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Revision of Geometry and topology front page from Thu, 24/06/2010 - 07:50

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 Euclidean geometry). However, there is also the possibility of using algebraic reasoning (as is done in classical 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 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 tensor analysis , the methods of algebraic topology (such as homology and cohomology groups, or homotopy groups), the exploitation of group actions, and many others.

Algebraic geometry is one modern outgrowth of analytic geometry and projective geometry, and uses the methods of modern algebra, especially 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 Diophantine equations, where geometric methods (often based on algebraic geometry) are a key tool. Also, investigationss in commutative algebra and group theory are often informed by geometric intiution (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 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 iconIncomplete This article is incomplete. We should add more subareas, and problem solving methods
Note iconAttention This article is in need of attention. Is there some general category such as "highly symmetric geometry" into which one could put Euclidean, spherical, hyperbolic, Lorentzian, projective, etc.?
Note iconAttention This article is in need of 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?

Comments

I added a brief description

I added a brief description of geometry. I was motivated here by the somewhat standard tripartite division of mathematics into algebra/analysis/geometry, so my conception of geometry includes topology, for example (i.e. not just metric geometry). This seemed justified, since 3-manifolds was one of the existing links.

I added a short list of modern methods in geometry, together with some discussion of how geometry can be applied in other areas of mathematics. It would be good to flesh this out by adding more examples, and creating pages that illustrate them, with appropriate links.

I have updated this page

I have updated this page somewhat in line with the discussion on the ``Front pages" page. I also added links, matching those on the ``Front pages" page. (For some reason, I couldn't achieve the nice indentation that is on that page. If someone can fix my formatting, please do so.)

The reason your indentation

The reason your indentation didn't work was that you were leaving a line between the items in the list. It seems that the asterisks are interpreted in a relative rather than an absolute way and that once you start a new paragraph everything starts again.

Inline comments

The following comments were made inline in the article. You can click on 'view commented text' to see precisely where they were made.

As a possible answer to the

As a possible answer to the question this note raises, I will suggest that we should have an (or potentially more than one) article on symmetric spaces, which include all these highly symmetric geometries. These are central to a whole host of fields (number theory and automorphic forms, representation theory, geometric group theory, and many others), and there is a lot that could and should be said about them.

Perhaps the first such article could explain that these ``highly symmetric geometries" have a natural group-theoretic origin, which one can exploit when studying them.