/sci/ - Science & Math

[Anonymous No.12753603]

Before I go through this meme chart is it a good chart to gain a foundational comprehension of mathematics? Or are some books not good/excludable?

[Anonymous No.12753634]

What are your interests? You want to improve your college level or you're interested in math for its own sake? If so, any goals you want to achieve?
Math book charts are meaningless without context.

[Anonymous No.12753827]

>>12753634
I just want to be good at math for its own sake. I'm very flawed in my math abilities and want to understand it to gain a better worldview/knowledge sake. Although I do have a leaning interest in Geometric shapes(they're cool) and Mathematical logic, I've been told by friend's that are math majors that "proofs" are a far more creative output to hone/train your math skills and that college-level math is nothing like the linear highschool type which I hated(not that it's not necessary by any means though).

Mathematical logic because I wish to understand philosophy which originates from it mainly.

[Anonymous No.12753834]

>>12753603

[Anonymous No.12753839]

>>12753603
>>12753827
Skip the meme chart, it was written by an undergrad. Read cohen's "set theory and the continuum hypothesis" for the foundations stuff. It's super short and accessible, and it's masterful pedagogy. Read Lang for everything else, he is the pedagogical king. There's a Lang chart somewhere around here.

[Anonymous No.12753841]

>>12753834
>Philosophy
>Engineering
Kek

[op No.12753867]

>>12753839
I'm going through a mega with a bunch of charts and can't find the Lang chat you speak of. Most of the charts are like pic rel or >>12753834

[Anonymous No.12753881]

>>12753603
For this chart, depending on where you all you need to read are basic mathematics, book of proof, calculus and dummit and foote

IMO studying linear algebra in its own right before abstract algebra is retarded but you're gonna wanna watch this youtube series before reading dummit and foote (https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab))

[Anonymous No.12753887]

>>12753867
you did an incompetent search, use warosu, here this is the one

[op No.12753894]

>>12753881
This youtube series looks great, I'll keep those three in mind too
>>12753887
These covers together are so aesthetically pleasing and all these books seem to cover everything I could possibly want

thanks, guys, going to bookmark this thread

[Anonymous No.12753933]

>>12753894
>someone actually fell for the Lang chart
I cannot believe a human being can achieve this stupidity.

[Anonymous No.12753945]

Is this any good?

[Anonymous No.12754056]

>>12753827
The problem with mathematics is that it's hard to get a general picture of what a particular field is about unless you delve deeply into it. So whichever fields you're intrigued by may appear pretty dull, while the fields you ignored can turn out to be fun once you get into it.
Keep in mind that different fields of mathematics can be very disconnected. Two mathematicians can have little in common if their area of expertise differ.
To get a vague impression of what math is browse through advanced lectures, seminars, arxiv papers just for the vibe of it to feel the culture without the intent to understand a lot, read mathematicians' biographies. This will give you a wider outlook on what it is all about.
Once you get to this layman level, it will be pretty obvious for you which books to read and you wouldn't feel like asking.
Curiosity about mathematical logic is a good thing in the beginning. You can unironically learn its fundamentals by watching youtube vids, reading blogs, wiki and stack exchange. You have to be comfortable with induction, the idea of cardinality (infinities coming in different sizes), Russel paradox, Cantor's diagonal argument, ordinal numbers, being able to express everything in terms of sets.
>>12753839
"set theory and the continuum hypothesis"
This book would likely not fit you now even if you're an aspiring model theorist. Develop an intuition about set theory first by implementing it different contexts. Actually pretty much everything you really need about set theory is in the appendix of Munkres' topology.

Everybody has different taste, but you may give a shot at Allufi's algebra. It's a thourough text and I feel it's beginner-friendly. Also it familiarizes you with categories right away, which may play into your interest about the philosophy behind.

[Anonymous No.12754166]

>>12754056
>To get a vague impression of what math is
Since you know more about it then can you recommend any of the "lectures, seminars, papers and biographies" you personally find fun?

[Anonymous No.12754181]

>>12753834
holy shit this still exists

[Anonymous No.12754296]

If I want to be an actuary how much math do I need

[Anonymous No.12754611]

>>12754166
>"lectures, seminars, papers and biographies"
I meant you should feed your curiousity by just researching more, widening your perspective, so you can stumble over things which will grab your interest. You should also realize how wide math is and you can only focus on a small subset of it. Hence you'd get different advices.

