Overview of Mathematics .pdf
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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
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