/sci/ - Science & Math

No, but this one is:

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.

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).

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).

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.

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).

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).

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.

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)

I want to learn math but I'm 37. Is it too late for me?

>>9735907
I want to learn math just because I am interested in it. But I am a med student. So assuming I study 2-3 hours a day, will I make any significant progress in a few years?

>>9735948
memelist

>>9735948

I think it is better.

but I started form [Tom Apostol] calculus vol 1

>>9735925
If you consistently study 2-3 hours a day for a few years, you will make significant progress in just about anything

>>9735907
it's a list likely made by an undergrad based on things that undergrad did or recommendations that undergrad found on /sci/.
actually, looking closer, it's garbage. if you're recommending something like velleman, then an "optional" first chapter of some set theory book, then you clearly haven't read one of those books. many tracks just outright skip bridge books like velleman or discrete math books, and cover all necessary set theory within the first chapter or appendix of their linear algebra book.
>although ideally one would like to study algebra first, linear algebra is so important...
what? this is either poor writing or just stupidity, as is everything else in this paragraph. linear algebra comes first because it's easy, well-known, and very useful (so easy ti motivate). it has nothing to do with helping you understand examples in later algebra books and there is no smart idealist out there saying algebra should come first.
>recommending dummit and foote
>before you encounter these structures, you want to be able to identify them
no, you don't have to. it's entirely unnecessary. jump into analysis if that's what you want, you have a time harder in any notable way than those who took algebra first. there are more concise and interesting algebra books to recommend than dummit and foote's, not to say it's a bad book.
>real analysis after analysis, and Tao's books no less
Tao is an amazing mathematician. however, his books for undergrads aren't anything special and a bit too easy, so i personally find them overrated. i think his reputation, not the quality of the book, is what's made them popular.
also, picking up a hard, more general analysis text (like rudin) completely sidesteps the need for an undergrad real analysis course.

there's more to say but i need to poop

>>9735907
if there is an equivalent for physics someone please post it

>>9736312
>if there is an equivalent for physics someone please post it
http://www.staff.science.uu.nl/~gadda001/goodtheorist/index.html

>>9735921
No, start right now:

1+1? Why?

>>9736215
pls continue after pooping, be sure to include ur own meme list by the end of it

I agree with everything you've said so far, but it's all for nothing without a list.

>>9735907
Start with Spivak/Apostol (with Apostol use the relevant MIT OCW notes from Munkres, Spivak use the Clark Honors Calc 2400 lecture notes), if you get stuck consult Book of Proof/InfDesc.pdf and/or Axler's "Precalculus: Prelude to Calculus" and Courant's "What is Mathematics?". Use whichever text you didn't choose as a supplement to the other.

Check this as you progress:
>https://math.vanderbilt.edu/schectex/commerrs/

Then move on to linear algebra, using any proof based (as opposed to computational) book you like, but I recommend Axler, H&K, and Valenza to begin with (particularly Valenza). Use letures from 3B1B, Wilderberger, Axler and Strang.

It's also worth mentioning here that Knuth's Concrete Mathematics is an excellent text that may be wise to read here, though this text tends to appeal more toward CS folk.

Then read Spivak's Calc on Manifolds with Munkres Analysis on Manifolds or Apostol's volume 2, using relevant MIT OCW material for either choice.

I can comment no further without backing it with experience, but I plan to move forward on to topics in abstract algebra using:
>https://www.math.upenn.edu/~ted/371F14/info.html#texts
>https://www.math.upenn.edu/~ted/371F14/hw-371SchedTab.html
>www.extension.harvard.edu/open-learning-initiative/abstract-algebra

As for analysis, Francis Su has excellent lectures on YouTube that I've already been using a bit, the canonical choice is Rudin, of which there is a plethora of supplementary material available, but Tao is also popular (though I agree with >>9736215, I found Tao much too easy personally).

After I finish the latter two topics up, I plan on exploring combinatorics through Lovasz, differential eq through Arnold and number theory through Apostol.

>>9736215
lmao
>many tracks just outright skip bridge books like velleman or discrete math books, and cover all necessary set theory within the first chapter or appendix of their linear algebra book.
>i didnt do it this way, therefore it is stupid
And this is not even considering that there is barely any intersection between Velleman or any other appendix/first chapter and naive set theory by halmos. Halmos in particular, is not a normal textbook, but rather, an easy and beautiful read that can be done in literally one or two afternoons, and has a more axiomatic view (ironically), and explains the axiom of choice, and its equivalents, and their importance. This does not appear in any introductory book in the subjects you mentioned!

