In your opinion, what are the fields that every self-respecting mathematician should know at least a little bit about?

oh and of course triple integrals

everything listed here: en.wikipedia.org/wiki/Glossary_of_areas_of_mathematics

all of them: you should have a good grounding in algebra, real and complex analysis, topology, probability, functional analysis and geometry by the time you graduate

You listed 4 different types of analysis yet otherwise just said algebra, topology, and geometry.

By algebra do you mean group theory? Galois theory? homological algebra? commutative algebra?

For topology, point-set topology? algebraic topology? differential topology?

For geometry, Euclidean? Hyperbolic? curves&surfaces? Riemmanian? Algebraic?

Underrated

so underrated the same joke was posted literally 13 seconds later

Computational complexity theory.

The basic fields of mathematics are Foundations, Algebra, Geometry, Analysis, Topology, and Applied Mathematics. Pretty much everything falls under those categories, is some kind of combination of them, or makes use of them to a significant extent, and you should be aware of all of them.

In no particular order (Asterisk on the ones where I think should be quite deep understanding)

>Elementary number theory/discrete
>*real and complex analysis
>functional analysis
>*topology
>*linear algebra
>*set theory/category theory
>*Algebra (Groups, rings, fields)
>Galois theory
>commutative algebra
>algebraic curves/surfaces
>*Geometries (Euclidean, projective, hyperbolic)
>Riemannian Geometry
>Basic algebraic and analytic number theory

Computer science
Economics
Statistics

Representation Theory
everything else is trivial

>no one mentioned graph theory
Brainlet board confirmed.

Bifurcation theory. Spectral theory. Pade approximants. Projective riccati equations. Lie group method. Homotopy analysis method. The h principle. Riquier janet theory.

Relations

Self referencing homological topology
Groethendieck knot theory
Algebraic lie theory in mandarin
Mapping ellipsoids on 4D spheres
Gaussian singularities in PDE

>Survival
>Hunting
>Farming
>Mechanic skills
>Shooting skills

Anything to be a proper man and not become the stereotypical smart but beta Maths guy.

Ethics

And preform an oath to do no hard to your fellow humans weather recognized by your governing laws or not.

barnett integrable functions

analysis on metric spaces, that's about it, everything else is just fancy and superfluous, whereas analysis on metric spaces is the only thing I would count on every mathematician I meet knowing.

statistics of race and biological sex

Precalculus
Single Variable Calculus
Multivariable and Vector Calculus
Ordinary Differential Equations
Applied Linear Algebra
Finite Vector Spaces
Complex Variables
Special Functions
Fourier Transforms
Calculus of Variations
Integral Equations
Asymptotics
Perturbation Theory
Fractional Calculus
Partial Differential Equations
Numerical Analysis
Numerical Linear Algebra
Approximation Theory
Numerical Ordinary Differential Equations
FEMs, FDMs, and Spectral Methods
Linear Programming/Optimization
Combinatorial Optimization
Convex Optimization
Proofs and Mathematical Reasoning
Introductory Set Theory
Introductory Logic
Intermediate Set Theory and Logic
Graduate Set Theory
Number Theory
Analytic Number Theory
Algebraic Number Theory
Computational Number Theory
Elliptic Curves
Cryptography
Information Theory and Coding Theory
Probability (Multivariable Calculus based)
Stochastic Processes
MCMs
Mathematical Statistics
Design of Experiments
Measure Theoretic Probability Theory
Stochastic Calculus
Abstract Algebra
Inequalities
Real Analysis (Metric Space based)
Analysis on Manifolds
Fourier Analysis
Complex Analysis
Graduate Real Analysis
Functional Analysis
Abstract Harmonic Analysis
Nonstandard Analysis
Point-set Topology
Algebraic Topology
Differential Topology
Non-Euclidean Geometry
Discrete Geometry
Smooth Manifolds
Riemann Geometry
Algebraic Geometry
Dynamical Systems Theory
Mathematical Control Theory

But the minefields of Croatia are so much nicer...

>No Lie theory, representation theory, or other advanced algebraic topics

Analysts are the worst.

What mathematics should you have learned after each stage on the way to getting a phd, from freshman to end of grad school?

Please give me a list of other important advanced algebraic topics and where they would go in that user's list.

Computer Science

IUT and finance all other options are meme fields.

Nice joke.

Rye

You don't have to know all of these, just 10-15 or so. Here are some key ones:

Precalculus
Single Variable Calculus
Multivariable and Vector Calculus
Ordinary Differential Equations
Linear Algebra
Partial Differential Equations
Set Theory
Logic
Number Theory
Abstract Algebra
Real Analysis
Point-set Topology
Algebraic Topology
Algebraic Geometry

Brainlet here. I have a question. Do you need to learn all of these in this order?

That's a good order to learn them. You should probably learn some differential geometry as well before algebraic geometry though

women's studies

Honestly it's most important that you build your way up to real analysis and abstract algebra, as they're extremely fundamental to nearly all high-level math

here's a good prereq chart i found

kek

basic programming/scripting

That's pretty neat. What does the red line mean?

It probably just means that real analysis is basically a much more rigorous study of calculus.

>real analysis to point-set topology

I mean, to be fair, isn't every single real analysis course in the world point-set topology based? Nice chart, anyway.

Topology is usually done after, but there's no reason they couldn't be done the other way around, I guess.

linear algebra should be learned with single variable calculus. I don't see any way how to understand multivariable calc and ODE without linear algebra.

You usually learn the necessary linear algebra in vector calculus and differential equations courses themselves, from what I've seen. It's not particularly difficult

that's because analysis is more rich and important than algebra or geometry.

Topology is simply a kind of analysis with fewer assumptions.

specifically, topology is simply the analysis of open sets and continuous functions.

But to solve systems of differential equations you need to know how to calculate [math]e^A[/math] for a matrix [math]A [/math] which requires you to convert it into Jordan Normal Form.
You need a linear algebra knowledge for that.

calculus courses are just mechanical computation anyway.

You can't actually understand vector calculus until you've done things like proven the implicit function theorem.

Do you do that in calculus 3?

No, you just learn crank the handle stokes theorem calculations.

So no, there's no point in understanding matrices in order to use them to calculate some jacobians in calculus 3 unless you also think that people should learn analysis before calculus.

for a proper first course on ODE I agree a first course on linear algebra should be done first.

but not calc 3. It's already so crank the handle and not about understanding.

like they're any harder than double integrals

>integrate once
>repeat x2

A lot of ODE courses seem to include that without linear algebra as a prerequisite.

>that's because analysis is more rich and important than algebra or geometry.
epic troll

triple integrals is a meme

not at all.
Of the great achievements in mathematics that have really let us conquer and do things we were not able to do before, very few have come from the last 50 years of pure algebra or pure geometry research.

discovering new finite simple groups or theorems about systems of solutions of polynomials is basically the mathematical equivalent of discovering miles and miles of sparse shrubland and tundra, while achievements in fields that fall under the analysis umbrella from the last 50 years are like discovering, rich , fertile farmlands an islands full of natural resources that will benefit the population.

thanks for htis post that literally doesn't mean anything

no need to get butthurt, kid

thanks for admitting you're trolling.

you don't have the motivation for topology without real analysis

yes i do

no you dont

topological spaces are interesting in themselves

Cooking is pretty important for mathematicians.
Also, organic synthesis.

Most of the interesting examples and 'visualizations' of open sets etc are from the real line, metric spaces, and metrizability is an important concept of topology

>Cooking is pretty important for mathematicians

That's sad. I bet his wife blamed herself for her husband's death.

Bump.