Mathematics Study Guide : A Foundational Approach

[Anonymous No.10106745]

Hello,

I am willing to get into mathematics by the path of self-study and found this reading list online. I think this approach is quite unusual but really interesting and I am willing to try it out. What do you think of this list ? Would you recommand it to someone with a high school level of education ? Would you make any change to the list ?

[Anonymous No.10106753]

>>10106745
You fell for the meme.

[Anonymous No.10106769]

>>10106753
Don't you have something a bit more insightful to say ?

[Anonymous No.10106813]

>>10106745

need more info, what is your motivation and endgoal?

[Anonymous No.10106854]

>>10106813
I am willing to learn math mostly because I find that it is a fascinating subject. I would like to have solid foundations so that one day I can go to college. I'm interested in this list because proofs and logic are some of the aspects that appeal to me the most.

[Anonymous No.10106865]

>>10106854

ok, so the objective is college, with a sprinke of unrelated things.

assuming you're getting into a proper STEM field, logic is not "that" essential above the basics, proofs are a whole different story and you can't learn to do proofs in general, since they are pretty linked to specific fields.

having said that, I don't know your starting level and I will assume you are still in highschool.
Do all of the proofs of every theorem you encounter, geometry and algebra/calculus alike.

but I'd need more info to give you a more detailed list of recommendations.

[Anonymous No.10106882]

>>10106745
Redundant as fuck
Either read "book of proof" or " transition to advanced mathematics"
"Elements of set theory" covers most of the material in "foundation of analysis" and "naive set theory"
Read "basic mathematics" or "principles of mathematics"
Courant>Apostol>>Spivak

[Anonymous No.10106891]

>>10106865
I have "graduated" from high school two years ago, but I am willing go back to school. If that can help you, topics that I'm looking forward to are topology, non-euclidian geometry and linear algebra.

[Anonymous No.10106899]

>>10106891

the only one you could start doing straight out from high school is linear algebra and maybe, maybe some bit of non euclidean geometry, but I'd not recommend it.

For linear algebra any textbook (not Lang) is good, also look at some courses online, the Strang from MIT is a good starter, very hands on.

[Anonymous No.10106916]

>>10106899
Thanks for your advice, any thoughts on this linear algebra book ?

https://www.amazon.com/Linear-Algebra-Dover-Books-Mathematics/dp/048663518X

[Anonymous No.10106940]

>>10106899
not op. I've been recommended Lang's several times is there any reason you don't?

[Anonymous No.10106949]

>>10106916

I have no idea about that particular book, but I'd recommend you start off with something not too hardcore and which has exercises, those never hurt.
See what they recommend at MIT's linear algebra class.
Also remind that reading multiple books and finding those who better suit your tastes and learning style is a viable solution.

>>10106940

Lang's algebra (not linear algebra) is by no means bad, but is probably a bit overrated and old, I have it but I also use other books along with it.

[Anonymous No.10106966]

OP here, any book recommendations on probability and statistics ?

[Anonymous No.10107005]

suh, how do i best learn basic trigonometry. unit circle and shit like that.

[Anonymous No.10107008]

>>10107005

any highschool book+internet resources are fine

[Anonymous No.10107092]

>>10106745
>Would you make any change to the list ?
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.10107094]

>>10107092
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.10107095]

>>10107094
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.10107099]

>>10106940
Lang's introduction to linear algebra is ok, the other one I don't know

[Anonymous No.10107103]

>>10107095
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.10107106]

>>10107103
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.10107107]

>>10107106
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.10107111]

>>10107107
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.10107113]

>>10107111
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.10107119]

>>10107092
This is a bit elementary, but I guess this might be sufficient in very limited circumstances.

[Anonymous No.10107653]

>>10106745
From my experience with self-study, learning which books are the best for you is also part of the experience. Asking for recommendations is always good, but try to get them from different sources, and don't 100% rely on them, because what you want on a book is going to be different from those people.

The Laws of Truth is a nice entry book for getting into the theory of classical mathematical logic, but it will in no way prepare you for 'alternative' nor a philosophical discussion for logic (it does good on semantics though).

>>10106882 has some points, but I'd honestly recommend reading "elements of set theory" AND "naive set theory". Enderton might be a little bit harsh if you're just getting into abstract reasoning. Halmos' is always a good companion for any set theory book. (By the way, Jech's is the standard book for more advanced foundational set theory).