>>recommending dummit and foote
Yeah, sure, it's long, but it has everything you need, tons of exercises and examples.

> linear algebra comes first because it's easy, well-known, and very useful (so easy ti motivate). it has nothing to do with helping you understand examples in later algebra books and there is no smart idealist out there saying algebra should come first.
Yeah, it doesn't say otherwise ??

>picking up a hard, more general analysis text (like rudin) completely sidesteps the need for an undergrad real analysis course.
rudin is terrible for self study


you're assuming this list is for math undergrads, but it's more for self studiers
The list could be like that other stupid anime list with 10 books before starting calculus by spivak, yes, but that defeats the point. In an ideal world, everyone would be looking at 5 different books at the same time on the same subject to compare and contrast, but it's not practical

>>9735920
>>9735919
>>9735917
>>9735914
>>9735913
>>9735912
>>9735911
>>9735910
>still no cute anime girl infographic
literally useless

>>9736320
>Proof 1 of 1'250'000
Yeah, useful!

>>9736215
>there are more concise and interesting algebra books to recommend than dummit and foote's
such as?

>>9736215
obvious dumb undergrad
rudin is a reference book

>>9736572
>you're assuming this list is for math undergrads, but it's more for self studiers
>rudin is terrible for self study
so why is it included in this list at all? why is it full of books with no solutions to exercises?

Let's play a game of spot the freshman

>>9737034
>so why is it included in this list at all? why is it full of books with no solutions to exercises?

Spotted

>>9737055
I finished my education years ago, retard

>>9737059
i guess a high school diploma counts

>>9735925
Where are you from?

>>9737065
I've got masters actually
>sophomore thinks he's all grown up now
my sides

>>9735907
>James Stewart calculus
>A taste
NIGGA THAT BOOK OVER 1000 PAGES

>>9737170
sure you do bud

>>9737034
>>9737059
What would exercise solutions look like for a book/course based entirely around rigorous proof-writing? They would be proofs, and they would be longer than the text itself. They would only show you one specific way of establishing the result, and could not tell you if, or why, your proof is invalid.
You have to remember that "real" math is not about computing things and getting an answer.

>>9737171
And the vast majority of it is repetitive exercises.

>>9736546
best guide

How over is it if you have to start learning from arithmetic, starting at log division/multiplication? Should I just rope and not even bother? I'm a NEET so I have a lot of time.

>>9737077
Germany. Does that make a difference?

>>9736572
i judged halmos by its cover, i admit. my experience with a proper set theory course was rather awful and no help for any of my other courses, which is why i said what i did. but halmos sounds fine.
the overlap of set theory books with velleman is enough to get started on upper div courses.
>it doesn't say otherwise?
it said algebra books have "countless examples which involve it." in my experience, that hasn't been the case - it's been just another subject in algebra books. there was the implication that it was ideal for algebra to come before linear algebra, unless i'm reading wrong, which is why i wrote that.
for what it's worth, i rarely went to lectures as an undergrad, so my experience overlaps with self studiers. on the other hand, i also only had a part time job and no great responsibilities.

>>9737026
i shouldn't have said "interesting" - i was in a hurry to poop. i've heard good things about artin - when i looked at it, it was kind of annoying, but i'd already taken algebra courses. my professor at the time had his own textbook that he wrote (i wonder if he'll ever publish it, it's great) and it was concise and touched on everything interesting.

>>9737029
>rudin is a reference book
it's not very motivating, sure, but if you can do all of the later exercises in each chapter then you'll have developed a very good foundation. i enjoyed using it for self-study (though it was incredibly hard). my job title is "mathematician" btw, i'm not an undergrad - i get to do math for a living all day!

