Here are examples of two questions which I would like to ask. I am afraid to ask them, because they might be closed. The question about 3-manifolds might survive. The question about sporadic groups would probably be closed as "not research level" question. This is my experience of asker in these two areas. My guess is that topologists are more flexible than algebraists. I might be wrong.
In area of 3-manifolds what are the possibilities of classifying them. I am interested in topological properties of manifolds. I see that there is focus on "geometrization" and hyperbolic manifolds. I am a bit skeptical about it. I admit it is interesting on one hand. On the other hand when we take one manifold, how do we see the geometry ? I think we first see the topology which is shape of the manifold. Of course this is just intuition or my brain preference. Maybe other person would see it in different way.
On this question I received answer from Bruno Martelli that nobody knows how to classify closed irreducible atoroidal orientable 3-manifolds. From this answer I conclude that "geometrization" didn't solve the problem for classification of 3-manifolds. I am stil in process of understanding the Haken manifolds and "virtual Haken conjecture". Can atoroidal manifold contain incompressible surface ? This is maybe "pub level" - as Gerhard suggests - my view is that there might be another path for 3-manifold classification. Since basic blocks left from JSJ decomposition - atoroidal ones - are not yet understood, then maybe we should look for another basic blocks.
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In area of finite groups I would like to ask how can we understand sporadic groups. They contain some symmetry - what symmetry it is ? I have read opinion that there is no one symmetry for all sporadic groups. It is rather different kind of symmetries which we can see. If we compare to exceptional Lie groups - these objects are connected to octonions. There is no similar one object which could explain existence of sporadic groups.