What do you learn after calculus? My math education ended at

>>8883797
>What do you learn after calculus?
Everything.

>is topology considered to be a meme?
Algebra, Analysis and Topology are the 3 pillars of modern mathematics. Without any of them you are not a mathematician.

>Is there a /sci/ approved branch of maths?
Check the wiki for that but a good start would be:

>Complete Calculus (which means go through Calc II and III)
>Then get some introductory book on Analysis, Algebra and Topology. One book for each. For each topic you can find introductions that are like 300 pages long.

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>>8883797
Linear algebra, and no, topology isn't a meme, but that's too advanced for you.

>>8883802
>topology isn't a meme, but that's too advanced for you.
WRONG.

Topology is not too advanced for anyone. The core concepts of topology (point-set) is literally a self contained theory, for which you need no previous background in anything but the formal language of set theory.

After that you decide if you want to learn analysis and/or algebra to be able to apply topology to more interesting problems.

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>>8883797
for continuous shit:
Linear algebra
Real analysis
Ordinary Diff Eq
Partial Diff Eq
Complex analysis
Vector Analysis
Differential geometry

some other important shit:
Statistics
Abstract algebra
Signal processing
Algorithms and complexity Analysis

>>8883867
What is signal processing? I've never heard of this before.

>>8883801
Thanks, I'll complete up to Calc III first and then check the wiki for the specifics.

>>8883797

Math branches out once you passed calculus. Learn what you want once you have the prerequisites.

>Calculus
Prerequisites: Precalculus
Useful: Proofs
>Matrix Algebra
Prerequisites: Precalculus
Useful: Proofs, Calculus
>Proofs and Mathematical Reasoning
Prerequisites: Precalculus
>Multivariable and Vector Calculus
Prerequisites: Single Variable Calculus
Useful: Matrix Algebra
>Differential Geometry of Curves and Surfaces
Prerequisites: Multivariable Calculus, Matrix Algebra
Useful: basic Analysis
>Ordinary Differential Equations
Prerequisites: Calculus, Matrix Algebra
>Applied Linear Algebra
Prerequisites: Calculus, Matrix Algebra
Useful: Finite Vector Spaces
>Finite Vector Spaces
Prerequisites: Calculus, Matrix Algebra, Proofs
Useful: Applied Linear Algebra
>Complex Variables
Prerequisites: Multivariable Calculus, Matrix Algebra
Useful: basic Analysis
>Special Functions
Prerequisites: Complex Variables, ODEs
Useful: basic Analysis
>Fourier Transforms
Prerequisites: Multivariable Calculus, Matrix Algebra
Useful: ODEs, Complex Variables
>Partial Differential Equations
Prerequisites: ODEs
Useful: Fourier Transforms, basic Analysis, Complex Variables
>Introductory Set Theory
Prerequisites: Proofs
Useful: Logic
>Introductory Logic
Prerequisites: Proofs
Useful: Set Theory
>Number Theory
Prerequisites: Proofs
Useful: basic Abstract Algebra
>Probability Theory
Prerequisites: Proofs, Multivariable Calculus
Useful: basic Analysis, Combinatorics
>Numerical Analysis
Prerequisites: ODEs, Multivariable Calculus, Matrix Algebra
Useful: Proofs, basic Analysis, PDEs
>Analysis with Metric Spaces
Prerequisites: Proofs, Calculus
Useful: Topology, mathematical maturity
>Topology
Prerequisites: Proofs, Calculus
Useful: Analysis, mathematical maturity
>Analysis on Manifolds
Prerequisites: Analysis, Topology, Finite Vector Spaces
Useful: Curves and Surfaces
>Abstract Algebra
Prerequisites: Proofs, Calculus, Matrix Algebra
Useful: Finite Vector Spaces, mathematical maturity

>>8883797
>he fell for the calculus = advanced math meme

>>8883797
>>8883966
For books see
http://4chan-science.wikia.com/wiki/Mathematics

>>8883939

It's a subject in EE that built on Fourier/Laplace/Z-Transforms.

take some real linear algebra and maybe also multivariable calc if you hate yourself.
if you really hate yourself, take diff eq.

absolutely positively take stats if you haven't already. basic stats are obnoxious and annoy math purists, but they're essential for any kind of empiricism.

>>8883966
>>8883972

Thank you for the comprehensive list. It seems a bit overwhelming at first glance but I suppose everyone finds their niche in time.

>>8884016
A lot of the stuff on his list could be put together, so it's not that scary.

>logic, set theory

You will have a good enough grasp of it after going through real analysis, number theory and topology.

>matrix algebra

Just put it together with linear algebra

etc

What uni course/career involves the most/hardest calculus and algebra?

>>8885627
https://en.wikipedia.org/wiki/Math_55

>>8884023
>>logic, set theory
>You will have a good enough grasp of it after going through real analysis, number theory and topology.
You never learned logic and set theory properly, did you?

But yeah, that list is kinda crap.

>>8883801
>Topology
What do you want topology for? I understand how calculus is very useful, but topology?

>>8885789

Topology has a lot of applications in Biology as well as certain problems in engineering.

>>8885789
Topology is useful even in algebra (search for Zariski and Krull topology).
In order to study PDEs you'll need lots of functional analysis, and some function spaces are not so nice that they are normalizable, so you'll need the general theory of topological vector spaces.
Topology is also a prerequisite for studying dynamical systems.

Topological spaces appear naturally in most of mathematics because of its relation to order theory, adjointness (or Galois connections) and Moore closure.

