I was exhausted this afternoon and went to CIW for dinner. I randomly placed my dish around a person sitting with a dog. Not long afterwards when I finished my meal I found him left with all trash on the table. I went straightforward to warn him to handle his own business, but he refuse to listen. A person jumped out of nowhere urged me to let him go. I was puzzled but granted that nevertheless. He then told me the thug I noticed is in fact a blind student using the dog as a his guidance, and it is to be expected that he cannot find the right place to put the trash. Suddenly I feel very ashamed of myself though I can say I did nothing wrong.
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.
Since I need to do community service, I went to the organization for the orientation program meeting. Unfortunately I came late by 5 minutes because of flat tire. I told the receptionist that I am very sorry, etc, but she told me policy cannot be changed. So I need to go to the meeting in a later date. This make me feel a little dejected, since I hope I can be done with this as soon as possible.
After I went back to school I found life is still very intense. Somanth give me an intense reading assignment from Bott&Tu to present in two weeks, and I have just finished reading commutative algebra. So I have to shift back to topology now. This is fine but I have too much to study at the moment. I managed to finish the commutative algebra notes tonight, and hoping to move on for in depth reading in Eisenbud tomorrow. If I can move by 20 pages a day, in a month I should be done with the majority of the book. But 20 pages a day seems a luxury as Eisenbud contains a lot of problems which I am not sure if I can solve right away.
Somanth also give a list of other reference books to read, and it is too numerous to list at here. It seems I have a lot to learn from now. While this is a good thing I felt deeply unhappy that I did not study as hard as I should when I was at Bard, so my knowledge/skill gap with other people is huge at the moment. There is no short cut and I have to move on very slowly but solidly.
I miss Daniela a lot.
I used to be quite unhappy with the grading process when I was an undergraduate. But now I am being assigned as a grader, with 60+ homework laid in front of me in a huge pile, I realize the task is not that easy. Students makes all kinds of careless mistakes from not noticing the condition given in the problem and omitting totally legitimate solutions. Overall they are not doing very badly but very few did really well. I am wondering if that is the same in other things, where shallow understanding is easy but deeper mastering of the subject remains rare. Mathematics is not the only subject that needs artistic creativity and masonic attention to detail to guarantee full rigor. Otherwise, like one of the erroneous homework solutions I graded – 1^3 could be 2 and we will be living in 1984 instead.
给父母亲打电话，才知道家乡的老房子要拆掉的消息。我心下很茫然。那是我长大的地方，那里有我最深的记忆，那里是我和外公，外婆度过一辈子的地方。然而因为追求经济发展，拆掉真的凝聚着历史的老建筑，造仿古的新建筑，它们就将消失 – 我真的不知道这样的意义在哪里。我非常想家。我知道自己回不去，但是我还是非常，非常想家。因为此时此刻的家已经不是一个具体的时间，地点，人物造成的位置，而是我心中自从离开后无法抹去的一切回忆的象征。这种人在外体会到的劳累和冷酷，我不知道怎么告诉他人。或许，我也并不在意，因为生活太现实，这样的想法太不重要，我们都在忙忙碌碌地活着。以至于我现在才用过电话知道这件事，而不是从朋友，乡亲或自己的经验得知。而我和镇上的领导一样，认为这样的生活方式是自然的：用新的建筑取代旧的建筑来指代更旧的建筑，无异于用新的记忆改变旧的记忆，而期待自己能够从某种角度回到过去 – 这当然是做不到的。
I really miss home at here. I do not know why. It is not simply that I do not see my parents for a while, but the sense of life is deteriorating slowly within me. Sometimes I am wondering if I am still the same person I used to be in the past, with curious ideas and ambitions to fulfill in real life. I miss home, but there is no way back and I have to move on now.
The task is immense. There is Eisenbud, and there is Conway. Loya is going to organized a seminar, and I need to work out Weibel – too much to do, too little time, life is always like that.