How to study college mathematics deeply!

Mathematics is very vast subject to study. Some parts are difficult and some are manageable. Primarily, it is divided into two parts: i. Pure Maths ii. Applied Maths.

i. Pure maths is mainly divided into four parts: a. Algebra b. Analysis c. Topology d. Geometry.

ii. Applied maths has many direct connections with Statistics, Economics, Computer Science, Cryptography, Mathematical Physics, Operational Research etc. An/A exhaustive/compact list does not exist.

After reading this article one should know and understand what are the basics and what should be done at least in undergraduate. The levels of difficulty with respect to a common undergrad are denoted by: Intermediate-Hard-Advance.

Some References: It is given so that one can understand what the bare minimum content is needed to pursue a career in maths. One can pick any book but should check the content accordingly.

Algebra:

1. Undergraduate Algebra by Serge Lang. [Text & Exercise] [Intermediate]
2. Graduate Algebra by Serge Lang. [Text & Exercise] [Part i & ii Hard, Part iii & iv Advance]
3. Topics in Algebra by I.N.Herstein. [Text & Exercise][Intermediate]
4. Linear Algebra Done Right by Sheldon Axler. [Text & Exercise][Intermediate]
5. Introduction to Linear Algebra by Serge Lang. [Text & Exercise][Intermediate]

Analysis:

1. Introduction to Real Analysis by Bartle & Sherbert. [Text & Exercise] [Intermediate]
2. Complex Analysis by Lars.V. Ahlfors. [Text & Exercise] [Hard]

Topology:

1. Topology by James Munkres. [Text & Exercise] [Intermediate & Hard][General Topology: First 8 chapters]

Geometry:

1. Elementary Differential Geometry by Christian Bar [Text & Exercise] [Intermediate]
2. Elementary Differential Geometry by Andrew Pressley. [Text & Exercise][Intermediate]

Multivariable Calculus:

1. Functions of Several Real Variables by Moskowitz & Paliogiannis.[Text & Exercise] [Intermediate & Hard]

Discrete Mathematics:

1. Discrete Mathematics and its Applications by Kenneth H. Rosen. [Text & Exercise] [Intermediate]

Combinatorics:

1. Principles and Techniques in Combinatorics by Chen Chuan-Chong & Kho Khee-Meng. [Text & Exercise] [Intermediate]

Number Theory:

1. Elementary Number Theory by David.M.Burton. [Text & Exercise] [Intermediate]
2. Elementary Number Theory with Applications by Thomas Koshy. [Text][Intermediate]

Probability:

1. A First Course in Probability by Sheldon Ross [Text & Exercise] [Intermediate]

Statistics:

1. Statistical Inference by Casella & Berger. [Text & Exercise] [Intermediate]

Cryptography:

1. Cryptography Theory and Practice by Douglas R.Stinson. [Text & Exercise][Intermediate & Hard]

Automata:

1. Introduction to Automata Theory, Languages and Computation by Hopcroft-Motwani-Ullman. [Text & Example] [Intermediate]

Set Theory:

1. Notes on Set Theory by Yiannis Moschovakis. [Intermediate & Hard & Advance]

Minimal Technical Knowledge: LaTeX

How to Study Deeply: Just choose one book for Text and Exercise. That’s it! Solve it by using Pen & Paper and draw pictures. Use internet wisely. Intuition & Intelligence are not enough. If one wants to go deep, that person has to be rigorous which can be obtained only by writing maths rigorously. Do it smartly and do not work hard unnecessarily. Eat healthy and sleep at least 8 hours. First go slow and learn things. Then, grab momentum. Do as many Reading Projects as you can.

Forgetful Memory: If someone has a forgetful memory like me, s/he can do several things: Discuss and share ideas, Teach someone else, Make explicit notes if time permits.

Post Scriptum: While writing this article, I thought of Geometry as Algebraic Geometry. But I changed that to Differential Geometry later on. It is highly unlikely and way more difficult to study Algebraic Geometry in undergrad.

The thing that makes you say, “I want to do something”- that is the beginning of talent -Stella Adler

Good luck & All the best,

Sayan

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