On credentialism and selection systems

I’ve mentioned before that an Asian-American friend of mine, who is quite smart, disapproves of the whole campaign against Asian quotas spearheaded, or at least advocated, by Steve Hsu and others.

His words are the following:

  1. I don’t believe in legitimizing the credentialist culture of modern academia
  2. I don’t generically feel much kinship with Asian-Americans (who are the most affected by purported discrimination in admissions), even if I might feel more kinship with them on average than I would with any other large ethnic group in America (which is itself not necessarily true)
  3. I don’t find it implausible that there are legitimate reasons to discriminate against Asian-Americans in the admissions process, if by ‘discriminate’ we mean ‘weigh their formal accomplishments less than one would for a member of a different race’
  4. At the end of the line, I believe that persistent whining about this is a reflection of emotional immaturity on the part of Steve et al., in that they seem to have a ‘chip on their shoulder’ which they are incapable of overcoming, and if they were actually taking a principled approach, they would come together and try to create a superior alternative to the radically broken university system, which will likely not be saved by any infusion of Asian students

Here’s what I think.

On 1), I don’t like the credentialism culture of modern academia either. Much of it is a superficial and soulless arms race. Not that grades, test scores, publications, citations, impact factor aren’t strong signals but they are prone to manipulation and artificial inflation, and that there are qualities of work not well-captured by those metrics. People are more or less compelled to single-mindedly play this game, often at the expense of actually substantial scholarship, if they are to survive in academia nowadays.

On 2), I hate to say that this country has become more toxically consumed by identity politics over the years, not to mention that people are judged at least subconsciously by who one is associated with. So collective bargaining is crucial for a group’s position on the status hierarchy.

On 3), there is that due to Asian-Americans’ and Asians in general having traditionally been the underdog, as well as their lack of media presence, which is intimately tied to the alienness of their names in the Western linguistic context, some people are inclined to view Asians are grinds who aren’t actually as capable as they might appear on paper. Especially with the whole tiger mother phenomenon that Amy Chua popularized with her infamous book. Of course, China’s rise over the recent years has altered this perception somewhat, especially the one that Asians are smart but not creative, though surely, it does seem that controlling for grades and test scores, or IQ, Asians do seem less creative, though that may be due to environmental factors, such as de facto or implicit quotas imposed by diversity mandates and economic circumstances.

On 4), I mostly disagree. Asian-Americans don’t really have the power to create a sufficiently credible alternative in a world that runs so heavily on associating with prestigious, usually long-established, institutions like Harvard and Goldman-Sachs. In their ancestral countries, China and India, Asians can improve the university and research system and the economic and technological competitiveness of the country as a whole, so as to make their universities more credible as well. In America, all Asian-Americans can really do is make more noise around the issue to exert more pressure on the elite universities, and also donate more and enhance their media and political presence as their socioeconomic position improves, especially at the elite end, improves, so that the elite universities perceive themselves as having more to lose from discriminating against Asian-Americans based on race.

This is all I have to say as pertains exclusively to Asian-Americans. I shall now give my thoughts on credentialism and selection in general.

The job of admissions and hiring committees and HR is astronomically harder than in the pre-internet age. So many people apply for positions they are grossly under-qualified for, now that it’s so easy to shoot off a resume or application online. There are, of course, application fees for college and grad schools, but they are not enough to deter. This means in the selection process can be afford now significantly less time per candidate, and one can argue that as a consequence, the process becomes more bureaucratic and easier to game. Often, people will in the pre-screening stage eliminate all applicants who do not meet certain formal criteria, such as minimum GPA/test scores or a certain degree from a certain set of sufficiently credible universities. In the case of academia, to my limited second-hand knowledge, committees will look at publications lists with a focus on citation count and impact factor of the journals on which the papers were published and also verify the candidate against senior, tenured faculty in the same or at least similar area of research. In the case of industry jobs, what matters more is the interview, where for technical roles, technical questions will be asked to further test the technical aptitude and knowledge, as well as, the softer aspects of communication and personal chemistry. For non-technicals, I can only say it’s even more about credentials (school, companies, job titles, dates of employment) and how you present yourself. I can only conclude that way more energy is expended now in aggregate on application and selection than before, which is quite costly really. In the career world, people are mostly out for themselves and don’t really care about wasting other people’s time, so long as they can get away with it with impunity more or less.

