Here is some targets which I want to acomplish in recent research:
Orbifold Fundamental Group Of 2-Orbifold
Orbifold is a natural generalization of manifold. One can define
orbifold fundamental group for an orbifold, as we do for a
manifold.
Though it has been almost half a century since Thurston gave this
conception, no one gives a total description of the fundamental groups
for all 2-orbifolds.
I believe I can calculate it in recent days.
Hurewize Theorem for Orbifold Fundamental Group and Weighted Homology
The motivation for me to calculate the orbifold fundamental groups
for all 2-orbifolds, is that I want to prove there exists Hurewize
isomorphism between orbifold fundamental group and the weighted homology
group.
There are many attempts to build a homology theory for orbifolds, which
can reflect the information about singular points of orbifolds. Weight
homology is one of them. Try to build some connections with classicial
conceptions will show that weight homology is a suitable tool to study
orbifolds.
Which Cohomology Element Can Be Realized as Euler Class of Some Vector Bundle
The KO-groups and cohomology groups of toric manifolds are clear. I want to figure out which cohomology element of cohomology element can be realized as Euler class.
Build A Class of Non-positive Curvature Spaces
I am not sure whether I can figure out this. But I think this is an interisting topic.