No.9781341]

>>9781333
>I don't even know if logic should be one of the starting points
Probably not
From that pic of yours you should definitely check out one of (or both) "How to prove it" and "Book of proof" and I think that Lang's book goes well with either of these, introducing you to rigorous math from high school topics. I *really* like Halmos "Naive Set Theory" so I would also advise to check it out, I do like Spivak's Calculus, really don't think it's a "meme" book that's not suitable for studying but you should see for yourself ofc. After that you could pick up some 'computational' linear algebra while studying multi-variable calculus, maybe from Apostol's book, or from whatever really. And then get a taste of some algebra, maybe with Fraliegh's First course in algebra book. That should be enough to figure out which fields you develop a taste for, or if math is your thing at all I think.

[Anonymous No.9781609]

>He's not studying from BASED Shelah
Not going to make it!

[Anonymous No.9781779]

>>9781323
>>9781333
An extensive treatment of mathematical foundations is not at all necessary to approach abstract mathematics. You won't get much out of it, either, without a certain level of background/perspective in rigorous math.
That said, some basic set theory definitely is necessary. Most "self-contained" books will cover this in their preliminary chapter, but the intro chapter of Munkres' topology book is basically all you'll ever need. It's a MUCH more thorough treatment than comparable books offer in their on intros and certain sections can be skipped, like the axiom of choice, but above all it offers a manageable first encounter with proofs for a bright student.
If it's too hard or you're lost on what a proof is supposed to be, Smith's Introductory Mathematics has a much gentler but still thorough and rigorous coverage of the same material (sets, functions, relations, etc.) with some brief coverage of algebra and analysis to get your feet wet. It's a perfectly valid starting point which I highly recommend for beginners. The target audience is more or less high school seniors encountering abstract math for the first time.
I've never read Velleman so I'm not sure exactly what role it fills and who ought to read it, but I've seen it recommended as well.
Although, virtually any "self contained introduction to X at the advanced undergraduate level" is a beginner book, even baby Rudin, provided you're willing to work like a dog and bang your head against the wall. The prerequisite for math books below the graduate level is almost never *content* but a certain level of sophistication. Work on developing that and when you start learning algebra/analysis/topology/linalg choose books that challenge but don't baffle you, so that you're always growing in sophistication.

[Anonymous No.9782247]

nothing wrong with onions

[Anonymous No.9782620]

>>9782247
If you're a woman or a faggot, that is.

[Anonymous No.9783122]

>>9775986
>>9775988
>>9775993
>>9775995
>>9775997
>>9775999
>>9776000
>>9776003

>no probability
soiboi detected

[Anonymous No.9783342]

>>9780166
not an argument

[Anonymous No.9783631]

onions

[Anonymous No.9786029]

>>9776003
>>9776000
>>9775999
>>9775997
>>9775995
>>9775993
>>9775988
>>9775986

How can math fags go through all this and STILL not able to solve stuff like the Collatz conjecture?

literally LMAOing at u brainlet mathfags.

[Anonymous No.9787043]

What about Introduction to calculus by Kazimierz Kuratowski