what AMS is not doing, but probably should be

American Mathematical Society is the dominant professional organization for mathematicians working in research-active places, – mostly, departments of mathematics with PhD programs. This is a large organization, counting around 30 thousands members (this includes however folks outside of the US, retirees, and many others).

AMS is doing many things that other professional organizations do: it runs a publishing house, serves as a conference platform, lobbies the US government, awards various prizes and so on, just like the IEEE or the ACM. And I believe this is not enough. The AMS could do more to serve the community, and, probably, it should do more, – just to survive.

I outline below three areas where AMS could be actually useful. There are surely other ways, but I wanted to start with the most obvious opportunities.

Graduate Programs Ranking

The lowest hanging of all fruits is the ranking of math departments, or, rather of their graduate programs. Currently, and infamously, this ranking is done by a commercial enterprise1They also rank mattresses., the US News and World Report. It is a rather controversial venture, using murky methodology, open to manipulations and relying on polling of the graduate program heads2i.e., converting unpaid labor (a surprising common thread of this post) into their profit..

There is no conceivable reason why to relegate the task to anybody outside of the profession. The AMS is perfectly positioned to create a transparent procedure, and to execute it. One would need to work out rigorously defined criteria, aggregation algorithms, and methods of collecting the data. (I note that the most labor intensive items, the educational outcomes, – graduation rates, placements etc., – or publication activities, – are collected by the AMS anyway.) Polling the programs is also easy (and, from experience, the directors of graduate studies are much more willing to respond to a professional nonprofit organization than to a sloppy commercial outlet.

One might argue that the outsider status makes the US News more objective; I would respond that it makes it more indifferent; objectivity and fairness are achieved by a collective action, which the AMS should be perfectly able to arrange.

I estimate this project can be done within a year, with ease, by a volunteer committee of mathematicians, plus a few IT folks for software implementation.

Graduate admissions

Admissions to graduate programs in the departments of mathematics are stressful, – not just to the applicants, but to the graduate programs as well. Every Fall, thousands of prospects send their applications to the departments. Application fees notwithstanding, a large department, like ours, receives hundreds of dossiers. The committees spend weeks reading the documents and ranking the applicants.

The problems start when the programs are ready to send the offers to the selected prospects. How many letters to send? Just as the prospects hedge their bets applying to many schools, the departments offer many more admissions than they are able to accommodate. Indeed, only a fraction, – will accept. And this fraction (yield) is a big unknown.

True, past performance is an indication of what to expect, but the fluctuations are inevitable, and high, and the mistakes can be very costly. By sending an offer letter, the program takes legal obligations, and commits significant resources. A severe overshoot will deplete the department’s budget, while undershoot will leave it without research and teaching assistants.

So, a wild situation emerges. Once the offers are sent, the programs are in the state of anxiety. The prospects offered the admission are under no obligation to respond before April 15th, the shared across grad schools deadline. This uncertainty is not just driving directors of graduate studies crazy. It incentivizes strategic admission policies, haggling and is in general counterproductive, and inserts wild uncertainty element into the graduate program planning.

This is clearly both quite bad, and quite avoidable.

The problem of allocating students to schools has long history, and a highly effective solution is known for at least 70 years. I am talking (as you might have guessed) about the Match, the National Resident Matching Program (NRMP).

In preparation of practicing medical doctors, the residency is the last, critical step, not unlike PhD study in training of research mathematicians3The medical trade also shares the somewhat bizarre business model most of the departments of mathematics in the US (especially those at large public universities) are operating. Namely, we provide graduate education for free, while underpaying the student’s work (whether teaching or research assistanships), compared to what tenure-track faculty makes. Essentially the same model is deployed by the residencies3The doctors spend several years working under supervision, under a lot of pressure (in real hospital setting, addressing real medical challenges), for far smaller salaries than their established colleagues. The Match is a centralized mechanism of pairing the applicants to the hospitals that mathematical graduate education in the US can, and I believe, should, adopt.

The input data of the NRMP process are the rankings, of the applicants by schools, and of the schools, by the applicants. These are the results of the visits and interviews. Hospitals also indicate how many residencies they plan to fill.

Once the data are collected, a matching algorithm kicks in, using the famous stable matching approach, formally introduced and analyzed by Gale-Shapley4In fact the Match algorithm is essentially equivalent to Gale-Shapley one, and predates it.. It provides a stable outcome (no alternative pairing of a program and an applicant could beat the outcome for both of them), and is the best possible for each applicant among all such stable matches.

