Major modern research areas at the intersection of algebra and geometry/topology

[Grassman12]

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? Is there a list of all such topics? 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. Is this true? In particular there are many areas in group theory such as geometric group theory, topological groups and Lie groups that also fit this requirement. Are these areas related to algebraic geometry/algebraic topology in a deep way; is one a subset of another?

Thanks!

Edited November 28, 2015 by Grassman12