Gaining Broad Math Knowledge: What's Left to Learn?

  • Thread starter Reedeegi
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In summary, the speaker is seeking to gain a broad mathematical base and has already studied foundational topics such as Axiomatic ZFC Set Theory and Category Theory, as well as modern topics such as Analysis, Algebra, Differential Geometry, and Point-Set Topology. They are seeking suggestions for other areas to study, including Algebraic Topology, Lagrangian and Hamiltonian mechanics, differentiable manifolds, Riemannian geometry, geometric topology, differential topology, mathematical logic, discrete mathematics, and number theory. Other areas mentioned are geometry, numerical/applied analysis, asymptotic analysis, calculus of variations, finite calculus, difference equations, ordinary and partial differential equations, integral and integrodifferential equations, model theory, and topos
  • #1
Reedeegi
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I'm currently trying to gain as broad a mathematical base as possible, and here's what I've done:
Foundational:
Axiomatic ZFC Set Theory, Category Theory

Modern:
Analysis (Real, Complex, and Abstract), Algebra (Abstract, Linear), Differential Geometry, and Point-Set Topology

What are some other areas of mathematics that would be useful in gaining a very broad level of knowledge in mathematics?
 
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  • #2
Although it often fits in with differential geometry, I think algebraic topology is rather important to know. In addition, since it plays such a large part in pure mathematics, it wouldn't hurt to learn Lagrangian and Hamiltonian mechanics.

Also, it's difficult to gauge your actual level of mathematics from your description. It could be the case that you need to go a bit more in depth in certain categories to have more of a mathematical base. For example, have you studied differentiable manifolds and Riemannian geometry?
 
  • #3
phreak said:
Although it often fits in with differential geometry, I think algebraic topology is rather important to know. In addition, since it plays such a large part in pure mathematics, it wouldn't hurt to learn Lagrangian and Hamiltonian mechanics.

Also, it's difficult to gauge your actual level of mathematics from your description. It could be the case that you need to go a bit more in depth in certain categories to have more of a mathematical base. For example, have you studied differentiable manifolds and Riemannian geometry?

I'm well familiarized with most of the topics of all the areas I mentioned except except differential geometry, which I'm still learning. I have looked into Riemannian Geometry and I think I'll study it after Algebraic Topology. Also, would Geometric Topology be useful? Or Differential Topology? Or Mathematical Logic?
 
  • #4
In terms of elementary topics, I notice that you seem to be missing both discrete mathematics (e.g. combinatorics, graph theory) and number theory.
 
  • #5
Just don't go lunatic, that's my only advice. :-)
 
  • #6
Hurkyl said:
In terms of elementary topics, I notice that you seem to be missing both discrete mathematics (e.g. combinatorics, graph theory) and number theory.

I tried studying number theory, but I lost interest rather quickly. There is something about number's I've always seemed to dislike...
 
  • #7
What other have said plus

Geometry
Euclidean
Non-Euclidean
Projective
Analytic
Algebraic
Symplectic
Convex

Applied
Numerical/Applied Analysis
Asymptotic Analysis
Calculus of Variations
Finite Calculus
Difference Equations
Ordinary Differential Equations
Partial Differential Equations
Integral Equations
Integrodifferential Equations
 
  • #8
Reedeegi said:
I tried studying number theory, but I lost interest rather quickly. There is something about number's I've always seemed to dislike...

Have another look. Most books and courses concentrate on dull matters. Number theory is very broad and deep. It draws on many other areas of mathematics. There are transendential, computational, algebraic, analytic, elementary, and other areas.
 
  • #9
If you are interested in the foundations of mathematics, it may also do well to look into Model theory (cf the text by Bruno Poizat) and topos theory.
 
  • #10
I second the suggestions to study geometry and combinatorics, you've overlooked some of the most interesting branches of math!
 

What is the importance of gaining broad math knowledge?

Gaining broad math knowledge allows individuals to better understand and analyze the world around them, as math serves as the foundation for many fields such as science, technology, and economics. It also helps with problem-solving skills, critical thinking, and decision-making.

What are some ways to gain broad math knowledge?

Some ways to gain broad math knowledge include taking advanced math courses, practicing regularly, seeking out additional resources such as books or online tutorials, and applying math in real-life situations.

What are some key concepts that are important to learn in math?

Some key concepts that are important to learn in math include basic operations (addition, subtraction, multiplication, and division), fractions, decimals, percentages, algebra, geometry, and statistics. These concepts serve as the building blocks for more advanced math topics.

How can I overcome challenges in learning math?

Some ways to overcome challenges in learning math include breaking down complex concepts into smaller, more manageable parts, seeking help from a tutor or teacher, practicing regularly, and finding real-life applications for math concepts.

What can I do after gaining broad math knowledge?

After gaining broad math knowledge, individuals can pursue various career paths in fields such as engineering, finance, data analysis, and computer science. They can also continue to expand their knowledge by learning more advanced math topics or applying their skills to solve real-world problems.

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