What is intersection homology theory?
There is a nice introduction of intersection homology theory from Wikipedia.

My work

In the above article, I developed a setting for treating variational problems on stratified pseudomanifolds with singularities, such as complex projective varieties. Rather than using the ordinary homology theory on the base space, I instead used a generalized ``homology theory '' ---the intersection homology theory introduced by MacPherson and Goresky. Such a theory turns out to be more suitable than ordinary homology theory for pseudomanifolds with singularities.
   
In variational problems, one needs to take various limits (e.g. of minimizing sequences), but a basic problem is that a limit of geometric intersection chains may fail to be a geometric chain; and even if it is, it may not satisfy the important perversity conditions of the approximating chains concerning intersection with singular set. This motivates my use of rectifiable currents with a suitably modified mass norm.
     
Here is a brief sketch of the results of mine. For a compact stratified subanalytic pseudomanifold, I showed  how to express the intersection homology groups in terms of integer multiplicity rectifiable currents. These are then isomorphic to the usual intersection homology groups defined by geometric or subanalytic chains with the corresponding perversity conditions. The key idea involved a technical modification of the proof of the Federer-Fleming's Deformation Theorem to accommodate the perversity condition of intersection homology theory. I studied properties of a ``safety function'' that was used to quantify the perversity condition for each simplex of the singular locus.
   
Then I introduced a suitably modified mass on rectifiable currents such that all rectifiable currents with finite modified mass and finite boundary modified mass automatically satisfy the given perversity conditions. Also, by using the Lojasiewicz' inequality for subanalytic sets, I was able to show that all allowable subanalytic chains have finite (modified) mass and finite boundary mass. This fact ensures that the category of rectifiable currents with finite modified mass is still rich enough to contain all the ``nice'' chains one might consider. Moreover, this modified mass satisfies an important theorem---an analogue of the compactness theorem of geometric measure theory which implies that each sequence of rectifiable currents with bounded modified mass and boundary mass will have a convergent subsequence and that the limit is a rectifiable current satisfying the perversity conditions of the approximating chains. This property of rectifiable currents overcomes the weakness of geometric chains stated earlier in the basic problem. The support of the currents I considered may intersect ( in a controlled fashion) the singular locus of the pseudomanifold.
   
In the last part of the article, I showed that there exists a modified mass minimizer in every intersection homology class on a compact stratified subanalytic pseudomanifold. Moreover, I showed that these mass minimizers are in fact almost minimizing currents. Thus, by a lemma of Almgren, I achieved a partial regularity theorem for these mass minimizers.

My future work in this field

---