Geometric Areas of Mathematics
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Here we consider all the fields which exercise our geometric
intuition: from Euclidean and analytic geometry to tilings and
tessellations, from the Klein bottle to knots, along with curvature,
soap bubbles and the very idea of dimension.
One of the oldest areas of mathematical discovery, geometry has undergone
several rebirths over the centuries. At one extreme, geometry includes the
very precise study of rigid structures first seen in Euclid's Elements; at
the other extreme, general topology focuses on the very fundamental kinships
among shapes. (There is also a more subtle notion of "geometry" implied in
Algebraic Geometry (14), which is frankly quite algebraic.)
Here the MathMap shows the Geometry areas in a yellow-orange and the
Topology areas in a yellow-green.
Other fairly geometric areas are K-theory (19), Lie Groups (22),
Several Complex Variables (32), and to some extent Global
Analysis (58) and the Calculus of Variations (49).
51: Geometry is studied from many perspectives! This
large area includes classical Euclidean geometry and synthetic (non-Euclidean)
geometries; analytic geometry; incidence geometries (including projective
planes); metric properties (lengths and angles); and combinatorial
geometries such as those arising in finite group theory. Many results in
this area are basic in either the sense of simple, or useful, or both!
52: Convex and discrete geometry includes
the study of convex subsets of Euclidean space. A wealth of famous
results distinguishes this family of sets (e.g. Brouwer's fixed-point
theorem, the isoperimetric problems). This classification also includes
the study of polygons and polyhedra, and frequently overlaps discrete
mathematics and group theory; through piece-wise linear manifolds, it
intersects topology. This area also includes tilings and packings in
53: Differential geometry is the language of modern
physics as well as an area of mathematical delight. Typically, one
considers sets which are manifolds (that is, locally resemble Euclidean
space) and which come equipped with a measure of distances. In particular,
this includes classical studies of the curvature of curves and surfaces.
Local questions both apply and help study differential equations; global
questions often invoke algebraic topology.
The remaining three areas are collectively known as Topology.
54: General topology studies spaces on which one
has only a loose notion of "closeness" -- enough to decide which functions
are continuous. Typically one studies spaces with some additional
structure -- metric spaces, say, or compact Hausdorff spaces -- and
looks to see how properties such as compactness are shared with
subspaces, product spaces, and so on. Widely applicable in geometry and
analysis, topology also allows for some bizarre examples and
55: Algebraic topology is the study of algebraic
objects attached to topological spaces; the algebraic invariants illustrate
some of the rigidity of the spaces. This includes various (co)homology
theories, homotopy groups, and groups of maps, as well as some rather
more geometric tools such as fiber bundles. The algebraic machinery
(mostly derived from homological algebra) is powerful if rather daunting.
57: Manifolds are spaces like the sphere
which look locally like Euclidean space. In particular, these are the
spaces in which we can discuss (locally-)linear maps, and the spaces
in which to discuss smoothness. They include familiar surfaces. Cell
complexes are spaces made of pieces which are part of Euclidean space,
generalizing polyhedra. These types of spaces admit very precise
answers to questions about existence of maps and embeddings; they are
particularly amenable to calculations in algebraic topology; they
allow a careful distinction of various notions of equivalence. These
are the most classic spaces on which groups of transformations act.
This is also the setting for knot theory.
The geometric areas share with the fields of algebra the tendency to distill their inquiry to the study of certain axioms and
their consequences; during the last half-century the ties between these
broad areas have increased. On the other hand, some of the geometric areas
remain close to analysis, particularly General Topology (to measure theory and
functional analysis) and Differential Geometry (to differential equations and
You might want to continue the tour with a trip through analysis.
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Last modified 1999/05/12 by Dave Rusin. Mail: firstname.lastname@example.org