The Gelfand books are short, sweet, clear, and well-written by an eminent mathematician. If you just want to study some good geometry books, try Kiselev's Geometry 1 and/or 2.
Alternatively you could forget Gelfand and Kiselev and just read either Simmons' "Precalculus Mathematics in a Nutshell", or Lang's "Basic Mathematics". Three routes to the same destination, take your pick.
>i wants to lurn mathh & sighence *brainlet wojak image*
Nice job setting a non-vague goal for yourself, OP. Math and science are kind of enormous human endeavors, if you try to be a "renaissance man" you won't end up with more than a freshman undergraduate understanding of anything.
>QFT before "more specialised phusics"
idk dude.
I'm teaching myself orgo and I'm committing to finishing it in 2 weeks. Make sure that I stay on the track /adg/.
I'm guessing it depends on your background since:
Gelfand/Kiselev route is ~900 pages
Lang's Basic Mathematics is ~420 pages Simmons' book is ~120 pages
Ya when I think basic math I think Grad students go fuck yourself you nigger
I see. Well, I'll break it down. Start by learning to be able to count to 100. Once you can do that without fucking up, start with arithmetic, but just adding the natural numbers. After you've done that for a few years, come back and ask for more trivial advice.
Also, enroll in a class you fucking pompous unsocialized 'autodidact.'
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 [math] C [/math] 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).