Mathematics Learning FlowChart

[KingKai]

I am trying to visualize the correct learning curve for mathematics.

Currently I know a Calculus 1 and 2 and basic linear algebra.

But as far as post-secondary math courses go, that is all I have taken.

I want to advance my knowledge of mathematics to the point where I can look at math PhD research papers and appreciate them, because they currently look like foreign gibberish to me.


I ultimately want to learn (to a respectable level of fluency) Vector Calculus, ODE, PDE, Set Theory, Group Theory, Real Analysis, Complex Analysis, Discrete Math, Abstract Algebra, Combinatorics, Dynamical Systems, Number Theory, Differential Geometry, Topology, Chaos Theory, Graph Theory, Category Theory, Order Theory, Measure Theory, & Fractal Geometry.

I would also like to learn enough mathematics to understand and appreciate Quantum Field Theory.

http://s12.postimg.org/gesxh6viz/Mathematics.jpg


^ Here is the rough copy of my flow chart thus far, what changes/suggestions would you make.

 

[jedishrfu]

There is a website by Gerard T'Hooft that outlines a path to become a theoretical physicist and that may help you with your math mapping.

http://www.staff.science.uu.nl/~Gadda001/goodtheorist/index.html

 

[homeomorphic]

It's good to aim high, but I think you're being a bit ambitious. Traditionally, you are expected to specialize a bit more than you have in mind, and although I hate specialization, there is a reason for it, since you can only learn a certain amount in a finite lifetime. I do think it's possible to read and understand one book about each of those subjects over the course of many years, but it might not fit very well into a traditional academic career where you have to reach the frontiers relatively quickly and start publishing.

I want to advance my knowledge of mathematics to the point where I can look at math PhD research papers and appreciate them, because they currently look like foreign gibberish to me.

I have a PhD in math and most of the research literature still looks like foreign gibberish to me, unless you are talking about my sub-sub-sub-field that I studied in grad school, and even then, it's not that easy to read. I know the some of the basics of all the main branches of math, but generally not more than the basics.
I made a few mind map things like that for certain branches of topology because I didn't necessarily agree with the order I was taught things in. But overall, it would get to be sort of convoluted and a bit ambiguous at some points, if you wanted all those subjects. I don't think I would be brave enough to even attempt it. I'm not even sure about what order I would learn the undergraduate material in, in hindsight because I'm always changing the way I think about things and what looks like the right way to me now will probably be different a year from now.

It's hard to compare a subject like topology with calculus because although, technically calculus could be seen as a bigger field, it's usually covered in a set way in the curriculum with 3 or 4 standard classes. Topology breaks down into point-set, algebraic topology (including homotopy theory, homology, cohomology, each of which could be its own class), differential topology, and it's less standardized. Each of those topology classes also would have to be its own part of the flow chart because algebraic topology will require abstract algebra, for example, but other topology classes might not. There are other problems, such as the issue with point-set topology that it doesn't have any logical prerequisites, but it would be a bad idea to take it without having some familiarity with proofs and preferably real analysis.

Also, you'd run into issues like what books you are going to learn from and how that might affect the order in which you'd learn things. Something like Visual Complex Analysis would be a good book to read before taking real analysis because it's good preparation for the kind of thinking you need to do in order to really understand proofs, but normally complex analysis would probably be best taken after real analysis because it's a little more complicated. So, the flow chart would get very subjective and very convoluted.

 

[HakimPhilo]

There's the excellent Chicago mathematics bibliography:

http://www.ocf.berkeley.edu/~abhishek/chicmath.htm

which is divided into 3 parts:

• ELEMENTARY
  • Algebra (4)
  • Geometry (2)
  • Foundations (1)
  • Problem solving (4)
  • Calculus (6)
  • Bridges to intermediate topics (2)
• INTERMEDIATE
  • Foundations (5)
  • General abstract algebra (7)
  • Linear algebra (3)
  • Number theory (5)
  • Combinatorics and discrete mathematics (1)
  • Real analysis (10)
  • Multivariable calculus (2)
  • Complex analysis (5)
  • Differential equations (2)
  • Point-set topology (5)
  • Differential geometry (4)
  • Classical geometry (3)
• ADVANCED
  • Foundations (1)
  • Problem solving (1)
  • General abstract algebra (1)
  • Group theory and representations (5)
  • Ring theory (4)
  • Commutative and homological algebra (5)
  • Number theory (5)
  • Combinatorics and discrete mathematics (3)
  • Measure theory (2)
  • Probability (1)
  • Functional analysis (5)
  • Complex analysis (6)
  • Harmonic analysis (5)
  • Differential equations (4)
  • Differential topology (3)
  • Algebraic topology (7)
  • Differential geometry (6)
  • Geometric measure theory (4)
  • Algebraic geometry (5)

( ) refers to the number of books contained in each section. There's also this list taken from math.SE:

I hope this is what you're looking for.

