It has been three weeks since school started, and I do not feel I did much. I finished reading some rudimentary commutative algebra notes and read a chapter of Vikal, but this does not really mean anything. I need to quickly catch up with others because what I know soon become obsolete in a few days.
It is not entirely clear to me why understanding a more general concept is better than understanding a simpler, easy concept. I did not thought much about it until today’s index theory class, during which the professor rightly pointed out that index theory in Euclidean spaces are just too simple to delve with. This is certainly true, but when I think about it from a topologist’s point of view, I do not really believe the intrinsic approach working with coordinate patches from a local – global basis is a better route. When we get a much generalized statement that holds for all manifolds, we simultaneously lost the clarity and control we had in Euclidean case. There is certainly the difficulty of writing out an explicit equation of the surface in a Euclidean space make differential geometry be forced to work with the intrinsic as opposed to the extrinsic (Nash Embedding, for example) point of view. But I want to question the merit of using manifold as a concept itself when there are basic questions in Euclidean case needs to be answered.
One way to argue about this kind of generalization is justified is that they are necessary because they are inevitable – no one will disagree that the sphere is a two dimensional surface, and over the projective complex plane the inversion is just a reflection of the inside and outside of the sphere. Another way is to argue that there are simply too many such examples exists in mathematics. What is a line, what is a surface, what is a ball after all? The understanding of dimension is so fundamentally important in all fields of mathematics that a generalization of Euclidean space itself become a powerful theoretical unification language to connect problems which seems totally disconnected from each other. But both points are not really helpful; mathematically there are always a myriad way to generalize a familiar concept, and the fact that one can find easy examples to serve as base cases for an overly generalized concept is usually not supportive of its relative merit in comparison with other concepts. Consider the people living in the flatland world – why would they need to look at the possibility of the existence of the three dimensional space, if they are forever being trapped to the flatland? Why not they focus on their distinct internal problems in flatland first, and then try to move on to other cases?
Let us go back to the situation of a manifold. It seems plain to me that it will not help us to understand the situation of the Euclidean case. Certainly naturally as aforementioned we have seen phase spaces, integral curves, etc all naturally correspond to some kind of topological object that is best to be characterized, even if we are only studying the relative simple case of a linear ODE system like Newtonian mechanics. But there are deep question buried within the Euclidean case; when can two regions in the space be deformed smoothly to each other? What if we impose a smooth condition? What if we impose the maps to be conformal with certain boundary conditions? Can we conclude anything as easily as we defeat the laplace’s equation?
The answer is no, because we do know the situation in dimension 2 (again, thanks to generalizations – Riemann’s mapping theorem), and there are counter examples in dimension 3. But beyond dimension 3 no one knows what is happening. The fact that a relatively simple concept can be properly generalized to a universal object which is harder to analyze but more applicable to real life situations, is the source of the astray; there is a deep difference between the generalization and the original concept in that because of the wiggle room of extra control and extra clearness the problems we left in the base case are extremely hard. It is always nice to look ahead to see a problem that might be be solved via methods totally unexpected in the first place; but beside connecting dots, we also need people who are extremely focused on how to explore further things we thought we know but do not really know very well. This is what I think really important for a mathematican, not because there are already a multitude of people interested in the non-technical side of our lives, but because hard, intricate problems often need deep, innovative ideas to tackle. I still think Euclidean spaces can be a challenge to anyone, in the sense the easy part of the game is finished – and the rest had just begun.