/sci/ - Science & Math

>>9747087
>tell me where to start?
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.

>>9747096
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).

>>9747098
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).

>>9747100
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.

>>9747101
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).

>>9747102
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).

>>9747103
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.

>>9747105
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)

>>9747096
>>9747098
>>9747100
>>9747101
>>9747102

what's the point in even posting this. he wants to learn calculus and will pretty much just ignore these posts. anyone in op's boat will do that same as well. these posts would only appeal to a very small minority which would include ramanujan and even then not really because he would want to study specific things. this is also ignoring the fact that it's incredibly unrealistic and has literally no time schedule for each subject. you just arbitrarily threw them together and then spam them while everyone skips it.
protip: anyone who hasn't already studied at university wouldn't be able to read this

>>9747107
>>9747105
>>9747103
>>9747102
>>9747101
>>9747100
>>9747098
>>9747096

Whoa Whoa Whoa- Thank you-
I won't study this all overnight, But I'll save this list, and progress alongside it- I feel like, at this point (dry 24 y.o.) It'll take me my whole life to know everything- But I'll try- Damn, I wish my parents had exposed me sooner to math, though...

>>9747111
Trips of truth- this is a good guideline for research, though


hopefully I don't fall flat on my face

note: my englisch sucks ass so good luck reading this.

first of all good for you setting that goal.

math is, in my experience, like a ladder. were ever you want o go. you start at the bottom step. for calculus this means first to understand the language of mathematics. that language has, so to speak, a syntax (set theory) and of course a grammar (logic). followed by the property of different sets of numbers. then it's on to equations and functions. Last step would be basic euclidean geometry to gain an understanding of trigonometry. and than it's time for calculus! most precalculus books follow this pathe. understanding the concepts is of far more importance then solving exercises. the internet offers an intimidating number of free courses. there is khanacademy.org and patrickJMT to name some i like. any book will do. just pirate some and look what appeals to you.

to become good a math, just do a lot of math. there is no royal road to calculus. you have to put in work.

here some books for free

https://openstax.org/subjects/math

>>9747087
Get a few introductory level math books, start with basic arithmetic ones even.

Now upen two or three, and get through them page by page, use internet as back up to help you understand said books and augment/compliment the learning.

Also, have two notebooks, in addition to alot of scratch papper, the scratch papper is self explenatory, the second notebook serves to write down interesting things you came about while you were solving excersizes, the first notebook serves to document everything you learn as you go, so you can easily revisit, so a few excersizes, references, and processeses explained.

And that is basically the way, open two or three books on the same matters and at the same levels, go through them, and write down your summary as you go, also do not be afraid to go back to basics, arithmetic itself is simple but rich and complex, soon you will realize maths are very tricky beyond the first pale of simplicity, and starting with the basics ensures you have very early roots into the heart of darkness per se

Oh and do this studying/writing down/googling/video watching at least four hours a day, nonstop, no rests, math is ruthless to brainless like ourselves trust me, no mercy.

PS math books, buy or print, make annotattions in them, highlight them, whatever you need to do, they are your punching bags, and if you do not break your mental knucles into them it means you are not doing it right

>>9747122
This squiggly line is the symbol for integral, in other words, the area under a curve. The sigma sine means sum ex) [math]\sum_{k = 1}^{5}k[/math] is the same thing as saying 1+2+3+4+5

>>9747087
Watch some 3blue1brown videos, they're very visual and intuitive.

>>9747087
>I really want to learn math.

Out of curiosity, why do you want to learn? You think it's fun or just for a cushy job?

>>9750735
Fun, if you give me those two options. I want to think in a more complex way about the world and change my perspective by knowing math- I think understanding math is a way to really push your mind- A cushy job would be ideal, but that's just a bonus-

>>9751199
Good for you anon, I admire people who have interests outside of the normie realm of football and Game of Thrones.