Quick 1-paragraph summary of all of math

  • Context: Undergrad 
  • Thread starter Thread starter Fletcher
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SUMMARY

To efficiently self-learn higher-level mathematics from a calculus foundation, one should follow a structured path that includes Logic and Foundations, Algebra, and Analysis, progressing through advanced topics such as Abstract Algebra and Functional Analysis. The discussion emphasizes the importance of aligning study with personal goals, particularly for aspiring mathematicians. A comprehensive overview of these topics can be found in specialized texts aimed at individuals with a Bachelor of Science in Mathematics.

PREREQUISITES
  • Calculus knowledge
  • Understanding of Algebra
  • Familiarity with basic Logic and Foundations
  • Basic concepts of Geometry
NEXT STEPS
  • Research "Abstract Algebra" textbooks for a deeper understanding of algebraic structures.
  • Explore "Real Analysis" to grasp advanced calculus concepts.
  • Study "Functional Analysis" to learn about operator algebras and their applications.
  • Investigate "Mathematical Physics" to see the application of mathematics in physical sciences.
USEFUL FOR

This discussion is beneficial for mathematics students, educators, and self-learners aiming to deepen their understanding of advanced mathematical concepts and their applications.

Fletcher
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Go!

Actually I want to pose this hypothetical question:

If you wanted to self-learn most of the higher level mathematics most efficiently, starting with a calculus level knowledge, how would you go about doing it?
 
Mathematics news on Phys.org
* 01. Logic and Foundations
* 02. Algebra
* 03. Number Theory
* 04. Algebraic and Complex Geometry
* 05. Geometry
* 06. Topology
* 07. Lie Groups and Lie Algebras
* 08. Analysis
* 09. Operator Algebras and Functional Analysis
* 10. Ordinary Differential Equations and Dynamical Systems
* 11. Partial Differential Equations
* 12. Mathematical Physics
* 13. Probability and Statistics
* 14. Combinatorics
* 15. Mathematical Aspects of Computer Science
* 16. Numerical Analysis and Scientific Computing
* 17. Control Theory and Optimization
* 18. Applications of Mathematics in the Sciences
* 19. Mathematics Education and Popularization of Mathematics
* 20. History of Mathematics

The best path of study depends on your goals with Math. If you want to be a Mathematician, you can't go wrong with Abstract Algebra and Analysis (advanced calculus).
 
Crosson,

Is there a book, understandable to a Mathematics B.S., which overviews the topics you list?
 

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