Like suppose you're really curious about mathematics of string theory. Then you would read about it from a non pop-sci source, say wikipedia. You see all these fancy mathy things which make little sense as for now, you will keep seeing words like "manifold", "variety" "homology", "sheave". Then you go on reading about them, and you quickly realize what math you have to learn to understand it, like smooth manifolds, topology. Google standart introductions to those subjects. Prerequisites to each field are well known and usually stated in the beginning of the book, so in case a book is too advanced, you'll easily know how to catch up. Eventually there'll be a book for your level.
Sometimes prerequisite material may not appear as fancy, but you have to get through it anyway.
Mathematical concepts should cause some emotional associations. Otherwise it's hard to develop intuition about something which has no meaning to you.
In my case I was curious about homotopy groups of spheres before I got into math, so it gave me a path of study for a few years.

If you're into esoteric schizo stuff, setting to understand the works of people like Grothendieck and Quillen might be interesting.


Also there seems to be a lot of pop math on youtube, check out PBS Infinite Series and 3blue1brown

[Anonymous No.12754640]

>>12754296
Look at the curriculum for exam P, exam FM, and exams MAS 1/2. You should be able to see all that in undergrad.

[Anonymous No.12754745]

>>12754296
A lot, but mainly calculus (up to 2 for FM) and probability.

[Anonymous No.12754949]

Mathematics program should be designed so

School program (exam Matshkolnik)

Euclidean geometry, complex numbers, scalar multiplication, the Cauchy-Schwarz inequality. Principles of Quantum Mechanics (Kostrikin-Manin). Group of transformations of the plane and space. Conclusion of trigonometric identities. The geometry of the upper half plane (hyperbolic). Properties of inversion. Effect of linear fractional transformations.

Rings and fields. Linear algebra, finite groups, Galois theory. Proof of Abel. Basis, rank, determinant, classical Lie group. Dedekind. Determination of the real numbers. The definition of the tensor product of vector spaces.

Set theory. Zorn's lemma. A well-ordered set. Cauchy-Hamel basis. Cantor-Bernstein theorem. Uncountable set of real numbers.

Metric spaces. The set-theoretic topology (defined continuous maps, compact, custom maps). Countable base. The definition of compactness in terms of convergent sequences for spaces with a countable base.Homotopy, the fundamental group, homotopy equivalence.

p-adic numbers, Ostrowski theorem, multiplication, and division of the p-adic numbers in a column

Differentiation, integration, the Newton-Leibniz. Delta epsilon formalism lemma policeman.

[Anonymous No.12754963]

The first course

Analysis on $ R ^ n $. Differential display. lemma contracting maps. Implicit Function Theorem. Riemann and Lebesgue. ("Analysis" of Laurent Schwartz, "Analysis" Zorich, "Problems and theorems of functions. Analysis" Kirillov-Gvishiani)

Hilbert spaces, Banach spaces (definition). The existence of a basis in the Hilbert space. Continuous and discontinuous linear operators. Criteria for continuity. Examples of compact operators. ("Analysis" of Laurent Schwartz, "Analysis" Zorich, "Problems and theorems of functions. Analysis" Kirillov-Gvishiani)

Smooth manifolds, submersion immersion Sard's theorem. Partition of unity. Differential Topology (Milnor-Wallace). Transversality. Degree of the map as a topological invariant.

Differential forms, de Rham operator, Stokes theorem, Maxwell's equations of the electromagnetic field. Gauss's theorem as a particular example.

Comprehensive analysis of a single variable (based on the book by Henri Cartan, or the first volume of Shabbat). Contour integrals, Cauchy, Riemann mapping theorem of any simply connected subset of $ C $ in a circle theorem on the extension of borders, Picard's theorem on the achievement of an entire function of all values except three. Multivalent functions (for example, the logarithm).

Category theory, the definition of the functors, equivalence, adjoint functors (McLane, Categories for working mathematician, Gelfand-Manin, the first chapter.)

Lie groups and algebras. Lie groups. Lie algebra as linearization. Universal enveloping algebra, the Poincare-Birkhoff-Witt. Free Lie algebra. Campbell-Hausdorff series and the construction of a Lie group by its algebra (yellow Serre, first half).

[Anonymous No.12754970]

The second course

Algebraic Topology (Fuchs-Fomenko). Cohomology (simplicial, singular, de Rham), their equivalence, Poincaré duality, the homotopy groups. Dimension. Bundle (Serre), the spectral sequence (Mishchenko, "Vector bundles ..."). Computation of the cohomology of classical Lie groups and projective space.

Vector bundles, connection, the Gauss-Bonnet, the Euler class, Black, Pontryagin, Stiefel-Whitney test. Multiplicativity of the Chern character. Classifying spaces ("Characteristic Classes", Milnor and Stashef).

Differential Geometry. Levi-Civita connection, the curvature of the algebraic and differential Bianchi identity. Killing Fields. Gaussian curvature of two-dimensional Riemannian manifold. Cell decomposition of the loop space in terms of geodesics. Morse theory on the loop space (based on the book of Milnor, "Morse theory" and Arthur Besse "Einstein manifolds"). Principal bundles and connections in them.

