Tricki

## Revision of Navigation pages for different areas of mathematics from Tue, 24/03/2009 - 16:36

Each main area of mathematics has its own top-level navigation page. If you go to the page for a particular subject you will find links to articles and navigation pages that are particularly relevant to that subject. We have (or will have when somebody creates them) pages for algebra, algebraic geometry, analysis, combinatorics, geometry, logic and set theory, mathematical physics, number theory, probability, statistics, theoretical computer science, and topology. Of course, these areas are very broad, so a typical page will contain a brief discussion of the area together with links to navigation pages for subareas such as group theory, partial differential equations, additive combinatorics, the geometry of 3-manifolds, model theory, general relativity, analytic number theory, stochastic processes, lower bounds for circuit complexity, and algebraic topology, several of which are split up further.

## Comments

### Geometry and topology

I think geometry and topology should go together as one subject area - often, the difference is a matter of context.

### I'm happy to do that and I

I'm happy to do that and I can see that it makes sense. Do others agree?

### I just added to the geometry

I just added to the geometry front page, and (as I wrote in a comment there) I took the view-point that topology is a part of geometry. (I hadn't seen this comment thread at that time, but I seem to have acted in accord with its sentiment in any case.)

### In the light of that, why

In the light of that, why don't we go ahead and fuse the pages? It seems to me that the best approach is to make an algebraic topology front page a child page of the geometry front page. I had originally imagined that Topology would split into General Topology and Algebraic Topology, but from the point of view of the Tricki, with its focus on types of problems, it makes much more sense to make general topology a subbranch of analysis.

I'd do the reorganization myself, but I'd rather leave it to one of you two, since you know more about the area and may not agree with everything I've just said.

### Topology as part of geometry

Historically, topology was geometria situs...
However, most low-dimensional topology seems to have little to do with geometry (it might sometimes even be math.QA). I certainly don't know much geometry.
Maybe with further subdivisions it makes more sense:
Geometry and Topology having "Low-dimensional topology" as a subcategory, which in turn has "4-manifolds", "3-manifolds", "knots and links", and "surfaces" as sub-discplines... what do you think?

### Geometry can mean many

Geometry can mean many things, clearly. It can mean metric geometry (with roots in Euclidean geometry, going all the way through to Riemannian geometry), but also projective geometry, say (which is one precursor to modern algebraic geometry), symplectic geometry, etc.

In 3-manifold topology, hyperbolic geometry can often play a big role (as well as the other Thurston geometries).

Of course, as you wrote, other quite different methods can also play a role.

It might make sense to change the title of this front page to "Geometry and topology" (which is in any case how currently it is listed on the Front pages for different areas of mathematics page. One could then write a slightly more elaborate blurb in the quick description, explaining some of what I wrote above.

As Tim wrote, general topology has a different, more analytic flavour, in comparison to the kind of topology we are likely to be linking to from here (whether it be low-dimensional, algebraic, differential, symplectic, or whatever), so we could include a remark to this effect, with a link to the general topology page.

What do other people think about this possible name change?

### Geometry and topology

I think it should be called "Geometry and topology". Clearly some subfields are distinctly one or the other, but there's a continuum between them: the use of hyperbolic geometry in 3-manifold topology is a good example.

Matthew's done a beautiful job starting the geometry front page, and I didn't want to meddle, but I'd be in favour of changing the title to "Geometry and topology front page".

### Based on Henry's reply, and

Based on Henry's reply, and the sentiments of Daniel's comment, I will go ahead and change the title to ``Geometry and topology front page'', and perhaps also edit the blurb a little to reflect this. (Thank you, too, for the kind words, Henry. But if I run out of steam, as is quite possible, then others should please go ahead and make their own edits in the same direction — or in any other direction that seems appropriate, for that matter!)

### I've made some additions to

I've made some additions to the list, but am running into an awkward problem. I put in "hyperbolic geometry", for example, but that would normally be thought of not so much as a branch of geometry but as a tool that is very useful in branches of geometry such as 3-manifolds. Nevertheless, it seems to me that there are useful tricks for how to answer questions in hyperbolic geometry and that it makes sense to have a front page for it. Maybe this isn't a serious problem: there is no reason for the Tricki classification of articles to be the same as a typical classification of mathematics in to research areas. (To give another example, one would clearly want a linear algebra front page, even though one would think of linear algebra as part of the background of all mathematicians rather than as a research area in itself – even if it is possible to come up with interesting problems about matrices and things).

### As I also wrote in a comment

As I also wrote in a comment on the geometry front page, one aspect of hyperbolic, projective, etc. geometries is that they are associated to symmetric spaces, and so have strong ties to group theory (especially Lie groups) and related topics. So I think that there will be a lot to say about them tricki-wise.

I might try to write an article discussing this aspect of things sometime soon. (Something that explains how the group theoretic view-point can be helpful in certain contexts.)