On grad school, science, academia, and also a problem on Riemann surfaces

I like mathematics a ton and I am not bad at it. In fact, I am probably better than many math graduate students at math, though surely, they will have more knowledge than I do in some respects, or maybe even not that, because frankly, the American undergrad math major curriculum is often rather pathetic, well maybe largely because the students kind of suck. In some sense, you have to be pretty clueless to be majoring in just pure math if you’re not a real outlier at it, enough to have a chance at a serious academic career. Of course, math professors won’t say this. So we have now an excess of people who really shouldn’t be in science (because they much lack the technical power or an at least reasonable scientific taste/discernment, or more often both) adding noise to the job market. On this, Katz in his infamous Don’t Become a Scientist piece writes:

If you are in a position of leadership in science then you should try to persuade the funding agencies to train fewer Ph.D.s. The glut of scientists is entirely the consequence of funding policies (almost all graduate education is paid for by federal grants). The funding agencies are bemoaning the scarcity of young people interested in science when they themselves caused this scarcity by destroying science as a career. They could reverse this situation by matching the number trained to the demand, but they refuse to do so, or even to discuss the problem seriously (for many years the NSF propagated a dishonest prediction of a coming shortage of scientists, and most funding agencies still act as if this were true). The result is that the best young people, who should go into science, sensibly refuse to do so, and the graduate schools are filled with weak American students and with foreigners lured by the American student visa.

Even he believes that now the Americans who go into science are often the ones who are too dumb or clueless to realize that they basically have no future there. I can surely attest to how socially inept, or at least clueless, many math grad students are, as I interact with them much more now. The epidemic described by Katz is accentuated by the fact that professors in science are not encouraging of students who seek a plan B, which everyone should given the way the job market is right now, and even go as far as to create an atmosphere wherein even to express a desire to leave academia is a no-no. I am finding that this type of environment is even corroding my interest in mathematics itself, which is sad. In any case, I sort of disagree with Katz in that I feel like the very top scientific talent of my generation still mostly ends in top or at least good graduate schools, though surely there are many who feel alienated or don’t find the risk worth taking, and end up leaving science. I myself am thinking of forgetting about mathematics altogether. So that I can concentrate my motivation and time and energy on developing expertise in some area of software engineering that is in demand, for the money and (relative) job security, and hopefully also find it a sufficiently fulfilling experience. There are a lot of morons in tech of course, but certain corners of it do provide refuge. I had always thought of mathematics as being a field with a much higher threshold cognitively in its content, enough to filter out most of the uninteresting people, but that’s, to my disappointment, less so than I expected. I do have reason to be scared, because one of the smartest and most interesting people I know took like five years following his math PhD to make his way into full employment, in a programming/data science heavy role of course, despite being arguably much better at programming than most industry software engineers with a computer science degree, which he lacked, an indicator of the perverse extent to which our society now runs on risk-aversion and (artificial) credential signaling. I can only consider myself fortunate that I do have a computer science degree from a reputable place, and with that, I have already made a modest pot of gold, despite being frankly quite mediocre at real computer stuff, which I have had difficulty becoming as interested in as I have been in mathematics. Maybe I was even fortunate to have not been all that gifted in the first place, which in some sense compelled me to be more realistic, as there is arguably nothing worse than becoming an academic loser, which academia is full of nowadays, sadly. This type of thing can happen to real geniuses too. Look at Yitang Zhang for instance, the most prominent case to come to mind. Except he actually made it afterwards, spectacularly and miraculously, with his dogged belief in himself and perseverance under adversity. For every one of him, I would expect like 10 real geniuses (in ability) who were under-nurtured, under-recognized, or even screwed, left to fade into obscurity.

I’ll transition now to a problem that I’ve been asked to solve. Its statement is the following:

Let f be holomorphic on a simply-connected Riemann surface M, and assume that f never vanishes. Then there exists F holomorphic on M such that f = e^F. Show that harmonic functions on M have conjugate harmonic functions.

Every p_0 \in M corresponds to an open connected neighborhood U =  \{p : \lVert F(p) - F(p_0) \rVert < F(p_0)\}. Let \{U_{\alpha}\} be the system consisting of these neighborhoods, (\log F)_{\alpha} a continuous branch of the logarithm of F in U_{\alpha}. From this arises a family F_{\alpha} = \{(\log F)_{\alpha} + 2n\pi i, n \in \mathbb{Z}\}.

In Schlag, there is the following lemma.

Lemma 5.5. Suppose M is a simply-connected Riemann surface and

\{D_{\alpha} \subset M : \alpha \in A\}

is a collection of domains (connected, open). Assume further that these sets form an open cover M = \bigcup_{\alpha \in A} D_{\alpha} such that for each \alpha \in A there is a family F_{\alpha} of analytic functions f : D_{\alpha} \to N, where N is some other Riemann surface, with the following properties: if f \in F_{\alpha} and p \in D_{\alpha} \cap D_{\beta}, then there is some g \in F_{\beta} so that f = g near p. Then given \gamma \in A and some f \in F_{\gamma} there exists an analytic function \psi_{\gamma} : M \to N so that \psi_{\gamma} = f on D_{\gamma}.