Commutative algebra (Atiyah-Macdonald). Noetherian, Krull dimension, Lemma Nakayama adic completion, integrally closed, discrete valuation ring. Flat modules, local criterion plane.

Beginning of algebraic geometry. (Chapter Hartshorne either green or Shafarevich Mumford). Affine variety, projective variety, projective morphism, the image of a projective variety is projective (via resultants).Beams. Zariski topology. Algebraic variety as a ringed space. Hilbert's theorem on zeros. The spectrum of the ring.

[Anonymous No.12754981]

Beginning of homological algebra. Group Ext, Tor for modules over a ring, resolutions, projective and injective modules (Atiyah-Macdonald). Construction of injective modules. Grothendieck duality (in the book Springer Lecture Notes in Math, Grothendieck Duality, number about 21 and 40).

Number theory, local and global field discriminant rule the ideal class group (blue book Cassels and Frohlich).

Reductive groups, root systems, representations of semisimple groups, the weight, the Killing form. Groups generated by reflections, their classification. Lie algebra cohomology. Computation of the cohomology in terms of invariant forms. Singular cohomology of a compact Lie group and its cohomology algebra. Invariants of classical Lie groups. (Yellow Serre, the second half, Hermann Weyl, "Invariants of classical groups"). Design of special Lie groups. Hopf algebra. Quantum groups (definition).

The third course

K-theory as a cohomological functor, Bott periodicity, Clifford algebra. Spinors (Atiyah's book "K-theory" or Mishchenko "the vector bundles and their application"). Spectra. Eilenberg-Mac Lane. Infinite loop space (the book Switzer or yellow book or Adams Adams "Lectures on generalized cohmology", 1972).

Differential operators, pseudodifferential operators, symbol, elliptic operators. Properties of the Laplace operator. Self-adjoint operators with discrete spectrum. Green operator and applications to Hodge theory on Riemannian manifolds. Quantum mechanics. (Book R.Uellsa analysis or Mishchenko "vectors of the bundle and its application").

[Anonymous No.12755002]

>>12754981
>he got bored midway
Cringe.

[Anonymous No.12755046]

The index formula (Atiyah-Bott-Patodi, Mishchenko), the Riemann-Roch. Zeta function of the operator with discrete spectrum and its asymptotic behavior.

Homological algebra (Gelfand-Manin, all in the last intermediate). Cohomology of Sheaves, derived category, triangulated category, derived functor, the spectral sequence of a double. The composition of triangulated functors and the corresponding spectral sequence. Verdier duality. The formalism of the six functors and perverse sheaves.

Scheme algebraic geometry, the scheme over the ring, the projective spectra, derivatives, Serre duality, coherent beams, replacement base. Own and separated scheme, valyuativny property and separability criterion (Hartshorne). Functors representability, the moduli space. Direct and inverse images of sheaves, higher direct images. With proper mapping higher direct images are coherent.

Cohomological methods in algebraic geometry, semicontinuity cohomology theorem of Zariski connectedness theorem Stein decomposition.

Kahler manifolds, Lefschetz, Hodge theory, the ratio of Kodaira, properties of the Laplace operator (zero head head Griffiths-Harris, clearly stated in the book André Weil "Kahler manifolds"). Hermitian bundle.Line bundles and their curvature. Line bundles with positive curvature. Kodaira-Nakano Vanishing on (Griffiths-Harris).

Holonomy, the Ambrose-Singer theorem, special holonomy holonomy classification, Calabi-Yau, hyper, theorem of Calabi-Yau manifolds.

[Anonymous No.12755052]

Spinors on the manifold, the Dirac operator, the Ricci curvature, the formula Weitzenbock-Lichnerowicz, Bochner's theorem. Bogomolov's theorem on the decomposition of manifolds with zero canonical class (Arthur Besse, "Einstein manifolds").

Tate cohomology class field theory (Cassels-Frohlich, blue book). The calculation of the factor Galois number field commutator. Brauer and its applications.

Ergodic theory. Ergodicity of billiards.

Complex curves, pseudo-conformal mappings, Teichmüller space, the theory of Ahlfors-Bers (the book Ahlfors thin).

11
User avatar
level 2
potato21
8 years ago

Fourth year.

Rational homotopy type and profinite étale cover nerve cell complex homotopy equivalent to its profinite type. Topological definition of étale cohomology. Galois action on profinite homotopy type (Sullivan, "Geometric Topology").

Étale cohomology in algebraic geometry, the comparison functor, Henselian, geometric point. Base change. Any smooth variety over a field locally in the étale topology is isomorphic to $ A ^ n $. Étale fundamental group (Milne, review of Daniel VINITI and SGA 4 1/2, the first article of Deligne).