I’ll say that there is a tradeoff between optimizing for one’s formal credentials and optimizing for one’s actual ability and knowledge. One loses out so much more now if one neglects the former too much due to more competition per position. Surely, there has been gross inflation of credentials. This is in its crudest form epitomized by college’s having become the new high school, thereby rendering prestige of institution a stronger signal. Furthermore, the largely consequent grade inflation and watering down of coursework has added more noise to school transcripts. Contest training, for math in particular, has become so much more popularized, that to not have credentials in those raises questions in some circles, and moreover, there is so much more of an obstacle course of summer programs and scholarships and grants and internships and jobs which one must pass through to some degree if one wants a reasonable chance of success at a specified level. In this sense, there is more pressure to conform to an existing, often complexity-ridden system. It may well be that people nowadays are not all that much better in terms of knowledge and proficiency than before, correcting for the positive effects of technology on learning, but they actually put in much more time and effort.

Now, if one expends much energy on actual substance, there is concern as to what would be lost if those translate not into formal credentials. Arguably more common is the other way round, where one turns into a soulless credential-chasing machine. I’ve been amazed at how many people manage to achieve much higher grades, test scores, and awards than what their knowledge and ability from interaction with them would reasonably indicate. Those people tend to be very boring and risk-averse, and they are often the types our current system selects for, like it or not.

I used to feel like to prove that one is actually smart, at least in STEM, one ought to do sufficiently well in one of those major math, physics, or computing olympiads or contests. I would say that for raw technical ability, that is probably still the strongest signal. Grades are somewhat noisy, because it’s not hard to copy or snipe homework solutions, and for tests, there is a large cramming and figuring out what’s gonna be on the test component. Perhaps they are more consequentially so as there are also some genuinely capable or even brilliant students who for related personality reasons have a hard time getting themselves to care too much about grades. I’ve personally seen some high GPA people, even in college, who signal in what they say or write complete idiocy that would make you wonder if they were pretending stupid, especially if said person were female. Some people learn much more deeply and also much more broadly, outside of what the system teaches them, to a high level of retention, much of which is not captured through any formal credential. From my personal experience, tests of a wide range of knowledge, sufficiently substantial but not too esoteric, are stronger signals since they cannot be crammed for, but they are, for the difficulty of organization, seldom administered.

In the real world and in academia though, what matters is the ability to deliver actual projects and conduct meaningful research, and those, while correlated with ability to learn, are not quite the same. Those are also way more context-dependent, which means more noise, both due to more variance and more ambiguity of judgment.

I will say that at times or even often, society is met with the problem of people finagling themselves into a position to judge what they are not really qualified to, per their ability and expertise, which means some resume-padding bozos rising up and actual competents being passed over. This problem I believe has been accentuated by the ever more credentialist culture that has emerged over the recent years. What’s kind of sad is how the more conformism and risk-aversion rises, the more these traits are pressured and selected for.

I’ve come to notice that there tends to be some difference between maverick genius and the conformist first-rate professional. If one looks at history, real genius, the ones who create paradigm shifts, tends to have more very lopsided profiles, though surely, it might go too far to say that *most* of the real geniuses were out of it in a Stallman or Galois like fashion, especially as it’s the deranged ones which garner more attention. But one can say with high level of confidence that there were many real geniuses who had a hard time fitting in even into the elite mainstream of his profession, who have even been marginalized. I’ve been told that the real genius mathematicians like Perelman, Langlands, and Shimura more or less cut contact with the mathematical community apparently out of disgust. There is also evidence that plenty would-be real geniuses did not actually make it, with their enormous potential having been thwarted by the system at some point and hardly realized. In an ideal world that optimizes for collective value, if somebody else can do the job much better than you and actually really wants to, you should let him do the job and get out of his way. Of course, reality is far from that. I have personally felt that way with regard to my mathematical ability, often feeling that I wasn’t good enough when I failed to derive something on my own, yet I see so many people worse than I am even so eager to play the whole credentialist game without recognizing how deficient they really are. This suggests that I am very partial towards a certain side of the spectrum. I even feel that in some sense, nothing is more embarrassing then formally being much higher than what one’s ability actually merits, since it demonstrates not only incompetence but poor character. However, I am, regrettably, or not, feeling that circumstances are pressuring me ever more towards the opposite direction.

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.