What are some of the main directions and trends in modern (let's say within the last ~10 years) algebraic topology? What are some major open problems or recent results?
In a more specific direction, to what extent does modern algebraic geometry interact with modern algebraic topology? How easily would a grad student or postdoc working in algebraic geometry be able to work on a problem in algebraic topology?
Note: I'm looking for answers for specifically algebraic topology as opposed to say low-dimensional geometric topology (which certainly uses algebraic methods heavily).
Thanks!
Best Answer
One of the more important recent results is the solution of the Kervaire invariant one problem by Hill, Hopkins, and Ravenel.
Here is their paper.
The chromatic view-point, which studies stable homotopy theory via its relationships to the moduli of formal groups, and related topics such as topological modular forms, use a sizable amount of (fairly abstract) algebraic geometry. And Lurie's work on derived algebraic geometry was motivated in part by establishing foundations adequate to the task of defining equivariant forms of TMF.
There are also relationships between more classical algebraic geometry and algebraic topology, e.g. as in Hirzebruch's book and his proof of his version of the Riemann--Roch theorem. These sorts of connections don't seem to be as much in vogue right now (although I am not an expert in algebraic topology by any means, so may be wrong on this). However, one contemporary researcher whose work straddles the two fields, and which has a flavour which reminds me of Hirzebruch's work, is Burt Totaro; you may want to look at some of his papers.