Diversity

These days our institution is quite concerned with diversity and
broadening participation. Recently we had a presentation devoted to a
single question: Why are there so few women at the
top US mathematics departments?

Popular culture identifies three factors that contribute to individual's success: perseverance, will power and hard work;  talent and natural abilities; and supportive environment.

American mythology puts premium on hard work and natural abilities, while modern thinking (as in Gladwell's book "Outliers") points to the environmental factors.

Clearly a tenure at Math Department at Harvard requires plenty of talent, hard work and as much support as one can get. Yet I will argue that environmental factors alone can account for the observable disparity between men and women at the top academic tier of mathematical sciences, and one does not need to look further.

In this context environment relates to a myriad of stimuli, some positive and some negative, that one receives while pursuing professional  goals. Each of these stimuli is not significant, and they do not raise to the level of being a behaviour changing factors, yet collectively they shape one's direction, particularly over a long period of time.

The main point is that the overall effect of the environment seems to be exponential. For example, in   a large group of students learning a difficult subject one will generally observe that certain percentage of them will drop out after one semester, similar percentage after another semester and so on. In other words, the rate of change is proportional to the quantity that is measured which means that  the attrition rate follows an exponential curve. In such situation, even tiny difference in the way one interacts with the environment leads to a dramatic difference twenty years later - a time it takes to climb near the top of the ladder in mathematical sciences.  Another simple example illustrating this phenomenon is as follows - suppose that getting  tenure is akin to getting certain number of heads in a long sequence of coin tosses (with asymmetric coin). If the probability of getting a head in a single toss is larger than some threshold value then the probability of success is very high and it goes exponentially to one as the length of the trial increases. On the other hand, if the probability of success is lower than the threshold value then the odds of success go exponentially to zero at a rate that depends on the difference between the threshold value and the number of trials.

How would one test the effect of the environment?
We need to find an activity that is somewhat similar to doing mathematics, but where it takes much less time to reach super-star status and (ideally) where the environmental factors are more gender blind.  Mathematicians may be upset with this choice but I think that spelling is the right comparison and it proves the point.

Spelling is a curious activity which has a well organized support system. No US kid can escape being pushed to the maximum level that  their ability and patience will allow. Spelling talent is easily recognizable, training is straightforward and the very ability to do super-human spelling is not
intimidating. Year after year the group of top 100 US spellers has the numbers of boys and girls, minorities, poor and rich kids, etc., that conform to the anticipated frequency. Statistics that deviate from the expected values are mostly genetic - having super-spellers in the family, etc.
Inescapable conclusion is that both ability and will power to be a super-speller is distributed in the population quite evenly and even-handed system aimed at identifying top performers and eliciting from them several years of hard work to reach the finals of the  National Spelling-Bee works well.

Back to mathematics - the path to the comparable status in mathematics takes about twenty years. Mathematics is far more intimidating than spelling and performing at the higher level will not automatically get you in touch with likewise gifted people. Quite the opposite, mathematically gifted kids endure years of boredom at school and often question their own sanity. Assessing mathematical talent and nurturing it is not easy and in this respect the system works moderately well only at the graduate school level. Overall the path is long,  steep and arduous.

The educational system favors men when mathematics and other sciences
are concerned. It is not a strong bias but rather a persistent push
which gives a sense of swimming against the current. Women are
constantly discouraged from pursuing mathematics - by teachers,
families, friends and media. Because of the gender imbalance there are far
fewer role models and support networks. On the path to mathematical
pantheon, one has to go through graduate school and one or two
postdoctoral positions. While men can move their families with relative
ease, women often face the phenomenon of an immovable spouse. Dream job requires tremendous level of accomplishment and related commitment of time and energy. It is safe to say that at a time these efforts are needed women have more on their plate than men.  To get a job one needs a number of publications in top journals. These days a lot of work is collaborative and it is fairly common that in cases of joint work involving  male and female mathematicians, the men's work is often viewed as more substantial.   Professional success of female mathematicians that happen to be  good looking and attractive often brings another type of speculations - yet another small insult that is painful and difficult to confront.

