# Learning pseudo-differential operators

I found learing pseudo-differential operator quite difficult.

# 245C, Notes 4: Sobolev spaces

Very nice article!

As discussed in previous notes, a function space norm can be viewed as a means to rigorously quantify various statistics of a function $latex {f: X \rightarrow {\bf C}}&fg=000000$. For instance, the “height” and “width” can be quantified via the $latex {L^p(X,\mu)}&fg=000000$ norms (and their relatives, such as the Lorentz norms $latex {\|f\|_{L^{p,q}(X,\mu)}}&fg=000000$). Indeed, if $latex {f}&fg=000000$ is a step function $latex {f = A 1_E}&fg=000000$, then the $latex {L^p}&fg=000000$ norm of $latex {f}&fg=000000$ is a combination $latex {\|f\|_{L^p(X,\mu)} = |A| \mu(E)^{1/p}}&fg=000000$ of the height (or amplitude) $latex {A}&fg=000000$ and the width $latex {\mu(E)}&fg=000000$.

However, there are more features of a function $latex {f}&fg=000000$ of interest than just its width and height. When the domain $latex {X}&fg=000000$ is a Euclidean space $latex {{\bf R}^d}&fg=000000$ (or domains related to Euclidean spaces, such as open subsets of $latex {{\bf R}^d}&fg=000000$, or manifolds), then another important feature of such functions (especially in PDE)…

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# 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.

# Qual and other things

I passed, but did not really do very well in probability. As a result they persuaded me to take the class with others. This is fine.

I really feel muddled.

# Qual review in process

I am really nervous for the qual, hopefully everything will go through.

# The difference between weak and norm topology

I was asked to prove the weak and the strong topology on $l^{p}$ is different when $1, in the sense that a sequence converges weakly to $0$ on $l^{p}$ probably not have its norm converge to $0$ as well. The hint is the proof on $l^{1}$ does not work at here.

Conway suggested that for most Banach spaces, $H^{*}$ is almost never separable unless $H^{*}$ is reflexive and separable in the first place. So since we know $l^{p}$ spaces (or in general, $L^{p}$ spaces over $\sigma$-finite measure space) should be separable, we reach a tautology on the weak star topology on $l^{q}$. Since we already know $l^{p}$ spaces are reflexive, this does not tell us anything as the weak star topology coincide with the strong topology by considering the double dual.

The problem at here, though, is weak topology itself is not the same as strong topology on $l^{p}$. So we had two strategies:

1) To construct a sequence $a_{n}=a_{i}^{n}$ such that $|a_{n}|_{p}\not\rightarrow 0$, but nevertheless satisfies $\langle a_{n},\phi\rangle$ for all $\phi\in l^{q}$.

2) Find a gap in Conway.

1) can be done explicitly in the $l^{p}$ case by considering sequences whose $n$th term be a constant and the rest be $0$. In this case we know the sequence’s $l^{p}$ norm is $C$, while it is obviously weakly convergent to $0$. But I am wondering a deeper reason why this fails. So here is a discussion on 2).

Conway’s proof on $l^{1}$ is complicated, but can be roughly summed up in 3 steps. First he constructed a family of sets $F_{m}$ such that $\bigcup F_{m}$ is the whole space. Second he managed to construct an equivalent metric on $l^{\infty}$ by $d(\phi,\psi)=\sum^{\infty}_{j=1}2^{-j}|\phi(j)-\psi(j)|$. This strange twisted metric can be proved to be equivalent to the metric induced by weak star topology (namely, induced by the action of elements in $l^{1}$) without much effort. Now since we assumed the original sequence is weakly convergent in \$$l^{1}$ in the first place, we can emulate the classical $\frac{\epsilon}{3}$ type proof by producing a suitably bad enough element in $l^{\infty}$. This element in effect “chop up” the irrelevant finite part and give a bound on the infinite part. So by manipulating all the 3 parts together he was able to bound the norm of the original sequence altogether.

The essential part of the proof is the construct of the metric. The first part used Baire’s Category Theorem, which carries over in any complete metric space. The third part is non-trivial in the $l^{p}$ case, since then we do not have an obvious bound on the finite part, but since we know $|a|_{p}\ge \max |a_{i}|$ we can manipulate it in some ways by fixing the finite order part. However, it is not clear to me how can we construct a similar metric to make the proof work (even though we know this is a wrong direction). The curious question to me is:

Since we know such a metric does not exist, what kind of metric on $l^{p}$ can be equivalent to the weak star metric? And does this carries over to $L^{p}$ case as well when the measure is $\sigma$-finite?