There is an enormous follow-up literature to that paper addressing many twists and subtle points in the choice of implementation and various add-ons, – see, e.g., this article. What is important to us is the existence of a robust and tested implementation of the algorithm. I won’t discuss the various merits and demerits of that approach, – but it is clear that the approach met the test of time.

Setting it up, however, would not be easy. For starters, the graduate programs would need to negotiate with the graduate colleges in the respective universities. Some legal issues almost certainly will emerge. One hardly can expect a consortium of PhD math programs to pull it off on their own. And above all, any implementation needs to be centralized.

Here, the position of the AMS as the national society of research mathematicians5with a federal lobbying arm makes it uniquely suitable to be the driver of the process, and the center of the resulting implementation. Doing this would dramatically improve and streamline the graduate admission process in the US math departments.

Mathematical Reviews reform

Mathematical Reviews is one of the most important endeavors of the AMS. Subscriptions to its current electronic incarnation, MathSciNet account for about 42% of its total revenue (all other publishing combined is another 28%). More importantly, MathSciNet is widely relied upon by the mathematicians in their research.

MathSciNet is also under threat of losing any relevance within a decade. To begin with, there is competition, – not only from zbMATH (the electronic version of Zentralblatt für Mathematik, Math Reviews’ older sibling) which is now supported by the German government and is free to use, but also from the general indexing services, like Semantic Scholar. Unlike MathSciNet and Zentralblatt, startups like Semantic Scholar rely not on the free labor of its reviewers, but on AI, and, to be frank, TL;DR’s they produce are perfectly fine, and are free6Sure, their business model will change, but as of now they are extremely inexpensive to operate compared to the traditional outlets.. Does AMS expect the university libraries will keep subscriptions as the news about free equivalent products spreads?

For the MathSciNet to stay relevant, it needs to find ways to be useful in the changing world. They start in a good position: great name recognition, institutional knowledge, remarkably clean database of authors, but these is not the same as having a product the universities will be ready to pay for.

Search for a new business model is never an easy one. I believe the MathSciNet should start by becoming more open, to enable to data analytic explorations. Creating APIs7Application Programming Interfaces, tools that allow an application to access the database on the background. Right now, any automated download from the MathSciNet requires a manual intervention of the Executive Director…, and allowing the community to use it (for free or for a nominal fee) would go long way to creating intellectual property that will be hard to reproduce elsewhere.

Once the curious hordes get access, one could only imagine what uses they will bring to the table. My personal favorite would be to restructure the Mathematical Subject Classification (MSC).

Currently, the MSC is a taxonomy updated once a decade, by a committee. The updates are very protective to legacy rubrics, and keep the overall structure of a tree with very few levels and a lot of children at each branching. It does have a warm, nostalgic feel, but is increasingly useless8Remember how often choosing the MSC for your paper feels like an unnecessary chore?. This slide into irrelevance happens at the background of an explosion of theoretical understanding and algorithmic development of clustering techniques. Ideally, we should be analyzing the current state of the trade9As an experiment, try finding out how many articles are published at a few top level MSC categories, and marvel at the disparities in numbers., and create clusters reflecting existing interrelationships between areas in mathematics (and beyond), not necessarily as a tree-like taxonomy.

Whether MSC, or other use cases, – it is clear to me that MathSciNet should drastically expand its interface with the mathematical and other communities. Otherwise, the operation of this mighty enterprise will turn awry and lose the name of action, and, as I mentioned, without the MSN the existing business model for the AMS as a whole just isn’t viable.

Over the past couple of decades, the AMS seemed to attempt to become a version of its engineering counterparts, introducing named lectures and fellowship programs. I feel this cannot be the way to become more (or even to stay) relevant. Rather, the AMS should serve the community, and the society in general, by introducing programs that materially help mathematical community by leveraging its status and clout in innovative ways. A few ideas outlined above might be not the best ones, but one needs to start somewhere, – to keep the American Mathematical Society the powerful institution it is now.

Update: I sent this little note to AMS, offering to publish it in the Notices. They told me that “it was decided” I need to shorten it to one third of its size. Not sure anyone will benefit from reading a mangled version, and certainly I am not eager to publish one.

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