PDF Document Overview of Mathematics.pdf

Overview of Mathematics .pdf

Original filename: Overview of Mathematics.pdf
Title: arXiv:gr-qc/9704009v2 1 Dec 1998

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GENERAL RELATIVITY
LORENTZ
GROUP

QUANTUM FIELD THEORY

(Operator-valued fields on R 4obeying certain Lorentzinvariant PDE’s and commutation relationships,
acting on an abstract Hilbert space)

(3+1-dimensional pseudo-Riemannian
manifolds with tensor fields obeying PDE’s of, say,
Einstein-Maxwell theory with perfect fluid component)

SU(2)

Specialize

LIE
GROUPS

LIE
ALGEBRAS

DIFFERENTIAL
OPERATORS

ALGEBRAS

LINEAR
OPERATORS

Define linear
vector
mappings

TRIPLE
FIELDS
Add 4th
binary op.
etc.

Add 3rd
binary operation, etc.

Add new class
(vectors), define
vector sum &
scalar product

Add commutativity
Add inverse

METRIC
MANIFOLDS

COMPLEX
MANIFOLDS

HILBERT
SPACES

Define charts and
atlases

REAL
Add ordering,
least upper bound NUMBERS
property

Define .
=, + &
for pairs

COMPLEX
FUNCTIONS
Add
norm &
completeness

Define =, + & . for equivalence classes of pairs

INTEGERS
Define =, + & . for
negative numbers

RINGS
Add commutativity and 2nd
assoc. & distrib. binary operation

SEMIGROUPS

MEASURABLE
SPACES
Add
measure

METRIC
SPACES
Add
metric

TOPOLOGICAL
SPACES

NATURAL
NUMBERS
Add number theory
symbols & axioms

Add = & associative
binary operation with identity

REAL
FUNCTIONS

COMPLEX
NUMBERS

RATIONAL
NUMBERS

Add multiplicative
inverse

Define linear
functionals

BANACH
SPACES

Define =, + & . for
Dedekind cuts

ABELIAN
FIELDS

DISTRIBUTIONS

Add inner
product

Cn

Rn

FIELDS

GROUPS

DODECAHEDRON
GROUP

REAL
MANIFOLDS

Add commutativity

ABELIAN
GROUPS

Specialize

TENSOR
SPACES
Define linear
functions of
n vectors

VECTOR
SPACES

DOUBLE
FIELDS

MANIFOLDS WITH
TENSOR FIELDS

Add
open
set
axioms

ABSTRACT
GEOMETRIES

Add symbols
LOWER
and axioms for
SETS
RELATIONS
PREDICATE =, in, subset,
Add
cartesian
CALCULUS union & intersection
products
Add quantifiers

BOOLEAN
ALGEBRA
Add symbols & axioms for
brackets, not, and, etc.

MODELS
FORMAL
SYSTEMS

FIG. 1.
Relationships between various basic mathematical structures. The arrows generally indicate addition of new
symbols and/or axioms. Arrows that meet indicate the combination of structures — for instance, an algebra is a vector space
that is also a ring, and a Lie group is a group that is also a manifold.

2