Regarding proofs, there is yet a book that leaves me satisfied on the topic. I'd suggest learning from different sources. Velleman's my favourite though.

The following books are all skippable until A Transition to Advanced Mathematics. If you plan to get into serious undergrad pure math, I honestly recommend you reading books of this type. If you plan to study "mathematics" for the sake of it and not just preparing for college, For later on algebra I'd suggest Fraleigh's and Herstein's as introductions. For Linear Algebra, get yourself Axler's (to-the-point) and Hoffman's (actually goes more in-depth) I'd suggest skipping calculus and learn analysis from the get-go. Prerequisites are only a solid grasp of set theory and you being comfortable with abstract thinking.

[Anonymous No.10107675]

>>10107653
>I'd suggest skipping calculus

Everyone hates aspies that do this.

[Anonymous No.10107722]

>>10106966
Intro to statistical methods. It shows you all the different types of tests as well as correlation and such. Other than that mathematical statistics is good for all the theory of combinations as well as expected value and probability density functions and such

[Anonymous No.10108298]

>>10106916
I’ve worked through that book, and I can say it’s not for folks starting out in learning higher math. The content isn’t the difficult part, linear algebra is a really good gateway subject into rigorous math. The problem is that this book doesn’t present its material well for someone trying to build mathematical maturity. The text is really dense and doesn’t have much in the way of exposition. It wouldn’t be impossible to tackle as a first book; you could bust your ass trying to slowly trudge your way through, but ime it’s painful and demoralizing. You might like to start with Axler’s Linear Algebra Done Right as a way to learn linear alt and acclimate to higher math.

[Anonymous No.10109399]

how do i permanently filter these threads?

[Anonymous No.10109944]

>>10108298
It's true that Shilov's book is quite dense I'll look into Axler first it's really more newbie-friendly
>>10107653
Thanks for your detailed answer, really helpful

[Anonymous No.10110030]

>>10108298
I feel like a lot of the concepts and insight are buried under the notation in shilov, and it has that "using formalism to show that the results 'happen' to hold, rather than showing 'why' " feel to it. Maybe it was because I hadn't taken a cookbook linear algebra class beforehand. Axler turned out to be a much better first exposure for me

[Anonymous No.10110668]

>>10107675
why

[Anonymous No.10110683]

Would you recommend skipping Logic Laws of Truth and going directly into Hammack/Velleman proof book ?

[Anonymous No.10110697]

>>10110683
ye

[Anonymous No.10110860]

https://discord.gg/KwVZCTn

[Anonymous No.10111145]

>>10110860
discordianists are falling right into TPTB hands.

[Anonymous No.10111957]

>>10111145
lmao, true.

[Anonymous No.10111963]

Not OP here. What amount of Geometry knowledge is sufficient to proceed with math? I was trying to learn calculus last year, but evey once in a while stumbled across some geometry facts and theorems, that I was supposed to know from school. So I had to distract from the subject at hand and dive into a particular geometry problem. This jumping back and forth was annoying and made me drop the whole thing eventually.

[Anonymous No.10112029]

>>10111963
I never took geometry in school and graduated with honors in math. Never needed anything other than [math]a^2+b^2=c^2[/math], [math]s=2 \pi r[/math], and [math]A=\frac{1}{2}bh[/math]. The rest you can get with algebra and calculus.

[Anonymous No.10112032]

>>10112029
Thanks, I will not bother with this shit then.

[Anonymous No.10112797]

>>10111963
Do you have examples of what was blocking you ?

[Anonymous No.10112818]

>>10106745 >>10112029 >>10107113 >>10111963

[Anonymous No.10112886]

>>10112818
The "Application" part always gets me

[Anonymous No.10112914]

>>10106745
Will this actually help you if you're a brainlet that doesn't know anything about math/logic, or is this just a meme?

[Anonymous No.10113185]

>>10106769
Here's something: If you want to self study go to a university's website and see the books they use for their courses and use those. Calc1? Get Stewart's book

[Anonymous No.10113228]

>>10112914
It does help, but you need to commit to it. Also, know that it'll take time. A lot of time.
t. former brainlet

[Anonymous No.10114441]

>>10112914
There's a lot of redundancy in that chart. Refresh your high school math, then grab first year uni math books.