In the early 1990's, Gil Kalai introduced me to a very interesting generalization of homology theory called intersection homology, which existed for like 10 years back then I believe. Defined initially by Goresky and MacPherson, this is a version of homology which agrees with ordinary homology on manifolds, but also retains crucial properties like Poincare Duality and Hodge Theory on singular (non-)manifolds. The original definition was combinatorial, but it was later re-interpreted in sheaf-theoretic terms (perverse sheaves?). 


Back then it certainly looked like an exciting new development. So, I'm curious - where does the field stand today? Is it still thriving, or has it been merged with something else, or just faded away? 



What (if anything) happened to Intersection Homology?
Intersection homology and cohomology are still around, but as a topic they have just substantially been renamed. They are part of the theory of perverse sheaves, which are widely used in the Langlands program, in algebraic geometry approaches to categorification, and elsewhere in algebraic geometry.


To the extent that intersection homology was intended for topology, it has stoked relatively less interest than in algebraic geometry. On the one hand, there has been a trend away from homological algebra in geometric topology. On the other hand, singularities are part of the structure of intersection homology. Singularies are more germane to geometric topology than to algebraic topology in the sense of homotopy theory. Both singularities and homological algebra are major aspects of algebraic geometry.