>>8885809
>certain problems in engineering.
examples? im finishing my engineering degree and ive only ever heard about topology on /sci/

I'd recommend going through in this order

>Lang, Basic mathematics (precalc, conics, etc)
as at least a refresher
>Stewart, calculus
as a refresher + calc 2,3 + some other basic stuff
>How to prove it, Velleman and/or how to solve it, Polya
As a basic introduction to proofs
>Strang, Linear algebra
Basic linear algebra is needed in EVERY field
>Spivak, Calculus
To get a taste of real analysis, you can skip this though
>Pinter, A book on abstract algebra
A taste of basic abstract algebra, explained in a very intuitive way, with a LOT of exercises, as well as interesting exercises (you develop some code theory on the side)
>Halmos, Naive set theory
Short book, gives you an idea on the importance of set theory. You can skip this, but I really recommend it, and it's really short.
>Apostol, Mathematical analysis or Abbot, Understanding analysis
Intro to real analysis, it is important to give you the notion of why rigour is important, and gives a lot of motivation for other fields (especially topology).
>Hoffman/Kunze, Linear algebra or Halmos, Finite Dimensional Vector Spaces. Axler, Linear Algebra done right to supplement
More rigourous linear algebra, useful for differential geometry
>Artin, Algebra
Rigorous abstract algebra
>Jones, Elementary number theory
Skippable although it is a very beautiful subject


Do you prefer analysis or algebra? You can skip what you don't like, but some are a must

>Analysis
>Munkres, Topology
A very good, self contained text, you can skip the algebraic topology section
>Complex analysis: no real classic book, but Needham, Visual complex analysis is good and supplement with Lang or Stein/Sakarchi
>Tao, Intro to measure theory
>Evans, PDEs
>Do Carmo, Differential geometry of curves and surfaces
Intro to DG
>Milnor, Topology from a Differentiable viewpoint
Very good book on Differential topology
>Lee, Introduction to smooth manifolds
Differential geometry, alternatively, Spivak's 5 volumes on differential geometry are a classic

>>8885988


>Algebra
>Steward/Tall, Algebraic number theory with a view towards Fermat's Last Theorem
optional, but it is a satisfying subject
>Eisenbud, Commutative Algebra with a view towards algebraic geometry
>Serre, A course on Arithmetic
>hatcher, Algebraic topology
classic, and you can get it for free
>Shafarevich, Basic Algebraic Geometry, followed by Hartshorne


Obviously this is not a complete list, since I'm missing notably geometry, probability, analytic number theory, etc, but it should be something to strive for.

>>8885789
topology shows up in almost every branch of mathematics, minimally as a tool

in logic - stone spaces, topological sheaves for semantics
algebra - ring spectra, lattice theory (coherent locales)
analysis - metric spaces, differential topology

"continuous variation" shows up everywhere in math, and you often need special notions of continuity which topology makes precise

>>8885825
https://en.wikipedia.org/wiki/Topology_optimization
is the only one I know.
I don't even see how it's topology (no open sets?), but I'm just getting into that

>>8885988
Halmos is the worst book on set theory, use The Joy of Sets by Devlin or Elements of Set Theory by Enderton if the former is too hard for you, and Jech and Hrbaceck if that's too easy.

Munkres is also not great, although I learned from it first, it has no uniform spaces, filters and nets, try Willard instead.

Ahlfors is the classic complex analysis textbook, but no one likes it. I'd recommend Conway though.

Starting PDEs with Evans is too much, but I don't know any other book on PDEs...

Lee is good, but Bredons Geometry and Topology (Topology and Geometry?) is better.

I'm still learning algebraic geometry but Hatcher looked horrible. I'm enjoying Homology Theory by Vick and A Concise Course in Algebraic Topology by Peter May (which you can download for free).

Also, after measure theory and before PDEs, add:
>Functional Analysis
Elements of the Theory of Functions and Functional Analysis - Kolmogorov and Fomin
A course in functional analysis - Conway

>>8883797
>I went on to pursue a liberal arts degree
wew lad

>>8885627

Functional Analysis and Algebraic Geometry

kind of off topic but I'm starting inear algebra in the fall. my skills are definitely weak. what should I study over the summer to prepare me to dominate linear algebra? I'm thinking if I brush up on at least basic calculus and algebra

>>8886345

Calculus doesn't come up much. Get a book on Linear Algebra and read ahead.

>>8884016
that's not NEARLY a comprehensive list, it's a list on engineering classes and then a couple of random math classes

>>8883801
>Analysis and Topology
What's the difference? Honest question here

>>8887327

Analysis is all about putting you epsilon into that cute sigma. Topology is the art of getting the bra off.

>>8887351
I don't know what you're getting at. From what I've seen until now, analysis is usually thought of as normed + metric spaces, which is just a special case of topological spaces so I don't understand the distinction

>>8883809
I wanted to emphasise this. Nearly every point set topology book contains an introduction to set theory even.

>>8883999
double trips

>>8887351
lmao tru tho

Linear algebra, differential equations and partial differential equations, linear algebra 2, stat and prob. That's the start of it.

>>8887357
Topology is too general to be used as an introduction to higher level mathematics in the way analysis is. What you say is true but come on. An introductory topology class has a very different tone than an introductory analysis class.

In introductory analysis you revisit ideas you already almost completely understand and see how the process of turning your ideas into reality works. In introductory topology you for the first time see many ideas that you have no idea where they come from nor why specifically they were invented.

>>8883797
You learn Mathematical Analysis.

https://en.wikipedia.org/wiki/Mathematical_analysis

>>8883802
Wtf? Undergrads learn this in their 1st semester, right after set theory

>>8883797
>Is X a meme?
You're a fucking meme and deserve to be recommended nothing.