Geometry and topology front page

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Geometry and topology front page

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[QUICK DESCRIPTION]

Very roughly speaking, geometry is that part of mathematics that
studies properties of figures.  Often, the reasoning used in geometry
itself is of geometric nature, i.e. one reasons with properties 
of figures (as say is done in classical 
[[w:euclidean geometry | Euclidean geometry]]).
However, there is also the possibility of using algebraic reasoning
(as is done in classical [[w:analytic geometry | analytic geometry]]
or, what is the same thing, Cartesian or coordinate geometry), combinatorial
reasoning, analytic reasoning, and of course combinations of these
different approaches.

In contemporary mathematics, the word ``figure'' can be interpreted
very broadly, to mean, e.g., curves, surfaces, more general manifolds or
topological spaces, algebraic varieties, or many other things besides.   Also, the word ``space''
is used more often than the word ``figure'', as a description of
the objects that geometry studies.

There are also many aspects of figures, or spaces, that can be studied.
Classical Euclidean geometry concerned itself with what might be called
metric properties of figures (i.e. distances, angles, areas, and so on).
Classical [[w:projective geometry | projective geometry]] 
concerned itself with the study of properties invariant
under general linear projections.  Topology is (loosely speaking) the study of those properties of spaces that are invariant under arbitrary continuous distortions of their shape. 
In general, several of these different aspects
of geometry might be combined in any particular investigation.
In particular, although topology is less ancient than some other aspects
of geometry, it plays a fundamental role in many contemporary geometric investigations, as well as being important as a study in its own right.

There are many techniques for studying geometry and topology.  Classical methods
of making constructions, computing intersections, measuring angles,
and so on, can be used.  These are enhanced by the use of more modern methods
such as [[w:tensor | tensor analysis ]], the methods of 
[[w:algebraic topology | algebraic topology]] (such as
homology and cohomology groups, or homotopy groups), the exploitation of
[[w:group actions | group actions]], and many others.

[[w: algebraic geometry | Algebraic geometry]] is one modern outgrowth of 
analytic geometry and projective geometry, and uses the methods of modern [[w:algebra | algebra]],
especially [[w: commutative algebra | commutative algebra]] as an important
tool.

Geometry and topology are important not just in their own right, but as tools for
solving many different kinds of mathematical problems.  Many questions that
do not obviously involve geometry can be solved by using geometric methods.
This is true for example in the theory of
[[w:diophantine equation |Diophantine equations]], where geometric methods (often based on algebraic
geometry) are a key tool.  Also, investigations in commutative algebra
and group theory are often informed by geometric intuition (based say
on the connections between rings and geometry provided by algebraic
geometry, or the connections between groups and topology provided by
the theory of the fundamental group).  Certain problems in combinatorics may
become simpler when interpreted geometrically or topologically.  (Euler's famous
solution of the [[w:seven bridges of konigsberg | Konigsberg bridge problem]] gives a simple example of a topological solution to a combinatorial problem.)  There are many other examples of this phenomenon.

===Subareas===

[note article incomplete]  We should add more subareas, and problem
solving methods [/note]

*[[Algebraic topology front page | algebraic topology]]
**[[Sheaf theory front page | sheaf theory]]

*[[Algebraic geometry front page | algebraic geometry]]

*[[Differential geometry front page | differential geometry]]
**[[Riemannian geometry front page | Riemannian geometry]]
**[[Symplectic geometry front page | symplectic geometry]]

*[[Elementary geometry front page | elementary geometry]]

* [[General topology front page | general topology]]

* [[Geometric topology| geometric topology]]
** [[3-manifolds front page|3-manifolds]]
** [[4-manifolds front page | 4-manifolds]]

* [[Geometric measure theory front page | geometric measure theory]]

*[[Symmetric spaces front page | symmetric spaces]]
**[[Euclidean geometry front page | Euclidean geometry]]
**[[Hyperbolic geometry front page | hyperbolic geometry]]
**[[Lorentz geometry front page | Lorentz geometry]]
**[[Projective geometry front page | projective geometry]]
**[[Spherical geometry front page | spherical geometry]]

[note attention] Is there some general category such as "highly symmetric geometry" into which one could put Euclidean, s<comment thread="159" />pherical, hyperbolic, Lorentzian, projective, etc.?  [/note]

[note attention] In response to the comment on the first note I've put a general category called "symmetric spaces". Does that look OK? Also, is Lorentz geometry (as opposed to the geometry of Lorentzian manifolds) worth including? And if so, should it be called something else, like Minkowski geometry? [/note]

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