Mathematics

Kinds of mathematical thinking - Algebraic, Geometric, Analytic

1. Algebra vs Geometry - More symbolic vs more pictorial/visual
2. Algebra vs Analysis - Non-limiting exact formulas vs limiting process estimates (equalities vs inequalities)
   These words (algebra, geometry, analysis) are also used to denote particular branches of mathematics in addition to general methods of thinking, subbranches of which can be more or less algebraic, analytic or geometric in character.

Main branches of mathematics:

1. Algebra - number systems, polynomials and more abstract structures like groups, fields, vector spaces, and rings.
2. Number theory - properties of the set of positive integers.
   1. Equations in integers - algebraic number theory
   2. Prime numbers - analytic number theory
3. Geometry - study of manifolds.
   1. Topology - two manifolds same if they can be continuously morphed/deformed into one another.
   2. Differential geometry - studying the precise nature of distances between points on a manifold.
4. Algebraic geometry - study of manifolds defined using polynomials and the singularities that arise.
5. Analysis
   1. Partial differential equations and dynamics.
   2. Vector spaces such as Hilbert spaces, Banach spaces, Von Neumann algebras, C*-algebras.
6. Logic - fundamental questions about mathematics itself. Includes set theory, category theory and model theory.
7. Combinatorics - counting problems with mathematical structures that have "few constraints" as opposed to, say specific constrained problems in number theory.
8. Theoretical computer science - efficiency of computation.
9. Probability - includes information theory and other macroscopic statistical questions.
10. Mathematical physics