 

[KingKai]

There's the excellent Chicago mathematics bibliography:

I hope this is what you're looking for.

This is great yes thank you.


It is hard to find a direction in the sequence of mathematical learning topics after calculus. This will help me strategize my learning process.

 

[KingKai]

It's good to aim high, but I think you're being a bit ambitious. Traditionally, you are expected to specialize a bit more than you have in mind, and although I hate specialization, there is a reason for it, since you can only learn a certain amount in a finite lifetime. I do think it's possible to read and understand one book about each of those subjects over the course of many years, but it might not fit very well into a traditional academic career where you have to reach the frontiers relatively quickly and start publishing.

Yes, this will no doubt be a lifelong pursuit, but part of the attraction to me is to be able to see connections between various branches of math and to be able to see symmetries is seemingly unrelated fields.

I get really inspired when I read about equations like euler's identity which combines complex numbers with trigonometry and exponents (This equation hit me like a ton of bricks when I first saw it, and I attribute it heavily to my interest in math) and for example the link between elliptic curves and modular forms in solving fermat's last theorem.

It makes me think that all sub-specialties of math are unified in many mysterious ways we have yet to discover. Perhaps if everyone chose to specialize in an extremely unique field without understanding others than maybe certain unsolved mysteries like the Riemann Hypothesis will never be solved due to the difficulty of one specialist being able to effectively collaborate and understand the workings of another specialist, almost like a person who speaks Russian trying to collaborate with another person who speaks Icelandic. A greater breadth of understanding would lead to the ability to notice undetected patterns between different fields, no?

That being said, if anyone was to solve the Riemann hypothesis it certainly wouldn't be me because I am no Gauss or Euler, but I want to be able to at least attempt to tackle these mathematical mysteries with a vast and broad mathematical understanding in many different fields.

 

[homeomorphic]

I get really inspired when I read about equations like euler's identity which combines complex numbers with trigonometry and exponents (This equation hit me like a ton of bricks when I first saw it, and I attribute it heavily to my interest in math) and for example the link between elliptic curves and modular forms in solving fermat's last theorem.

I also attribute a lot of my interest in math to Euler's equation, but the reason it attracted my attention was that I found it puzzling and even irritating when I first saw it, since I didn't know the motivation for it. That lead me to discover complex analysis to dispel the mystery of Euler's equation. The thing is, you can't just think of more advanced math as just more of the same. It's very different from the math that you've probably learned so far. The scale is much greater. It doesn't take that long to learn stuff like complex numbers and trig and exponents. When you think of research level math today, you have to think in terms of thousands and thousands of pages of books and journal articles. Then, suddenly, it doesn't seem so easy to make connections between different subjects. Using Euler's equation as a model gives you a very misleading picture I think.

It makes me think that all sub-specialties of math are unified in many mysterious ways we have yet to discover. Perhaps if everyone chose to specialize in an extremely unique field without understanding others than maybe certain unsolved mysteries like the Riemann Hypothesis will never be solved due to the difficulty of one specialist being able to effectively collaborate and understand the workings of another specialist, almost like a person who speaks Russian trying to collaborate with another person who speaks Icelandic. A greater breadth of understanding would lead to the ability to notice undetected patterns between different fields, no?

Well, it is good to know something about different fields, but most mathematicians are not capable of learning more than one thing in depth. They learn a little bit about a lot of subjects, which is helpful in their research, but they only have real expertise in one thing. Part of the problem is "publish or perish". That, together with the extreme complexity of today's world leads to a bit of a specialization contest because no one really has time to master a lot of different subjects in depth. Over the course of a whole career, you might get a little bit more flexibility, but especially at the beginning, there are too many constraints to be able to branch out much. Freeman Dyson had some article he wrote about birds and frogs, where the birds cover a wide area, but not in much depth, and the frogs just knew their little area very well. He said we needed both birds and frogs. But even a bird has limitations in how far they can see.

Most people end up studying a good chunk of your list, but usually not all the more advanced topics.