>>9737706
sauce textbook and provide memelist

>>9736509
my own meme list would just be what i've taken - i only commented because i thought some of the things that list said were silly.
here's some of what i read (of course, not usually the entire text) in my undergrad (in no particular order), all of which i was able to find a way to enjoy
>numerical analysis, burden&faires
>advanced calculus, buck
>nonlinear dynamics and chaos, strogatz
>elementary analysis, ross
>principles of mathematical analysis, rudin
>princeton lectures in analysis, comples analysis and real analysis, stein & shakarchi
>applied combinatorics, tucker
>enumerative combinatorics, bona
>notes on set theory, moschovakis
>linear algebra, hoffman&kunze
>lectures on abstract algebra, elman
https://www.math.ucla.edu/~rse/algebra_book.pdf
and also lots of c++ stuff
the internet is your friend, but if you didn't sit down and try a problem for several hours first, don't look up a solution. see general remarks here:
http://www.math.ucla.edu/~tao/resource/general/131ah.1.03w/assign.html

>>9737718
oh, and of course i read velleman, and dummit&foote too, as previously mentioned. i had an interesting experience with "concrete mathematics" - when i bought it and first tried to read it, i couldn't get through the first chapter (my overall math background was very poor, relatively, but i think it was oribably worth starting at chapter two in hindsight since things got easier). later, as i neared graduation, i looked through it again and it was too simple to be worth spending time on. funny how that works

>>9737718
>c++ stuff
nice try, CS-fag

>>9737726
i'm evaluating some CS papers as part of my job right now. trust me, i hate it as much as the next guy because the average quality is poor and there's a lot hand waving that goes on.
however, programming is fun and i enjoy knowing a lot about the c++ language for some reason. i still do a lot of programming in my free time because it's "fun"

>>9737706
in an ideal world, a vector space would be defined as a module over a field. Sadly this is not an ideal world and a module is defined like a vector space over a ring.

Same kind of lista for physics?

>>9737730
>and i enjoy knowing a lot about the c++ language for some reason

>>9737694
No

>>9737860
>in an ideal world, a vector space would be defined as a module over a field
Why?

>>9737860
>a vector space would be defined as a module over a field.
Isn't that how it is defined?

>>9737600
Good point. Worked through the entire content of the book in my uni classes in not very long... Probably could've done it in a few months on my own.

>>9736745
>cute anime girl infographic

>>9735921
>37
>Is it too late for me?
Definitely not. If you really want to put in the effort, it's never to late. Don't be put off by the naysayers on this board.

Also, don't waste your time on holy wars over which book is best for what &c. The book you actually do use and work through is always better than the book you don't.

A good curriculum is sketched on /sci/'s own Wiki,
http://4chan-science.wikia.com/wiki/Mathematics

Also have a good look at this,
https://www.ocf.berkeley.edu/~abhishek/chicmath.htm

Finally, don't let yourself be discouraged be the self-styled "geniusses" on /sci/. With a few exceptions these people are show-offs and their mathematical skills not as impressive as they like to let you believe.

Godspeed, anon.

>>9738328
You are absolutely based.

>>9737725
Interesting, I had a similar experience with Concrete Mathematics.. hopefully when I return to it, it will be too easy as well :-) After all, it is just a math primer for TAOCP

>>9737718
Is that aa book you linked the one you mentioned earlier, that the author should publish?

>>9738549
that is the algebra book i was talking about; it seems to me that the logical end to writing something like that and having it proof-read is for it to be publisbed, but maybe that won't happen. i'm sure he'll end up doing what he wants with it.

>>9738593
Does it pair well with any particular book, or is it a complete book in its own right? At any rate, thanks for the replies - always a pleasure to talk with a working mathematician!

>>9738614
i would highly recommend against just sticking to one book. if you struggle with something and can't figure it out, try seeing how another book explains it.
the book is as complete as any undergrad algebra book, though - i found many of the exercises were taken from graduate texts (hungerford's algebra, lang's algebra, etc.) and courses

>>9735907
would yall recommend book of proof or how to prove it? Is the content essentially identical?

>>9738647
>don’t stick strictly to one book
As a general huerisitic when learning anything, I try to learn from as many sources as I can reasonably maintain.

But awesome, thanks for the resource.

>>9738758
They’re both good. I’d just try each (via libgen for HtPi, BoP is free) and see which you like. I personally preferred Book of Proof, but I already had taken a discrete math course before reading How to Prove It, so much of the first 4 or so chapters really came off as mundane review.

IMO, both are excellent texts and the content is close enough to being identical (but check the table of contents yourself to verify and peculiarities).

>>9738292
Now this really speaks to me

>>9738292
Now what about putting actual math books on the picture?

>>9735907
>>9735910
>>9735948
>>9736546
>>9737718
>>9738292
Is there a guide for reading various 4chan posts and guides?
Preferably with anime.

