Math Meme charts

[Anonymous No.10097475]

Which one is best and why?

[Anonymous No.10097476]

>>10097475

[Anonymous No.10097477]

>>10097476

[Anonymous No.10097478]

>>10097475
an university degree

[Anonymous No.10097479]

>>10097478
That's beyond the point, for XYZ reasons people might want to complement their education or just study maths for the sake of it.

[Anonymous No.10097485]

>>10097478
>>10097479
Is there a math meme chart that's equivalent (or greater) to a degree in content?

[Anonymous No.10097492]

>>10097475
This one.

[Anonymous No.10097538]

>>10097485
They're all equivalent or greater than a math degree

[Anonymous No.10097543]

>>10097485
They're all equivalent or greater than a math degree. Except this one: >>10097476

[Anonymous No.10097546]

>>10097485
I've seen some guides which go far beyond post doc level

[Anonymous No.10097577]

Autor this pic here >>10097477
OP pic is better >>10097475
Apostol Analyst is very good too and Complex Analysis by gamelin is very good for self-taugth.

[Anonymous No.10098022]

bump

[Anonymous No.10098027]

>>10097492
weird how set theory and mathematical logic are little dead end branches on this tree huh?

[Anonymous No.10098037]

>>10097546
no you haven't retard
there is no such thing as "beyond post doc level" mathematics. 30 year old postdocs don't spend all day reading textbooks. They write math that hasn't been written before.

[Anonymous No.10098170]

>>10097475
>Meme charts
Lrn2meme fgt pls

[Anonymous No.10098172]

>>10098027
>implying they aren't
>>10097546
Fuggin post one.

[Anonymous No.10098380]

>>10097477
i dont like munkres' analysis on manifolds that much, sometimes seems a bit too computational for me but it's not bad

[Anonymous No.10098429]

>>10098380
Spivak better? Something else?

[Anonymous No.10098468]

>>10097475

>>10097476 >>10097477 >>10098170 >>10098172 >>10098380 >>10098429 >>10098037 >>10097577 >>10097546

[Anonymous No.10099098]

>>10098468
Based.

[Anonymous No.10099454]

>>10099098
And redpilled.

[Anonymous No.10099468]

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.10099470]

>>10099468
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.10099472]

>>10099470
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.10099481]

>>10099472
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.10099484]

>>10099481
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.10099485]

>>10099484
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.10099489]

>>10099485
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.10099491]

>>10099489
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.10099493]

>>10099468
>>10099470
>>10099472
>>10099481
>>10099484
>>10099485
>>10099489
>>10099491

[Anonymous No.10101049]

>>10099468
>>10099470
>>10099472
>>10099481
>>10099484
>>10099485
>>10099489
>>10099491
god i hate this meme

[Anonymous No.10101055]

>>10101049
poor little brainlet can't handle the Rigorous Russian Curriculum

[Anonymous No.10101072]

>>10097475

[Anonymous No.10101201]

Why do most guides not have measure theory or probability theory?

[Anonymous No.10101256]

no love for category theory? y'all are heartless

category theory is great especially if you're fucking around with the weirder parts of CS, like type theory (if you're good enough at it, HoTT gets a bit more tolerable)

just do "joy of cats" and Mac Lane's book, and you can shitpost about the Yoneda lemma and cofree comonads with the best of 'em

[Anonymous No.10101263]

>>10101072
what the fuck no CLRS this list is hot garbage

[Anonymous No.10101265]

>>10101256
this

category theory ought to be part of a well rounded modern math education

[Anonymous No.10101267]

>>10101072
wait also get that cooper & torczon shit outta here, Appel is how you actually make compilers (SICP is cool and all but sometimes you just want algorithms in a nicer language, like Java)

[Anonymous No.10101289]

>>10101201
and combinatorics/combinatorial constructions! that shit's hard

also if you want to dirty your hands a bit, a bit of statistics etc wouldn't hurt - knowing the definition of a beta distribution is nice but knowing what real-life scenarios you would use it for is nicer

[Anonymous No.10101630]

>>10101055
written by a no name russian larping as a gommie in brazil wow

[Anonymous No.10102249]

So which chart is king? who's willing to do the poll.

[Anonymous No.10102626]

>>10097475
Do you know where I can find PDFs of Gelfand's books ?

[Anonymous No.10102632]

>>10102626
No.

[Anonymous No.10102678]

>>10102626
Libgen.io

[Anonymous No.10102726]

>>10102678
Thank you very much

[Anonymous No.10102736]

>>10101267
>sicp
>a compiler book

>nicer language, like Java

>>>/g/tfo and never return here.

[Anonymous No.10104318]

>>10097475
Will these charts make me able to solve this? Because Im not able to solve this and I dont want to disappoint the anime ladies.

[Anonymous No.10104729]

>>10104318
solve what

[Anonymous No.10104731]

>>10104729
This.
She told me I should be able to solve this.

[Anonymous No.10104882]

>>10098429
I think that they complement each other nicely, have you seen how thin Spivak is? Sometimes it gets way too terse for me

[Anonymous No.10105303]

>How to Prove It / Book of Proof
When you don't understand the math, just look at the plethora of resources available online for basica algebra, trig and functions - this follows throughout.

>Spivak, Calculus
obvious GOAT material. There's other choices, but this a "best" guide, not an all inclusive brainlet friendly one.

Then you're basically good to go and above 95% of /sci/, but you can continue with

>Linear Algebra: An Introduction to Abstract Mathematics OR LADR by Axler
I really did not like H&K much compared to the above. I tend to agree Axler's weird autism about determinants too, but if you find yourself not compelled by his arguements (see: Down with Determinants! for an example) then feel free to opt out of it.