Using the families of analytic function F_{\alpha} as given above, it is clear that near p \in D_{\alpha} \cap D_{\beta}, (\log F)_{\alpha} + 2n_{\alpha}\pi i = (\log F)_{\beta} + 2n_{\beta}\pi i when n_{\alpha} = n_{\beta}, which means the hypothesis of Lemma 5.5 is satisfied by the above families.

I’ll present the proof of the above lemma here, to consolidate my own understanding, and also out of its essentiality in the construction of a global holomorphic function matching some function in each family. It does so in generality of course, whereas in the problem we are trying to solve it is on a specific case.

Proof. Let

\mathcal{U} = \{(p, f) | p \in D_{\alpha}, f \in F_{\alpha}, \alpha \in A\} / \sim

where (p, f) \sim (q, g) iff p = q and f = g in a neighborhood of p. Let [p, f] denote the equivalence class of (p, f). As usual, \pi([p, f]) = p. For each f \in F_{\alpha}, let

D'_{\alpha, f} = \{[p, f] | p \in D_{\alpha}\}.

Clearly, \pi : D_{\alpha, f}' \to D_{\alpha} is bijective. We define a topology on \mathcal{U} as follows: \Omega \subset D_{\alpha, f}' is open iff \pi(\Omega) \subset D_{\alpha} is open for each \alpha, f \in F_{\alpha}. This does indeed define open sets in \mathcal{U}: since \pi(D'_{\alpha, f} \cap D'_{\beta, g}) is the union of connected components of D_{\alpha} \cap D_{\beta} by the uniqueness theorem (if it is not empty), it is open in M as needed. With this topology, \mathcal{U} is a Hausdorff space since M is Hausdorff (we use this if the base points differ) and because of the uniqueness theorem (which we use if the base points coincide). Note that by construction, we have made the fibers indexed by the functions in F_{\alpha} discrete in the topology of \mathcal{U}.

The main point is now to realize that if \widetilde{M} is a connected component of \mathcal{U}, then \pi : \widetilde{M} \to M is onto and in fact is a covering map. Let us check that it is onto. First, we claim that \pi(\widetilde{M}) \subset M is open. Thus, let [p, f] \in \widetilde{M} and pick D_{\alpha} with p \in D_{\alpha} and pick D_{\alpha} with p \in D_{\alpha} and f \in F_{\alpha}. Clearly, D'_{\alpha, f} \cap \widetilde{M} \neq \emptyset and since D_{\alpha}, and thus also D'_{\alpha, f}, is open and connected, the connected component \widetilde{M} has to contiain D'_{\alpha, f} entirely. Therefore, D_{\alpha} \subset \pi(\widetilde{M}) as claimed.

Next, we need to check that M \setminus \pi(\widetilde{M}) is open. Let p \in M \setminus \pi(\widetilde{M}) and pick D_{\beta} so that p \in D_{\beta}. If D_{\beta} \cap \pi(\widetilde{M}) = \emptyset, then we are done. Otherwise, let q \in D_{\beta} \cap \pi(\widetilde{M}) and pick D_{\alpha} containing q and some f \in F_{\alpha} with D'_{\alpha, f} \subset \widetilde{M} (using the same “nonempty intersection implies containment” argument as above). But now we can find g \in F_{\beta} with the property that f = g on a component of D_{\alpha} \cap D_{\beta}. As before, this implies that \widetilde{M} would have to contain D'_{\beta, g} which is a contradiction.

To see that \pi : \widetilde{M} \to M is a covering map, one verifies that

\pi^{-1}(D_{\alpha}) = \bigcup_{f \in F_{\alpha}} D'_{\alpha, f}.

The sets on the right-hand side are disjoint and in fact they are connected components of \pi^{-1}(D_{\alpha}).

Since M is simply-connected, \widetilde{M} is homeomorphic to M (proof given in the appendix). We thus infer the existence of a globally defined analytic function which agrees with some f \in F_{\alpha} on each D_{\alpha}. By picking the connected component that contains any given D_{\alpha, f}' one can fix the “sheet” locally on a given D_{\alpha}.     ▢

By this, we can construct an analytic F such that for all \alpha,

f_{|U_{\alpha}} = (\log F)_{\alpha} + n_{\alpha} \cdot 2\pi i, \qquad n_{\alpha} \in \mathbb{Z}.

from which follows e^F = f.

For the existence of harmonic conjugates, we do similarly. Take a connected open cover of M, \{U_{\alpha}\} where each U_{\alpha} is conformally equivalent to the unit disc, and v_{\alpha} is a harmonic conjugate of u in U_{\alpha} (which exists uniquely up to constant on the unit disc. Let F_{\alpha} = \{v_{\alpha} + c, \quad c \in \mathbb{R}\}. Then by the same lemma, there exists v such that for all \alpha,

v_{|U_{\alpha}} = v_{\alpha} + c_{\alpha}, \quad \text{some } c_{\alpha} \in \mathbb{R}

that is harmonic and conjugate to u since it is the harmonic conjugate to u on every element of the cover, again with choise of c_{\alpha}s to ensure that on intersection of cover elements there is a match.