Index Theory update

So far Paul Loya had asked me to type up the latex version of last semester’s lecture notes. To be fair I feel somehow being intimidated. The material is not really dense, but worth careful checking and double checking everywhere for miniature details. So I have to supplement every remark in the notes into a remark in the pdf file, and every “exercise” into something proved line by line so that any first year grad student can follow. I do not know if this is worth it – until I discovered I could not prove some trivial results I took for granted for a long time. Then I realized I actually learned something.

Paul’s plan this semester is to go through Riemann-Roch, Hirzebruch signature theorem, Spin geometry and principal bundles. Since I already know plenty of spin geometry and principal bundles, I guess I will just use the class as an opportunity to review and see more practice examples in real life. His proof of the signature theorem is extremely “pretty”, as it involved no topology but lots of elementary analysis. I am wondering how one can express the Todd class and Chern class in terms of det^{1/2}s. It seems to me that Paul’s approach is analysis to analysis, where the topological side(most nicely generalized by topological K-theory) is ignored. If I have time I should try to go through Lawson’s Spin geometry to see the other perspective.

While I passed the master exam, the department obviously is not happy with me. I was aghast when the chair asked me for five minutes in private, and he expressed concern that “people say you are not good at teaching”. He suggested me to make some calculus presentations to correct this impression, otherwise I would have trouble to get funding next year. I did heard of similar rumors in the past that I am not good at giving presentations or tutoring students, but I did not know that people could make such important decision based on dubious evidence. So far I have taught no class at all at here. While I am not nervous about making a calculus presentation, I feel very awkward to be treated this way. It is true that if I am in his position I probably will act in the same way, but I felt uncomfortable nevertheless. I do not really know what is wrong at here, but if the department chair could fire a well performing graduate student because he or she did not do well in a calculus presentation, then I think something is fishy at here. On the other hand, I decided not to have a fight on this. My instinct tells the situation would be even worse if I asked for evidence, rules, law, etc to support his decision. In the end even if I appeal to Dean of Students it will boil down to the same thing, that I have to show I can teach to win their trust for a teaching assistantship. I think I do not like this school’s rigid administration structure and top down approach to its students. They informed me my master degree is not available because some of the classes I took for the exam I have never registered; they informed us we might be moved to Whitney hall because in this way we can hire more professors and have more room for other things. What I saw is not just a lack of consideration of the students’ voice at this school, but a lack of appreciation of students’ importance and personal interest in comparison with myriad administration needs and regulations. I found the situation frustrating.

I really miss Leah, and I do not know what to do about it. Sometimes I really want to talk to her, but I realized it is impolite to call her any further after she asked very pointedly not to contact her social circle in future. I also realized she must have a new boyfriend or at least is dating someone. But this does not address the core of the problem, that now I no longer love her and we no longer share an intimate friendship, such that if I called on her without notice we would not know what to say to each other. Our lives are now thousands of miles apart and our souls are no less distant away from each other. Slowly but inevitably I realized that my introspective, lazy, erratic life style is fundamentally different from her energetic, well organized, social life style. While these traits does not necessarily contradict each other and could even be complementing each other at times, very often when we were in an intimate friendship we tend to have petty disagreements on small things which I do not know how to address properly. Maybe I am still emotionally immature, but I was not able to address it on my own and she was not willing to help me address them, either. While I am still concerned about her, I do not know in whatever way I can act that is helpful for her life, and not being mistaken to be stalking her or obsessed with her. Maybe after I transferred to elsewhere I can contact her in future. I do not really know.

Sometimes I wish I can talk to someone on this, but I do not see how it might be helpful to address it. More importantly, I think hearing the other person complaining randomly on topics he or she does not have an immediate interest is mentally exhausting and emotionally uncomfortable. While I certainly cannot complain about this to my parents considering risk of them getting hysterical with the situation, I certainly want to talk to Daniela. I realized after all these years I still want to know her better and understand her beyond the formal level. I realized that I might also like her. But I do not know if my realization has any factual basis, or is just some psychological illusion based on incomplete information gathered over the last year since I graduated. I hope she will be doing well in her graduate school applications, and enjoying her life as usual. I also wanted to her that I have been reading Michael Foucault, in which Plato’s idea of the soul appeared very often. While I still feel uncomfortable that Leah considered me to be unattractive, sometimes I dreamed that maybe when people are in love not just their exterior elements are attracted to each other, but as Plato wrote love could help the soul to arise to a higher level of existence because of the divine element in it. But I do not know how to tell her (Daniela or Leah) about these things. I think Leah would be offended and try to explain to me I am mentally unattractive to her as well, even if I stripped of any romantic intention towards her. I wanted Daniela to know that whether she is attractive means very little to me, and she should not worried about it. But this sounds a bit too much because she only told me she cared about it a long, long time ago.

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