After that you're really ready to tackle whatever you want, you have the foundations to go into most undergrad stuff and as a plus it only takes three fuckin books ya wankers.

>>10098468
or this if you love memes

[Anonymous No.10105310]

>>10105303
can someone make this into a meme trilogy thing so people actually start listening to this sage advice

[Anonymous No.10105491]

>>10105303
Recently I've been looking for a good linear algebra book. Again, I've been recommended LADR, but the wiki ways it's bad because he doesn't deal with determinants until the very end of the book. Could you briefly explain both sides of the argument?

[Anonymous No.10105496]

>>10098172
that was the point of my post anon, aren’t mathfags supposed to be the second highest iq majors?

[Anonymous No.10105499]

>>10105491
go read shilov

[Anonymous No.10105508]

>>10105491
Basically, Axler's side is that determinants crop up early on linear algebra education and without much motivation and the intuitive understanding of them gets left in the dust too often. According to him, they are often used when simpler methods would work. His opinion is fully fleshed out here: http://www.axler.net/DwD.pdf

The other side basically says lmao dude stop being such a brainlet, determinants aren't that hard and they're very handy.

Personally, I think both sides are right, but I enjoyed the delayed presentation of determinants a bit more. I did feel they were a bit rushed normally, with Shilov being a good example that I had this feeling about (it was my first LA book and I quickly abandoned it as I was too much of a noob at that time). I too avoided it because of what the wiki says (and it's gay ass name), but ended up really enjoying it.

Beyond the pedagogical issues of determinants, I liked the type setting in edition 2 and found the problems early on in the book more engaging than Hoffman and Kunze's. I think the book by Valenza had some stellar problems too though, and I really loved that books prose and unification of abstract algebra and linear algebra from the get go, though I do think it's a bit more challenging because of that.

[Anonymous No.10105515]

>>10105508
>http://www.axler.net/DwD.pdf
Samefag here. One thing to consider when reading this is that he has a book to sell, and this might be accurately considered his sales pitch of some sort or another.

[Anonymous No.10105525]

>>10098468
>Pre-algebra
>A Course in Arithmetic by Serre
>Farming
>Sheaf Theory
>Queer studies
>Homology

The shit's hilarious.

[Anonymous No.10105603]

>>10097475
Optimized.

[Anonymous No.10105607]

>>10097476
Optimized.

[Anonymous No.10105611]

>>10097477
Optimized.
>>>/b/783616442

[Anonymous No.10105689]

>>10098468
IRL rofl

[Anonymous No.10105702]

Are there charts that just focus on a single field? I.e something like all books you should read to be an "algebraist" or similar for analysis, topology etc?

[Anonymous No.10107075]

>>10105607
>>10105603
Based

>>10105611
Not based

[Anonymous No.10107082]

>>10107075
Optimized.
>>>/b/783661187

[Anonymous No.10107097]

>>10107082
Based

[Anonymous No.10107104]

>>10099468
>>10099470
>>10099472
>>10099481
>>10099484
>>10099485
>>10099489
>>10099491

Now put this in a chart or I wont take you seriously.

[Anonymous No.10107480]

>>10107104
>Now put this in a chart or I wont take you seriously.
That's left as an exercise for the reader.

[Anonymous No.10107514]

>>10097475

Why is this a meme chart? The Algebra and Trig books are good, along with basic mathematics. Can't comment on the graduate texts because I'm a brainlet.

[Anonymous No.10108134]

>>10099468
>>10099470
>>10099472
>>10099481
>>10099484
>>10099485
>>10099489
>>10099491
Can you please make it a chart with an anime girl so I don't have to scroll past a wall of text every time this meme gets posted.

[Anonymous No.10108149]

>>10097475
Carmo's book is shit.

He ONLY focuses on surfaces and doesn't generalize to higher dimensions.

It was the first textbook I've ever regretted buying.

[Anonymous No.10108151]

>>10097475
Are Tao's books superior to Rudin?

[Anonymous No.10108161]

>>10101201
nigger royden has shitloads of measure theory in his massive book

[Anonymous No.10108167]

>>10108149
Duh, that's what "curves and surfaces" mean.

[Anonymous No.10108168]

>>10105491
pretty much what second anon said, but my 2 cents is:

>Axler puts a lot more emphasis on understanding linear maps than you could in a course that uses determinants

>however, determinants are really (REALLY) fucking useful, to the point that if you don't get to the chapter on determinants (or supplement it somewhere else), you will be completely left behind in almost any math book that requires even a bit of Linear algebra

[Anonymous No.10108170]

>>10108149
>the book on surfaces doesn't teach non-surfaces

[Anonymous No.10108221]

First, learn the tools (and their properties) of math.
1. Logic
2. Set
3. Symbols (Constants and Variables)
4. Functions
5. Numbers
6. Polynomials
7. Knowledge Representation (Cartesian coordinate, Fourier transform, etc)
8. Vectors and Matrices
9. Abstract Algebra

Once you learn those tools in order, you can pretty much understand every branch of mathematics.
- Elementary Algebra
- Linear Algebra
- Geometry
- Topology
- Calculus
- Physics
- Computer Science
- etc.

[Anonymous No.10109451]

>>10108151
depends on what you're looking for. they're way too easy imo, they're like a gentle, gentle introduction to analysis, maybe fit for someone who just finished stewart's calculus (which is when I read it)

[Anonymous No.10109482]

>>10101072
Where's the Dragon book? Where's SICP? And most important, where's Knuth's bible?

[Anonymous No.10110158]

>>10107514
Most of the elemental books are easily replaced by Lang, or even khanacademy.