Elliptic curves, j-invariant, automorphic forms, Taniyama-Weil conjecture and its applications to number theory (Fermat's theorem).

Rational homotopy (in the last chapter of the book or the Gelfand-Manin article Griffiths-Morgan-Long-Sullivan). Massey operations and rational homotopy type. Vanishing Massey operations on a Kahler manifold.

[Anonymous No.12755058]

Chevalley group of generators and relations (the book Steinberg). K_2 computation of the field (Milnor, Algebraic K-theory).

Algebraic K-theory of Quillen, $ BGL ^ + $ and $ Q $-structure (review Suslin in the 25th volume of VINITI, lectures Quillen - Lecture Notes in Math. 341).

Complex analytic manifold, coherent beams, the coherence theorem of Oka, Hilbert's Nullstellensatz for ideals in the sheaf of holomorphic functions. Noetherian ring of germs of functions, Weierstrass theorem on division, Weierstrass preparation theorem. Theorem on a branched covering. Grauert-Remmert (image of a compact analytic space under a holomorphic analytic morphism). Hartogs' theorem on the continuation of an analytic function. Multidimensional Cauchy formula and its applications (uniform limit of holomorphic functions is holomorphic).

Fifth Year

Kodaira-Spencer theory. Deformation of a manifold and solutions Maurer-Cartan equations. Solvability of the Maurer-Cartan and Massey operations on the DG-Lie algebra cohomology of vector fields. Moduli space and finite (see lecture Kontsevich, or the collected works of Kodaira). Theorem Bogomolov-Tian-Todorov on deformations of Calabi-Yau manifolds.

Symplectic reduction. Moment map. Kempf-Ness theorem.

Deformations of coherent bundles and bundles in algebraic geometry. Geometric invariant theory. Moduli space of bundles on a curve. Stability. Uhlenbeck compactification, Gieseker and Maruyama. Geometric invariant theory is the symplectic reduction (third edition of Mumford's geometric invariant theory, application, Frances Kirwan).

Instantons in the four-dimensional geometry. Donaldson theory. Donaldson invariants. Instantons on Kahler surfaces.

[Anonymous No.12755064]

The geometry of complex surfaces. Kodaira classification, Kahler and nekelerovy surface of the Hilbert scheme of points on the surface. Castelnuovo-Enriques criterion, the Riemann-Roch inequality Bogomolov-Miyaoka-Yau. The ratio of invariants of the surface. Elliptic surfaces, the surface Kummer K3 surfaces and Enriques.

Elements of the program Mori: Kawamata-Viehweg zero freedom theorem of base points, the theorem of Mori cone (Clemens-Kollar-Mori, "Multidimensional complex geometry", plus not translated Kollar-Mori and Kawamata-Matsuki-Massoud) .

Stable bundles as instantons. Yang-Mills equation on a Kahler manifold. Theorem Donaldson-Uhlenbeck-Yau metrics on the Yang-Mills theory on a stable bundle. Its interpretation in terms of the symplectic reduction. Stable bundles and instantons on hyperkähler manifolds, an explicit solution of the Maurer-Cartan equation in terms of the Green's operator.

Pseudoholomorphic curves in symplectic manifold. Gromov-Witten. Quantum cohomology. Mirror conjecture and interpretation. The structure of the group of symplectomorphisms (under the Kontsevich-Manin book Polterovich, "Symplectic geometry" green book of pseudoholomorphic curves and lecture notes McDuff and Salamon).

Complex spinors, Seiberg-Witten, Seiberg-Witten. Why Seiberg-Witten invariants are Gromov-Witten.

[Anonymous No.12755109]

>>12753603
Yes. This one is an okay start.
Someone bring out "The Arch-Wizard" chart and show this brainlet The True True.

[Anonymous No.12755146]

>>12755109
This?

[Anonymous No.12755270]

>>12753945
well actually pretty good

most of those are usually recommended in some of the charts on /g/

what I'm certain about is K&R (C prog) - it really reinforced my c knowledge although it is pretty outdated check https://wiki.installgentoo.com/wiki/Programming_resources for a bit more upto date C book
Book of proof is useful to build mathematical reasoning for other subjects
Computer Systems, Algorithm design manual, Intro to Theory of Computation are also great for an undergrad
Graph theory was mandatory for my Graph theory class so go figure

And of course TAOCP is the one to rule them all, unfortunately too brainlet to do it proper but Bill Gates said that if you go through it to give him a call and he will find a position for you at Google so...

[Anonymous No.12755280]

>>12755146
W-What the fuck is this?

[Anonymous No.12755285]

>>12755280
Autism. The purest autism.

[Anonymous No.12755345]

>>12755270
my nigga im a third worlder, how the fuck im gonna call him , thanks tho

[Anonymous No.12757436]

>Ahegao faces
reddit moment