## Analysis Thinking and The Branches of Mathematics and Pages 2 -7

Pages 2 – 7 up to Section I.2.

Algebra thinking is about finite things and analysis thinking is about the infinite.

The Main Branches of Mathematics are: Algebra, Number Theory, Geometry, Algebraic Geometry, Analysis, Logic, Combinatorics, Theoretical Computer Science, Probability and Methematical Physics.

Algebra – concerned with number systems, polynomials, groups, fields, vector spaces and rings.

Number Theory – properties of the positive integers, equations in integers or algebraic number theory and analytic number theory which is rooted in the study of prime numbers.

Geometry – study of manifolds, topology (exact distances don’t matter) and differential topology (distances matter).

Algebraic Geometry – manifolds described by polynomials.

Analysis – partial differential equations, infinite dimension vector spaces (Banach spaces, Hilbert spaces, C*-algebras, Von Neuman algebras) and dynamics (motion of planets).

Logic – fundamental questions about mathematics – set theory, category theory, model theory and logic concerning “rules of deduction”.  Results include Godel’s Incompleteness Theorems and The Independence of the Continuum Hypothesis.

Combinatorics – counting things, discrete structures (also discrete mathematics) and structures with few constraints.

Theoretical Computer Science – efficiency of computation an application is cryptography.

Probability – phenomena that are so complex they are best described by probabalistic statements.  Problems are often in fields of biology, economics, computer science and physics.  Examples include, phase transition – probabalistic models of critical phenomena and Brownian motion.

Mathematical Physics – pysicist often lead the way with non-rigourous aurguments that are then taken up and rigourously proved by mathematicians.  Examples are: vertex operator algebras, mirror symmetry, general relativity and the Einstein equations and operator algebras.