There is much more to say here but the point is perhaps clear - climbing to the top of mathematical profession is just a tad harder for women than for men. Those who succeeded devoted approximately half of their life to it. Just having a few extra hoops to jump through makes a lot of difference in this endurance race.

If one accepts this argument then there is a natural follow up supported by Bayesian statistics. If one succeeds against the odds then with high likelihood there is  a compensating factor.  Consequently, women at top math departments were more driven to succeed or they were simply better mathematicians than their male counterparts - both possibilities easily supported by anecdotal evidence.

Literature and Mathematics

Just recently I had an opportunity to witness a Ph.D. defence in comparative literature. The location was a well known university famous for trend-setting in the humanities and sciences alike. Its ascent to ivy-league status was arrested only by the location, where the acute problems of modern society are clearly visible within a half mile of the campus.
The Ph.D. candidate was was well prepared, professionally dressed and only odd looking shoes with heels looking like tree stumps may have betrayed a wilder side of her character.
The defense was a small gathering, four committe members, a handful of fellow students and friends and an older couple, who were most certainly the candidate's parents,  cowering in the corner.  The first question concerned the basic idea behind the thesis, its structure, origins and the thought process that led from the beginning to the final product. Given the open format of the proceedings I was assuming that the procedure is sort of like professional wrestling - lots of body slams, impressive chokes and eye gouging but in the end nobody gets hurt. Perhaps it is so, but nevertheless it felt like a real cliff-hanger.
The subject of the thesis was a comparison between two national literatures and the basis for it was several specific examples spanning several centuries. The candidate made an opening gambit by attempting to put in question the basic premises of her work, She indicated that the work was perhaps inconclusive, main claims were not supported by research as strongly as she would have liked it, key similarities between both societies may have been superficial, and so on. At this point, the father of the candidate sitting in the corner started viciously chewing on his fingernails providing much unneeded soundtrack. However, the committee took all this in stride. They put down their prepared questions and embarked on a forceful defense of the project, which at least in part must have been based on their input. Afterwards, the clouds parted and what followed was an interesting 90 minute discussion illuminating scientific methods of the field. For a complete outsider like myself, I was very impressed with the methodological issues - careful definitions of the terms, establishing the mechanism for the selection of sources, building support for conclusions based on findings, and ultimately explaining the link between two subject areas. The discussion moved slowly through establishing intellectual foundations for the work and surveying the landscape that came to view. All four committee members represented different countries and different national literatures so there was some tendency to make comparisons that were beyond the scope of this Ph.D. but in the end the work stood on its own and offered several interesting directions for continuation. At the conclusion, and after a short recess, the new Doctor of Philosophy was announced.

My own expertise, acquired late in life, is in evaluating and assessing projects in mathematical sciences and I was quite surprised at the intellectual rigor of this work in literature. There is a stark contrast with the quite fuzzy and vague ways in which modern mathematics is assessed. The projects that I am looking at are forward looking, exciting, beautiful, compelling, or alternatively they are narrow in scope, solid, incremental or unrealistic. When the fundamentals of what is being proposed are very precise and well rooted in the overall mathematical landscape, the  uncertain and speculative assessment of the outcomes invites soft and imprecise language which offers hedging against error in judgment. Apparently quite the opposite in literature - the subjectivity in which art is received  and the multitude of ways in which literary works can be interpreted asks for a precise methodology and analysis that takes things down to their fundamentals and dissects the literary works into bits and pieces of comparable nature.
It appears as if in terms of presence of precision and analytical thinking both literature and mathematics yield  a zero-sum game when the product and its evaluation are considered jointly. If it was indeed the case, rather than a superficial observation, it would support a post-modernist view that in the end truth is a cultural construct and regardless of the field of study the overall level of the scientific discourse is tied up to our cultural baggage rather than the intrinsic demands of the field.