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

Forum for the GRE subject test in mathematics.
Posts: 2
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.

Posts: 31
Joined: Sun Nov 09, 2014 1:02 am

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

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, 3-manifold topology, dynamical systems, etc.

If you're interested, check out Brian Bowditch's notes: https://www.math.ucdavis.edu/~kapovich/ ... course.pdf.

lonny_sti
Posts: 19
Joined: Fri Nov 13, 2015 9:12 pm

### 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?