The study of space originates with
geometry – in particular,
Euclidean geometry.
Trigonometry is the branch of mathematics that deals with relationships between the sides and the angles of triangles and with the trigonometric functions; it combines space and numbers, and encompasses the well-known
Pythagorean theorem. The modern study of space generalizes these ideas to include higher-dimensional geometry,
non-Euclidean geometries (which play a central role in
general relativity) and
topology. Quantity and space both play a role in
analytic geometry,
differential geometry, and
algebraic geometry. Within differential geometry are the concepts of
fiber bundles and calculus on
manifolds, in particular,
vector and
tensor calculus. Within algebraic geometry is the description of geometric objects as solution sets of
polynomial equations, combining the concepts of quantity and space, and also the study of
topological groups, which combine structure and space.
Lie groups are used to study space, structure, and change.
Topology in all its many ramifications may have been the greatest growth area in 20th century mathematics; it includes
point-set topology,
set-theoretic topology,
algebraic topology and
differential topology. In particular, instances of modern day topology are
metrizability theory,
axiomatic set theory,
homotopy theory, and
Morse theory. Topology also includes the now solved
Poincaré conjecture and the controversial
four color theorem, whose only proof, by computer, has never been verified by a human.
Change
Understanding and describing change is a common theme in the
natural sciences, and
calculus was developed as a powerful tool to investigate it.
Functions arise here, as a central concept describing a changing quantity. The rigorous study of
real numbers and functions of a real variable is known as
real analysis, with
complex analysis the equivalent field for the
complex numbers.
Functional analysis focuses attention on (typically infinite-dimensional)
spaces of functions. One of many applications of functional analysis is
quantum mechanics. Many problems lead naturally to relationships between a quantity and its rate of change, and these are studied as
differential equations. Many phenomena in nature can be described by
dynamical systems;
chaos theory makes precise the ways in which many of these systems exhibit unpredictable yet still
deterministic behavior.
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