>>9739097
thank you for responding :D

>>9735920
>>9735919
>>9735917
>>9735914
>>9735913
>>9735912
>>9735911
>>9735910
i know this is a meme but I think it's unironically a good guide and you need to have enough ability to actually follow this meme guide if you want to make it in mathematics.

>>9739629
>i know this is a meme
It's not.

>>9739633
either way,
>you need to have enough ability to actually follow this meme guide if you want to make it in mathematics.
is true. no point in even attempting pure mathematics if you are under 160 IQ imo, just do something easier like physics

>>9735907
It needs Hoffman and Kunze's linear algebra somewhere in there. Maybe not as a replacement to the other linear algebra text though since Hoffman and Kunze is a more abstract text.

It also needs a calculus text as there are many techniques covered in calculus texts that are not covered in analysis texts (like optimization and related rates). Though if you're just going for abstract theory then it makes sense to omit calculus (and if this is the case replace Adler's algebra text with Hoffman and Kunze). Personally I would cover topology before Analysis but that's just me (might be a good idea to include Korner's Metric and Topological Spaces as a quickstart to topology if you go that route).

Here's a list for non-brainlets who have done high school maths

Zorich - Mathematical Analysis 1
Lax - Linear Algebra and its Applications
Tenenbaum and Pollard - ODEs
Willard - General Topology
Zorich - Mathematical Analysis 2
Aluffi - Algebra: Chaper 0
Tu - An Introduction to Manifolds
Rudin - Real and Complex Analysis
Freitag - Complex Analysis
Evans - PDEs
Ireland - Classical Introduction to Modern Number Theory
Tu - Differential Geometry
Tourlakis - Lectures in Logic and Set Theory 1&2
Ciarlet - Linear and Nonlinear Functional Analysis
Robinson - Group Theory
Freitag - Complex Analysis 2
Tu - Differential Forms in Algebraic Topology
Klenke - Probability Theory
Mac Lane - Categories for the Working Mathematician
Gortz - Algebraic Geometry Part 1

>>9740280
And for those of us in the u.s without calc?

>>9735907
From the perspective of a physics person
>book of proof
come on lad, this should be intuitive. Reading a book about how to "prove" something is a waste of time, trying to teach something that shouldn't need to be taught
>set theory
completely useless
>abstract algebra
useless
>number theory
u s e l e s s
>algebraic topology
>algebraic geometry
unnecessary for any scientific application

This extremely abstract world that mathematicians live in divorces itself from things which are actually useful. Especially shit like abstract algebra is absolute cancer, you are disappearing up your own asshole. None of this is useful in any context

>>9740512
no one cares about your degenerated perspective :^)

>>9740512
holy shit you are a retard tier physics person. why are you even speaking about this when you don't know about abstract algebra in physics

neck yourself undergrad

Is there a single book that covers all math from basic algebra to precalculus? I'm very far behind the math requirments of my prospective major. I need to begin calculus this fall. I'd appreciate any recommended books.

>>9740512
>thinks abtract algebra and algebraic topology is useless in physics
>physics person

by physics person,do you mean you watch alot of documentaries with neil degrasse tyson in them?

>>9740512

Being an atheist does not make you a "physics person". GTFO

>>9736546
is apostol good for baby's first calculus book?

>>9740512
>this should be intuitive. Reading a book about how to "prove" something is a waste of time, trying to teach something that shouldn't need to be taught
Proof isn't intuitive for anyone. You need to have the right attitude towards proofs, what they are, how they work, and why something is or is not a proof. All of that is alien to most people's experience, and so it needs to be learned.
Although most smart people can pick that up from studying a subject like algebra or analysis and won't need a separate "intro to proofs" book

>>9740549
No, but I'd suggest ck12.org.

They have a bunch of free textbooks that are concise enough for self-study.

Above all, I'd suggest you just dive right into calculus with some youtube videos. You can learn calculus and just look up the stuff you don't know as you go along. This worked for me years ago.

>>9735921
Bro I stopped at ALG4 in HS and went back to college at 31 to get my BSc in physics/math. It is never too late.

>>9735921
Why ?
I'm not trying to make a joke, I can't math but can you explain me why you want to get into math ? Is it to understant something ? A scientific paper that looks like ancient magic to people like me ?
Please develop, I'm genuinely curious