本人的idea比较多，有的时候一秒钟一个。经验证明，这些idea一般都是可行的【虽然它们看上去很令人怀疑】，只是：基本上都给人做了，可能是太trivial了，来得太容易，呵呵。一点儿例子：

1) 许久以前说从morse complex恢复manifold，发现早给人做了，方法的确和我说的一样。不过还好，最近终于明白R2-valued理论了，可以正式动工了。

2) 把jones polynomial中的x换成e^x之类并相应修改以处理“巧合的取消”，一查果然做了，vassiliev。再进一步可以搞动力系统，L-function，这个好像没人做。

3) 俺说：steenrod和frobenius有关系吧。professor： 看不出来。我再说说。professor：哦，可能吧。俺一查，给做了。

4) 俺说：改改variety的定义可以变成morse map然后系统研究singularity，一看，的确是基本的思想。

5) 俺看论坛帖子说搞non-abelian measure theory，用sheaf theory公理化measure，不知道connes有没有搞第一个，刚开始看他的东西。

下面本人陆陆续续把一些疑难杂症整理一下贴上来，供大家参考。

这里是部分非常raw的想法及问题集合【基本上全是1秒钟所得】，我没有精力个个挖掘，所以大都非常模糊，而且肯定有许多错。有兴趣的朋友可以在这沙里淘淘，看看有没有一些有意思的。待续。

*) how?: todd class <-> Baker-Campbell-Hausdorff <-> Bernouli number <-> zeta

*) group homomorhism G->H vs [BG, BH], explain the difference. example: pontryagin dual <-> group cohomology

*) how?: Spec(continuous X -> R) -> X. say, how to recover open sets from algebraic information.

*) what differeomorphism really is? [think exotic xxx]

*) how n-cell connect to (n-1)-cell: besides H_, still have what?

*) steenrod : frobenius = ? : zeta

*) simplest proof of poincare duality, bott periodicity, class number formula

*) how?: coefficient ring <-> formal group law

*) how?: coefficient ring <-> functor between theories

*) systematical way of studying [X, Y], in both the stable & unstable case. say, when giving the postnikov tower.

*) the simple way of studying [X, Y] is to study induced map on cohomology theories. what's the limit of information? only care for stable maps? all stable [X, Y] give cohomology theory?

*) suspension, loop, classification: not exact functor, to what extent?

*) dold-thom: homotopy -> homology，easy. how about inverse? may involve：groupoid & classification space & noncommutative.

*) link above to john baez's homology ~ homotopy equation.

*) knot space: knot as point，with extra structure. apply AT methods, at least on formal level

*) dynamic system for knot，get L-function

*) a measure theory where: open set -> GL(n)

*) postnikov tower for S^n: how it looks

*) example for: same pi_, diff H_; same H_, diff H*; etc.

*) minimum amount of atlas needed for a giving X?

*) pointwise=linear algebra; local=some 2-algebra?; global=?

*) realize hopf bundle on morse complex

*) sth which chi = zeta

*) good geometric explanation for zeros of a power series expansion

*) loop space of X has simple homotopy type. example?

补充I：

*) stable homotopy ~ infinite categorification of vector space

*) cohomology of Spec ~ homology of X

*) categorification of discriminant

*) the dynamics of operator: L, B, S

*) tangent groupoid <-> heat kernel

*) K^1 ~ Chern-Simons

*) (co)homology ~ 1-d representation of some ring?

*) ? ~ high-d representation of that ring