Major areas of mathematics at the intersection of algebra and geometry/topology

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 Joined: Sat Nov 14, 2015 8:11 pm
Major areas of mathematics at the intersection of algebra and geometry/topology
I am looking into possible research areas to specialize in and my main research interests are in connections between algebra and geometry/topology. What major areas of modern mathematical research deeply delve into the interactions between algebra and geometry/topology besides the canonical ones such as algebraic geometry and algebraic topology? Are there any others that either involve algebra/geometry or algebra/topology, or both? It seems like geometric representation theory is one such area, but it seems to just be a corner of algebraic geometry that interacts more heavily with representation theory rather than being a field of its own, more independent from algebraic geometry/algebraic topology.
Re: Major areas of mathematics at the intersection of algebra and geometry/topology
Hi roadrunner,
You should look into geometric group theory. Geometric group theory is mostly based on viewing groups as geometrical objects by considering their Cayley graphs and putting an obvious metric onto those graphs. One uses geometry to study the Cayley graphs and by doing so discovers properties of the corresponding groups. There are lots of connections with hyperbolic geometry, Teichmüller theory, mapping class groups, 3manifold topology, dynamical systems, etc.
If you're interested, check out Brian Bowditch's notes: https://www.math.ucdavis.edu/~kapovich/ ... course.pdf.
You should look into geometric group theory. Geometric group theory is mostly based on viewing groups as geometrical objects by considering their Cayley graphs and putting an obvious metric onto those graphs. One uses geometry to study the Cayley graphs and by doing so discovers properties of the corresponding groups. There are lots of connections with hyperbolic geometry, Teichmüller theory, mapping class groups, 3manifold topology, dynamical systems, etc.
If you're interested, check out Brian Bowditch's notes: https://www.math.ucdavis.edu/~kapovich/ ... course.pdf.
Re: Major areas of mathematics at the intersection of algebra and geometry/topology
Grad, thank you so much for these notes, I had an idea to check out the geometry group theory, and your post arrived just in time.

 Posts: 2
 Joined: Sat Nov 14, 2015 8:11 pm
Re: Major areas of mathematics at the intersection of algebra and geometry/topology
Grad, thanks for the suggestion, I'll look into it more. By the way, for the case of geometric representation theory, is it indeed considered a corner of algebraic geometry that intersects with representation theory? Or is it an independent field?
Re: Major areas of mathematics at the intersection of algebra and geometry/topology
To be honest, roadrunner, I've never really heard anything about geometric representation theory before (I've been a graduate student for a little over two years at this point). From what I can see online it does seem that the "geometry" in geometric representation theory refers to connections with algebraic geometry. My totally uneducated impression would be that geometric representation theory is not a very large field. But if anyone knows more about it than I do, please correct me.
Another uneducated impression: it seems to me that understanding basic geometric representation theory would require one to have a little bit of technical knowledge (about algebraic geometry, category theory, etc.). I have no idea what your background is. If you have the background necessary to read some introductory notes on geometric representation theory then you can probably find out exactly what the geometrical component of the subject is. You could also find out whether or not geometric representation theory is basically a subfield of algebraic geometry, a subfield of representation theory, or distinct enough to be considered a field in and of itself. On the other hand, if you don't have the background necessary to read some introductory notes on geometric representation theory, it might be best for you not to commit to that as something you'll likely want to study as a grad student (since you won't really be able to get a feeling for what the field is like).
Another uneducated impression: it seems to me that understanding basic geometric representation theory would require one to have a little bit of technical knowledge (about algebraic geometry, category theory, etc.). I have no idea what your background is. If you have the background necessary to read some introductory notes on geometric representation theory then you can probably find out exactly what the geometrical component of the subject is. You could also find out whether or not geometric representation theory is basically a subfield of algebraic geometry, a subfield of representation theory, or distinct enough to be considered a field in and of itself. On the other hand, if you don't have the background necessary to read some introductory notes on geometric representation theory, it might be best for you not to commit to that as something you'll likely want to study as a grad student (since you won't really be able to get a feeling for what the field is like).